%% @texfile{
%%     filename="Deform.tex",
%%     version="1.1",
%%     date="Sep-1996",
%%     cdate="19960908",
%%     filetype="AMSTeX",
%%     journal="J. d'Analyse Math. 70, 267-324 (1996)",
%%     doi="10.1007/BF02820446",
%%     copyright="Springer".
%%     }

\input amstex
\loadbold
\documentstyle{amsppt}
\magnification=1200
\pagewidth{14cm}
\baselineskip=15 pt
\NoBlackBoxes
\TagsOnRight

\font\myfont=cmsy10 scaled \magstep3
\def\arrowl{\text{\myfont \char"00\hskip-1.2mm\char"21}}
\def\arrowse{\text{\myfont \&}}
\def\arrowne{\text{\myfont \%}}

\def\smat#1#2{\hskip -3mm\spreadmatrixlines{-2mm}\matrix &#1\\&#2\endmatrix}
\def\tsmat#1#2{\hskip -4mm\matrix &{\ssize#1}\\ \vspace{-2mm}
&{\ssize#2}\endmatrix}
\def\subjclass#1{MSC: #1}
\def\keywords#1{Keywords: #1}

\def\gap{\vskip 0.1in\noindent}
\def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.}

%References
\def \baum {1}  % Baumgartner
\def\belok {2}  % Belokolos
\def\bggss {3}  % Bolle et al.
\def\bufi {4}   % Buys-Finkel
\def\crum {5}   % Crum
\def\dar {6}    % Darboux
\def\deift {7}  % Deift
\def\detr {8}   % Deift-Trubowitz
\def\eastkalf {9} % Eastham-Kalf
\def\ehkn {10}  % Ehlers-Knorrer
\def\erfl {11}  % Ercolani-Flaschka
\def\fit {12}   % Finkel-Isaacson-Trubowitz
\def\fir {13}   % Firsova
\def\flmcl {14} % Flaschka-McLauglin
\def\ggkm {15}  % Gardner et al.
\def\gellev {16}        % Gel'fand-Levitan
\def\gdekker {17} % Gesztesy Dekker
\def\gcam {18}  % Gesztesy Cambridge
\def\gjfa {19}  % Gesztesy JFA
\def\gkt  {20} % Gesztesy-Krishna-Teschl
\def\gnow {21}  % Gesztesy-Nowell-Potz
\def\gss {22}   % Gesztesy-Schweiger-Simon
\def\gstrans {23} % Gesztesy-Simon Trans.
\def\gsacta {24}        % Gesztesy-Simon Acta
\def\gst {25}   % Gesztesy-Simon-Teschl
\def\gsvi {26}  % Gesztesy-Svirsky
\def\gtproc {27}        % Gesztesy-Teschl Proc.
\def\gtdiff {28}        % Gesztesy-Teschl Diff.
\def\gwei {29}  % Gesztesy-Weikard
\def\gzh {30}   % Gesztesy-Zhao
\def\grma {31}  % Grosse-Martin
\def\iwa {32}   % Iwasaki
\def\jac {33}   % Jacobi
\def\kaymo {34} % Kay-Moses
\def\kumi {35}  % Kuznetsov-Mikhailov
\def\lei {36}   % Leighton
\def\levbook {37} % Levitan book
\def\levsbor {38} % Levitan Sbornik
\def\mar {39}   % Marchenko
\def\mckcpam {40} % McKean CPAM
\def\mckrev {41} % McKean Revista
\def\mckjsp {42} % McKean JSP
\def\mckspm {43} % McKean Symp. Pure Math.
\def\mcknew {44} % McKean CPAM 92
\def\mcktr {45} % McKean-Trubowitz
\def\mckmoe {46} % McKean-van Moerbeke
\def\nmpz {47}  % Novikov et al.
\def\potr {48} % Poschel-Trubowitz
\def\ratr {49}  % Ralston-Trubowitz
\def\schm {50}  % Schmincke
\def\simon {51} % Simon
\def\te {52}    % Teschl
\def\tpre {53}  % Teschl preprint

\topmatter
\title Spectral Deformations of One-Dimensional Schr\"odinger
Operators
\endtitle
\rightheadtext{Spectral Deformations of One-Dimensional Schr\"odinger Operators}
\author F.~Gesztesy$^1$, B.~Simon$^2$, and G.~Teschl$^3$
\endauthor
\leftheadtext{F.~Gesztesy, B.~Simon, and G.~Teschl}
\thanks$^1$ Department of Mathematics, University of Missouri, Columbia, MO
65211.
E-mail: gesztesyf\@missouri.edu
\endthanks
\thanks$^2$ Division of Physics, Mathematics, and Astronomy, California
Institute of
Technology, Pasadena, CA 91125. E-Mail: bsimon\@caltech.edu.  This material
is based
upon work supported by the National Science Foundation under Grant
No.~DMS-9401491.
The Government has certain rights in this material.
\endthanks
\thanks$^3$ Institut f\"ur Reine und Angewandte Mathematik, RWTH Aachen,
52056 Aachen, Germany. Current address: Institut f\"ur Mathematik,
Strudlhofgasse 4, 1090 Wien, Austria. E-mail: gerald.teschl\@univie.ac.at
\endthanks
\thanks {\it{J.~d'Analyse Math. {\bf 70}, 267--324 (1996)}}
\endthanks
\thanks
\subjclass{Primary 34B24, 34L05; Secondary 34B20, 47A10}
\endthanks
\thanks
\keywords{Spectral deformations, Sturm-Liouville operators, isospectral}
\endthanks
\date June 18, 1996
\enddate
\abstract  We provide a complete spectral characterization of a new method of constructing
isospectral (in fact, unitary) deformations of general Schr\"odinger operators
$H=-\frac{d^2}{dx^2}+V$ in $L^2 (\Bbb R)$. Our technique is connected to Dirichlet data, that
is, the spectrum of the operator $H^D$ on $L^2 ((-\infty, x_0))\oplus L^2 ((x_0, \infty))$
with a Dirichlet boundary condition at $x_0$. The transformation moves a single eigenvalue of
$H^D$ and perhaps flips which side of $x_0$ the eigenvalue lives. On the remainder of
the spectrum, the transformation is realized by a unitary operator. For cases such as
$V(x)\to \infty$ as $|x|\to\infty$, where $V$ is uniquely determined by the spectrum
of $H$ and the Dirichlet data, our result implies that the specific
Dirichlet data allowed are determined only by the asymptotics as $E\to\infty$.
\endabstract
\endtopmatter

\document

\flushpar{\bf \S 1. Introduction}
\vskip 0.1in

Spectral deformations of Schr\"odinger operators in $L^2 (\Bbb R)$,
isospectral and certain
classes of non-isospectral ones, have attracted a lot of interest over the
past three decades
due to their prominent role in connection with a variety of topics,
including the
Korteweg-de Vries (KdV) hierarchy, inverse spectral problems,
supersymmetric quantum
mechanical models, level comparison theorems, etc. In fact, the
construction of $N$-soliton
solutions of the KdV hierarchy (and more generally, the construction of
solitons relative
to reflectionless backgrounds) is a typical example of a non-isospectral
deformation of
$H=-\frac{d^2}{dx^2}$ in $L^2 (\Bbb R)$ since the resulting deformation
$\tilde H =
-\frac{d^2}{dx^2} + \tilde V$ acquires an additional point spectrum
$\{\lambda_1, \dots,
\lambda_N\}\subset (-\infty, 0)$ such that
$$
\sigma(\tilde H) =\sigma (H)\cup \{\lambda_1, \dots, \lambda_N\}
$$
($\sigma(\,\cdot\,)$ abbreviating the spectrum). On the other hand, the
solution of the
inverse periodic problem and the corresponding solution of the
algebro-geometric
quasi-periodic finite-gap inverse problem for the KdV hierarchy (and
certain almost-periodic
limiting situations thereof) are intimately connected with isospectral (in
fact, unitary)
deformations of a given base (background) operator
$H=-\frac{d^2}{dx^2}+V$\!.  Although
not a complete bibliography on applications of spectral deformations in
mathematical physics,
the interested reader may consult [\baum], [\belok], [\bggss], [\gjfa],
[\grma], [\nmpz], and
the references cited therein.

Our main motivation in writing this paper descends from our interest in
inverse spectral
problems. As pointed out later (see Remarks 4.5, 4.7, and 4.8), spectral
deformation methods
can provide crucial insights into the isospectral class of a given base
potential $V$\!, and in some
cases can even determine the whole class $\text{Iso}(V) =\{\tilde V\in
L^1_{\text{\rom{loc}}}
(\Bbb R)\mid\sigma (-\frac{d^2}{dx^2} +\tilde V)=\sigma
(-\frac{d^2}{dx^2}+V)\}$ of
such potentials. A particularly ``annoying" open problem in inverse
spectral theory concerns
the characterization of the isospectral class of potentials $V$ with purely
discrete spectra
(e.g., the harmonic oscillator $V(x)=x^2$).

In [\gsacta], we proposed a way to label the isospectral operators
for such
a situation
with Dirichlet data. Fix $x_0$ and let $H^D$ be the operator in $L^2
((-\infty , x_0))
\oplus L^2 ((x_0, \infty))$ with Dirichlet boundary condition at $x_0$.
$H^D = H^D_-
\oplus H^D_+$. The Dirichlet data are the pairs $(\mu, \sigma)$ with
$\mu\in\Bbb R$,
$\sigma\in \{+, -\}$ of eigenvalues of $H^D$ and a label of whether they
are eigenvalues
of $H^D_-$ or $H^D_+$. We showed in [\gstrans], Theorem 3.6, that for any
Dirichlet data,
there is at
most one $V$ in the isospectral class of a given $-\frac{d^2}{dx^2}+V_0$
with discrete
spectrum so that $V$ has the given Dirichlet data (in the degenerate case,
where any eigenvalues of $H^D$ and $H$ coalesce, one must include an
additional parameter in the Dirichlet data for each coincidence of
eigenvalues, see Remarks 4.9 and 4.10). That is, the map from
$V$ to Dirichlet
data is one-one when defined on the isospectral set of potentials. The
issue is
determining  the range of this map.

While this paper does not solve the inverse discrete spectral problem, it
will make one
important contribution. As a result of our principal Theorem 4.4, we obtain
the fact that
for any potential $V$\!, any finite number of deformations of Dirichlet
data (i.e., Dirichlet
eigenvalues together with their left/right half-line distribution, see
(2.7)) in spectral gaps of
$V$ produce isospectral deformations $\tilde V\in \text{Iso}(V)$ of $V$\!.
In particular,
there are no further constraints on these Dirichlet data (except, of
course, these deformations
are required to be finite in number and to stay within the spectral gaps in
question). Applied
to the inverse discrete spectral problem, this means that any constraints
 enforced on Dirichlet
data can only be asymptotic in nature, that is, can only come from their
``tail end" at infinity.
That such asymptotic constraints necessarily exist is a consequence of a
recently proved
general trace formula for $V(x)$ [\gsacta] (see Remark 4.8). The precise
nature of these
constraints, however, is unknown to date.

Mathematically, the techniques involved to produce isospectral $\tilde V$
or classes of
non-isospectral ones where eigenvalues are added or removed, but the
remaining spectral
characteristics stay identically to those of the base potential $V$\!, can
be traced back to
commutation methods. These commutation methods in turn are intimately
connected with factorizations of the Schr\"odinger
differential
expression $-\frac{d^2}{dx^2}+V(x)$ into products of first-order
differential expressions.
More precisely, one seeks a factorization of the type
$$\gather
-\frac{d^2}{dx^2} +V(x) =\alpha(\lambda) \alpha(\lambda)^+ + \lambda, \\
\alpha(\lambda) =\frac{d}{dx} +\phi (\lambda, x), \quad \alpha(\lambda)^+ =
-\frac{d}{dx} +\phi(\lambda, x)
\endgather
$$
for some appropriate $\lambda\in\Bbb R$. A subsequent commutation of the
factors $\alpha
(\lambda)$ and $\alpha (\lambda)^+$, introducing the differential
expression
$$
-\frac{d^2}{dx^2} + \tilde V(\lambda, x) =\alpha (\lambda)^+
\alpha(\lambda) +\lambda,
$$
then yields associated isospectral or special classes of non-isospectral
deformations
$\tilde V(\lambda, x)$ of $V(x)$ depending on the choice of $\phi (\lambda,
x)$ in
$\alpha(\lambda), \alpha(\lambda)^+$. In the following, we briefly outline
three different
instances of commutation techniques that occur in the literature.

We start with the single commutation or Crum-Darboux method (going back at
least
to Jacobi). In this method, $H=-\frac{d^2}{dx^2}+V$ is assumed to be
bounded from
below, $\inf\sigma(H) >-\infty$, and $\lambda\in\Bbb R$ is chosen according
to $\lambda
<\inf\sigma(H)$. One sets $\phi (\lambda, x)=\psi'_\nu (\lambda,
x)/\psi_\nu (\lambda, x)$,
where $\psi_\nu$ satisfies $\psi'' = (V-\lambda)\psi$ (cf.~(A.7)).
Depending on the choice
of $\psi_\nu (\lambda, x)$, the aforementioned commutation procedure yields
a spectral
deformation $\tilde H_\nu (\lambda)$ of $H$,
$$
\tilde H_\nu (\lambda) =-\frac{d^2}{dx^2} + \tilde V_\nu (\lambda, x),
\quad \tilde V_\nu (\lambda,x) = V(x) -2\{\ln [\psi_\nu (\lambda, x)]\}'',
\quad x\in\Bbb R, \  \lambda < \inf\sigma(H),
$$
which is either isospectral to $H$ or acquires the additional eigenvalue
$\lambda$ below
the spectrum of $H$, that is,
$$
\text{either } \sigma(\tilde H_\nu (\lambda)) =\sigma(H) \text{ or }
\sigma(\tilde H_\nu (\lambda)) =\sigma(H)\cup\{\lambda\}.
$$
Moreover, it can be proved that the remaining spectral characteristics of
$H$ remain
preserved in the sense that $\tilde H_\nu (\lambda)$ and $H$, restricted
to the
orthogonal complement of the eigenspace associated with $\lambda$, are
unitarily
equivalent.

A summary of this technique, as well as pertinent references to its
extraordinary history
and to more recent applications of it, will be given in Appendix A.

The fact that $\lambda$ is required to lie below the spectrum of $H$ is
clearly a severe
limitation. One possibility to avoid this restriction is provided by the
following second
technique, the double commutation method.

Formally, this method can be obtained from two successive single
commutations at a
point $\lambda\in\Bbb R\backslash\sigma_{\text{\rom{ess}}}(H)$
($\sigma_{\text{\rom{ess}}}(\,\cdot\,)$ the essential spectrum), or
equivalently, as the
result of two single commutations at $\lambda'$ and $\lambda''$,
$\lambda'\neq \lambda''$,
$\lambda', \lambda''\in\Bbb R\backslash \sigma_{\text{\rom{ess}}}(H)$ with a
subsequent limiting procedure $\lambda' \to \lambda$ and
$\lambda''\to\lambda$. The
final outcome can be sketched as follows. Pick $\gamma >0$, $\lambda\in\Bbb
R\backslash
\sigma_{\text{\rom{ess}}}(H)$ and real-valued $\psi_\pm (\lambda,
\,\cdot\,) \in
L^2 ((R, \pm\infty))$, $R\in\Bbb R$ satisfying $\psi'' (\lambda) =
(V-\lambda) \psi
(\lambda)$. The spectral deformation $\tilde H_{\pm, \gamma}(\lambda)$
of $H$ is
 then
given by
$$\align
\tilde H_{\pm, \gamma}(\lambda) &=-\frac{d^2}{dx^2} +
\tilde V_{\pm, \gamma}
(\lambda, x), \\
\tilde V_{\pm, \gamma}(\lambda, x) &= V(x) -2\biggl\{\ln \biggl[
1\mp\gamma \int\limits^x_{\pm\infty} dx'\, \psi_\pm (\lambda,
x')^2\biggr]\biggr\}'',
\quad \gamma >0, \
\lambda\in\Bbb R\backslash \sigma_{\text{\rom{ess}}}(H).
\endalign
$$
In this case, one can show that
$$
\sigma (\tilde H_{\pm, \gamma}(\lambda)) =\sigma (H) \cup \{\lambda\}
$$
and again $\tilde H_{\pm, \gamma}(\lambda)$ and $H$ are unitarily
equivalent upon
restriction onto the orthogonal complements of their eigenspaces
corresponding to
$\lambda$.

A summary of this method together with appropriate references to its
history, as well as
to recent applications of it, will be provided in Appendix B.

Finally, and most importantly in connection with the contents of this
paper, we shall describe
a third commutation method first introduced by Finkel, Isaacson, and
Trubowitz [\fit]
in 1987 in connection with an explicit realization of the isospectral
torus
of periodic
potentials. This method was again used by Buys and Finkel [\bufi] (see
also
Iwasaki [\iwa])
in the context of periodic finite-gap potentials and by P\"oschel and
Trubowitz [\potr] and
Ralston and Trubowitz [\ratr] for various boundary value problems on
compact intervals.
As in the previous case, this method formally consists of two single
commutations, but this
time at different values of the spectral parameter. The principal
contribution of this paper is
a generalization of the work of Finkel, Isaacson, and Trubowitz to
arbitrary (i.e., not
necessarily periodic) base potentials $V(x)$ and a complete spectral
characterization of
this commutation technique. As a result we obtain a powerful new tool in
constructing sets
of isospectral potentials for arbitrary base potentials.

We briefly sketch this approach. Suppose $\psi_\pm (z, \,\cdot\,)\in L^2
((R, \pm\infty))$,
$z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$, $R\in\Bbb R$ satisfy
$\psi''(z)=
(V-z)\psi(z)$, and pick a spectral gap $(E_0, E_1)$ of $H$ with $\mu,
\tilde\mu\in (E_0, E_1)$. Define
$$
W_{(\tilde\mu, \tilde\sigma)} (x) =(\tilde\mu -\mu)^{-1} W(\psi_\sigma(\mu),
\psi_{-\tilde\sigma}(\tilde\mu))(x), \quad \sigma, \tilde\sigma\in \{-,+\},
\  x\in\Bbb R,
$$
where $W(f,g)(x)=f(x)g'(x) -f'(x)g(x)$ denotes the Wronskian of $f$ and $g$
(taking
limits if $\tilde\mu =\mu$). The spectral deformation $\tilde
H_{(\tilde\mu, \tilde\sigma)}$
of $H$ is then given by
$$
\tilde H_{(\tilde\mu, \tilde\sigma)} =-\frac{d^2}{dx^2} + \tilde
V_{(\tilde\mu, \tilde\sigma)},
\quad \tilde V_{(\tilde\mu, \tilde\sigma)}(x)=V(x)-2\{\ln [W_{(\tilde\mu,
\tilde\sigma)}
(x)]\}''.
$$
In order to define $\tilde V_{(\tilde\mu, \tilde\sigma)}$, one needs, of
course, to show that
$W_{(\tilde\mu, \tilde\sigma)}(x)$ is non-vanishing on $\Bbb R$. Indeed,
the key to our
extension of this method to the whole line is precisely our proof in [\gst]
that this Wronskian
is non-zero. This proof avoids the indirect argument of [\fit], [\potr],
[\ratr] that relies on
compactness of the underlying interval. (Even if one is only interested in
the compact
interval case, our direct proof is simpler than their indirect argument.)
In addition to allowing the extension to whole line problems in
principle,
this paper provides explicit  calculations in the change of Weyl-Titchmarsh
and spectral
functions.

In our main result, Theorem 4.4, we shall prove that
$$
\sigma(\tilde H_{(\tilde\mu, \tilde\sigma)}) = \sigma(H),
$$
in fact, $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$ will turn out to be
unitarily equivalent.
Moreover, if $(\mu,\sigma)$ is a Dirichlet datum for $H$ with respect to
the reference point
$x_0$, then all Dirichlet data for $\tilde H_{(\tilde\mu, \tilde\sigma)}$
with respect to
$x_0$ are identical to those of $H$, except that $(\mu,\sigma)$ is removed
and $(\tilde\mu,
\tilde\sigma)$ is added instead. These results and a variety of extensions
 thereof constitute
the principal new material in this paper. Because the spectral types of all
operators in [\fit],
[\potr], and [\ratr] are explicitly known, the unitarity theorem is a
trivial consequence of the
determination of spectra. However, for general base potentials, the
spectral types can be
exotic so that the unitarity result is much stronger than a mere equality
result of spectra.
Our proof of the unitarity relies on the explicit formula of the changes in
the spectral matrix.

Section 2 provides the background needed in the remainder of this paper.
Section~3 treats
Weyl-Titchmarsh $m$-functions and spectral functions associated with
half-line Dirichlet
operators. Section 4 contains our principal results on isospectral
deformations and provides
a complete spectral characterization of this deformation method. In
particular, the
Weyl-Titchmarsh $M$-matrix and spectral matrix of the deformation $\tilde
H_{(\tilde\mu, \tilde\sigma)}$ are computed in terms of the corresponding
matrices of the base operator
$H$. A variety of additional results and possible extensions, including
limit point/limit
circle considerations, iterations of isospectral deformations, general
Sturm-Liouville
operators on arbitrary intervals, and scattering theory, are treated in
Section 5. Finally,
the single and double commutation methods are reviewed in Appendices
A and B,
respectively.

\vskip 0.3in

\flushpar{\bf \S 2. Preliminaries on the Dirichlet Deformation Method}
\vskip 0.1in

This section sets the stage for a complete spectral characterization
of the
Dirichlet
deformation method in the remainder of this paper.

Suppose
$$
V\in L^1_{\text{\rom{loc}}} (\Bbb R) \text{ is real valued}, \tag 2.1
$$
introduce the differential expression $\tau=-\frac{d^2}{dx^2}+V(x)$,
$x\in\Bbb R$, and
pick $\lambda_0 \in\Bbb R$ and $\eta_\pm (\lambda _0, x)$ satisfying
$$\gathered
\tau\psi(\lambda _0)=\lambda_0 \psi (\lambda _0), \\
\eta_\pm (\lambda _0, \,\cdot\,) \in L^2 ((R, \pm\infty)), \quad R\in\Bbb
R, \ \eta_\pm
(\lambda _0, x) {\text{ real-valued}}.
\endgathered
\tag 2.2
$$
Given $\eta_\pm (\lambda _0, x)$ we define the self-adjoint base
(background) operator
$H$ in $L^2 (\Bbb R)$ via
$$\aligned
Hf &=\tau f, \\
f\in\Cal D(H) &= \{g\in L^2 (\Bbb R) \mid g,g'\in
AC_{\text{\rom{loc}}}(\Bbb R);
\tau g\in L^2(\Bbb R); \\
&\qquad \lim\limits_{x\to\pm\infty} W(\eta_\pm (\lambda _0),g)(x) =0
\text{ if $\tau$ is l.c.~at $\pm\infty$}\}.
\endaligned
\tag 2.3
$$
Here $W(f,g)(x)=f(x)g'(x)-f'(x)g(x)$ denotes the Wronskian of $f,g\in
AC_{\text{\rom{loc}}}
(\Bbb R)$ (the set of locally absolutely continuous functions on $\Bbb R$)
and l.c.~and
l.p.~abbreviate the limit point and limit circle cases, respectively.
The
corresponding boundary
condition at $\omega\infty$ in (2.3) is superfluous and hence to be
deleted
whenever $\tau$
is l.p.~at $\omega\infty$, $\omega\in\{-, +\}$. The reader unwilling
to get
caught up in limit
circle situations may safely add the assumption that $\tau$ is l.p.~at
$\pm\infty$ which renders
$H$ independent of the choice of $\eta_\pm (\lambda_0, x)$. However, as
discussed in
Lemma 5.3, assuming $\tau$ to be l.p.~at $\pm\infty$ does not necessarily
dispose of all limit
circle considerations in connection with the deformation method at hand.

Given $H$ and a fixed reference point $x_0 \in\Bbb R$, we introduce the
associated Dirichlet
operator $H^D$ in $L^2 (\Bbb R)$ by
$$\aligned
H^D f &= \tau f, \\
f\in \Cal D(H^D) &= \{g\in L^2 (\Bbb R) \mid g\in
AC_{\text{\rom{loc}}}(\Bbb R), g'\in
AC_{\text{\rom{loc}}} (\Bbb R\backslash \{x_0\});
\lim\limits_{\epsilon\downarrow 0}
g(x_0 \pm\epsilon)=0; \\
&\qquad \tau g\in L^2 (\Bbb R); \lim\limits_{x\to\pm\infty} W(\eta_\pm
(\lambda_0), g)
(x)=0 \text{ if $\tau$ is l.c.~at $\pm\infty$}\}.
\endaligned
\tag 2.4
$$
Clearly, $H^D$ decomposes into
$$
H^D = H^D_- \oplus H^D_+ \tag 2.5
$$
with respect to the orthogonal decomposition
$$
L^2 (\Bbb R) =L^2 ((-\infty, x_0)) \oplus L^2 ((x_0, \infty)). \tag 2.6
$$
(For notational convenience, we shall later identify
$(x_0, \sigma\infty)$ with
 $(-\infty, x_0)$
or $(x_0, \infty)$ depending on whether $\sigma =-$ or $\sigma =+$.)
Moreover, for any
$\mu\in\sigma_d (H^D)\backslash \sigma(H)$  ($\sigma_d
(\,\cdot\,)=\sigma(\,\cdot\,)
\backslash\sigma_{\text{\rom{ess}}}(\,\cdot\,)$, the discrete spectrum,
$\sigma(\,\cdot\,)$
and $\sigma_{\text{\rom{ess}}}(\,\cdot\,)$, the spectrum and essential
spectrum, respectively),
we introduce the Dirichlet datum
$$
(\mu,\sigma) \in\{\sigma_d (H^D)\backslash\sigma_d (H)\} \times \{-,+\},
\tag 2.7
$$
which identifies $\mu$ as a discrete Dirichlet eigenvalue on the interval
$(x_0, \sigma
\infty)$, that is, $\mu\in\sigma_d (H^D_\sigma)$, $\sigma\in \{-,+\}$ (but
excludes it
from being simultaneously a Dirichlet eigenvalue on $(x_0, -\sigma\infty)$).

In some cases, for instance, if $V(x)\to\infty$ as $|x|\to\infty$, the
spectrum and Dirichlet data uniquely determine $V(x)$ [\gstrans],
Theorem~3.6 (cf.~also Remarks 4.9 and 4.10).

Next, we pick a fixed spectral gap $(E_0, E_1)$ of $H$, the endpoints of
which (without loss of generality) belong to the spectrum of $H$,
$$
(E_0, E_1) \subset \Bbb R\backslash\sigma(H), \quad
E_0, E_1 \in\sigma(H) \tag 2.8
$$
and choose a discrete eigenvalue $\mu$ of $H^D$ in the closure of that
spectral gap,
$$
\mu\in\sigma_d (H^D)\cap [E_0, E_1] \tag 2.9
$$
(we note there is at most one such $\mu$ since $(H^D -z)^{-1}$ is a
rank-one perturbation
of $(H-z)^{-1}$). According to (2.7), this either gives rise to a Dirichlet
datum
$$
(\mu,\sigma)\in (E_0, E_1) \times \{-,+\}, \tag 2.10
$$
or else to a discrete eigenvalue of $H^D_-$ and $H^D_+$, that is,
$$
\mu\in\{E_0, E_1\}\cap\sigma_d (H)\cap\sigma_d(H^D_-)\cap
\sigma_d (H^D_+) \tag 2.11
$$
since the eigenfunction of $H$ associated with $\mu$ has a zero at $x_0$.
In particular,
since $(H^D -z)^{-1}$ is a rank-one perturbation of $(H-z)^{-1}$, one infers
$$
\sigma_{\text{\rom{ess}}} (H^D) = \sigma_{\text{\rom{ess}}}(H), \tag 2.12
$$
and thus, $\mu\in \{E_0, E_1\}\cap\sigma_{\text{\rom{ess}}}(H)$ is
excluded by hypothesis (2.9). Hence, the case distinctions (2.10) and
(2.11) are exhaustive.

In addition to $\mu$ as in (2.9)--(2.11), we also need to introduce
$\tilde\mu\in
[E_0, E_1]$ and $\tilde\sigma\in \{-,+\}$ as follows: Either
$$
(\tilde\mu, \sigma)\in (E_0, E_1) \times \{-,+\}, \tag 2.13
$$
or else
$$
\tilde\mu\in \{E_0, E_1\}\cap\sigma_d (H). \tag 2.14
$$

Given $H$, we define solutions $\psi_\pm (z,x)$ of $(\tau-z)\psi(z)=0$
which satisfy
$$\gathered
\psi_\pm (z,\,\cdot\,)\in L^2 ((R,\pm\infty)), \quad R\in \Bbb R, \\
\lim\limits_{x\to\pm\infty} W(\psi_\pm (z), g)(x)=0 \quad \text{for all
$g\in\Cal D(H)$}.
\endgathered
\tag 2.15
$$
If $\psi_\pm (z,x)$ exist, they are unique up to constant multiples. In
particular, $\psi_\pm
(z,x)$ exist for $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$ and we
can (and
will) assume them to be holomorphic with respect to $z\in\Bbb
C\backslash\sigma(H)$ and
real-valued for $z\in\Bbb R$. One can choose,
$$
\psi_\pm (z,x) =((H-z)^{-1}\chi_{(a,b)})(x) \quad\text{for }
x\smat{>b}{<a}, \quad
-\infty <a<b<\infty \tag 2.16
$$
and uniquely continue for $x \tsmat{<b}{>a}$. (Here $\chi_\Omega
(\,\cdot\,)$ denotes
the characteristic function of a set $\Omega\subseteq\Bbb R$.) A finite
number of
isolated eigenvalues can be included in the domain of holomorphy of
$\psi_\pm (z)$ by
multiplying (2.16) with an appropriate function of $z$.

Next, we state a simple technical result which will be needed in the
context of (2.19) and
(2.20).

\proclaim{Lemma 2.1} Let $\psi_\sigma (\mu,\,\cdot\,)$,
$\psi_\sigma(\tilde\mu,
\,\cdot\,)\in L^2 ((R, \sigma\infty))$, $R\in\Bbb R$ be defined as in
{\rom{(2.15)}}.
Then
$$
\lim\limits_{\tilde{\mu}\to\mu} (\tilde\mu -\mu)^{-1} W(\psi_\sigma (\mu),
\psi_\sigma(\tilde\mu))(x) =-\int\limits^x_{\sigma\infty} dx' \psi_\sigma
(\mu, x')^2.
\tag 2.17
$$
\endproclaim

\demo{Proof} Since
$$
(\tilde\mu -\mu)^{-1} W(\psi_\sigma (\mu), \psi_\sigma (\tilde\mu))(x)=-
\int\limits^x_{\sigma\infty} dx'\psi_\sigma (\mu, x') \psi_\sigma
(\tilde\mu, x'), \tag 2.18
$$
((2.18) is easily verified by differentiating $W(\psi_\sigma (\mu),
\psi_\sigma
(\tilde\mu))(x)$ w.r.t.~$x$ and using (2.15)), we only need to justify
interchanging
the limit $\tilde\mu \to\mu$ and the integral in (2.18). By (2.16),
$$
((H-\tilde\mu)^{-1}\chi_{(a,b)}) (x') = c_\pm(\tilde\mu) \psi_\pm
(\tilde\mu, x') \text{ for }
\aligned x' &>x>b>a \\ x' &< x<a<b
\endaligned
$$
for some constants $c_\pm (\tilde\mu) \neq 0$, and hence
$$
-\int\limits^x_{\sigma\infty} dx' \psi_\sigma (\mu, x')
\psi_\sigma(\tilde\mu, x') =
c^{-1}_\sigma (\psi_\sigma (\mu) \chi_{(x, \sigma\infty)},
(H-\tilde\mu)^{-1}
\chi_{(a,b)})
$$
yields the desired continuity with respect to $\tilde\mu$. (This fails at
first sight if
$\mu\in\sigma_d (H)\cap \sigma_d (H^D)$. A proper factor removes the pole
at $z=\mu$
in this case.) \qed
\enddemo

Given $\psi_\sigma (\mu, x)$, $\psi_{-\tilde\sigma}(\tilde\mu, x)$, and
Lemma 2.1, we
define
$$
W_{(\tilde\mu, \tilde\sigma)} (x) =
\cases (\tilde\mu -\mu)^{-1} W(\psi_\sigma (\mu),
\psi_{-\tilde\sigma}(\tilde\mu))(x),
\quad & \mu,\tilde\mu \in [E_0, E_1], \  \tilde\mu \neq \mu \\
-\sigma \int^x_{\sigma\infty} dx' \psi_\sigma (\mu, x')^2, \quad &
(\tilde\mu, \tilde\sigma)
=(\mu, -\sigma), \ \mu\in (E_0, E_1) \
\endcases \tag 2.19
$$
and the associated Dirichlet deformation
$$\aligned
\tilde\tau_{(\tilde\mu, \tilde\sigma)} &= -\frac{d^2}{dx^2} +\tilde
V_{(\tilde\mu,
\tilde\sigma)}(x), \\
\tilde V_{(\tilde\mu, \tilde\sigma)}(x) &= V(x)-2\{\ln [W_{(\tilde\mu,
\tilde\sigma)}
(x)]\}'', \quad x\in\Bbb R, \\
&\qquad \mu, \tilde\mu \in [E_0, E_1], \  \mu\neq \tilde\mu
\text{ or } (\tilde\mu, \tilde\sigma) = (\mu, -\sigma), \  \mu\in (E_0, E_1).
\endaligned
\tag 2.20
$$
(We will show in Lemma 2.2 that $W_{(\tilde\mu, \tilde\sigma)}(x)\neq 0$,
$x\in\Bbb R$
and hence (2.20) is well-defined.)  In the remaining cases $(\tilde\mu,
\tilde\sigma) =
(\mu, \sigma)$, $\mu\in [E_0, E_1]$, and $\mu=\tilde\mu \in\{E_0, E_1 \}
\cap \sigma_d (H)$, we define
$$
\tilde V_{(\tilde\mu, \tilde\sigma)}(x) =V(x). \tag 2.21
$$
Equation (2.21) represents the trivial deformation of $V(x)$ (i.e., none at
all), and for
notational simplicity these trivial cases are excluded in the remainder of
this paper, unless  explicitly stated otherwise. For obvious reasons we
will allude to (2.20) as the Dirichlet
deformation method in the following.

If $\tilde\mu \in\sigma_d (H)$, then $\psi_- (\tilde\mu)=c\psi_+
(\tilde\mu)$ for some
$c\in\Bbb R\backslash \{0\}$, showing that $W_{(\tilde\mu,
\tilde\sigma)}(x)$ and hence,
$V_{(\tilde\mu, \tilde\sigma)}(x)$ in (2.19) and (2.20) become independent
of $\sigma$
or $\tilde\sigma$. In this case we shall occasionally use a more
appropriate notation
and write $\tilde V_{\tilde\mu}(x)$ and $\tilde\tau_{\tilde\mu}$ (instead of
$\tilde V_{(\tilde\mu, \tilde\sigma)}(x)$ and $\tilde\tau_{(\tilde\mu,
\tilde\sigma)}$).

The next result, taken from [\gst], shows that (2.20) is well-defined on
$\Bbb R$. For
the reader's convenience, we reproduce the proof of the special
case we need
of Theorem 1.6 in [\gst].

\proclaim{Lemma 2.2 [\gst]} Suppose $\mu,\tilde\mu \in [E_0, E_1]$
and $\psi_\sigma (\mu,x)$, $\psi_{-\sigma}(\tilde\mu, x)$, $\sigma,
\tilde\sigma \in
\{-,+\}$ are defined as in {\rom{(2.15)}}. Then
$$
W(\psi_\sigma (\mu), \psi_{-\sigma}(\mu))(x)\neq 0, W(\psi_\sigma (\mu),
\psi_{-\sigma}(\tilde\mu))(x)\neq 0, \quad \mu\neq\tilde\mu, \  x\in\Bbb R,
\tag 2.22
$$
and hence,
$$
\tilde V_{(\tilde\mu, \tilde\sigma)}\in L^1_{\text{\rom{loc}}} (\Bbb R)
\tag 2.23
$$
in {\rom{(2.20)}}.
\endproclaim

\demo{Proof} Since $W(\psi_\sigma (\mu), \psi_{-\sigma}(\mu)) =
\text{const.}\neq 0$
and (2.23) is clear from (2.19), (2.20), and (2.22), we only focus on
the case
$W(\psi_\sigma (\mu), \psi_{-\tilde\sigma}(\tilde\mu))(x)\neq 0$,
$\mu\neq\tilde\mu$,
$x\in\Bbb R$.

First, consider the case $\tilde\sigma=\sigma=-$, assume without loss of
generality
that $\tilde\mu >\mu$, and abbreviate
$$
W(x)=W(\psi_- (\mu), \psi_- (\tilde\mu))(x), \quad x\in\Bbb R.
$$
Suppose that
$$
W(x_1) =0 \quad \text{for some $x_1 \in\Bbb R$}. \tag 2.24
$$
Define
$$
\eta_1 (x) = \cases \psi_- (\mu, x), & x\leq x_1 \\
\gamma_1 \psi_+ (\tilde\mu, x), & x\geq x_1
\endcases ,
$$
where $\gamma_1 \in\Bbb R$ is defined such that $\eta_1 \in\Cal D(H)$ and
$$
\tilde\eta_1 (x) = \cases
\psi_- (\mu, x), & x\leq x_1 \\
-\gamma_1 \psi_+ (\tilde\mu, x), & x>x_1
\endcases .
$$
If $\tilde\mu \in\sigma_d (H)$, we define in addition
$$
\eta_0 (x)=\psi_+ (\tilde\mu, x) =-\tilde\eta_0 (x), \quad x_0 =-\infty
$$
and if $\mu\in\sigma_d (H)$,
$$
\eta_2 (x)=\psi_- (\mu, x) =\tilde\eta_2 (x), \quad x_2 =+\infty.
$$
Then
$$
(\eta_j, \eta_k) = (\tilde\eta_j, \tilde\eta_k) \quad \text{for all $j,k$}.
\tag 2.25
$$
Indeed, let $j<k$, then (2.25) just means that
$$
\int\limits^{x_k}_{x_j} dx\,\psi_- (\mu, x) \psi_+ (\tilde\mu, x) =0.
\tag 2.26
$$
But
$$
\int\limits^{x_k}_{x_j} dx\,\psi_- (\mu, x) \psi_+ (\tilde\mu, x) = (\mu -
\tilde\mu)^{-1}
[W(x_k)-W(x_j)]=0
$$
due to $W(x_1)=0$ and $\lim_{x\to\pm\infty} W(x)=0$ if $\tilde\mu$ or $\mu$
lie in $\sigma_d (H)$ since $\lim_{x\to\pm\infty}\mathbreak
W(g_1, g_2) (x)=0$
for all $g_1, g_2 \in\Cal D(H)$. (For $x_0=-\infty$ take $g_2 =\psi_+
(\tilde\mu)\in
\Cal D (H)$ and choose $g_1 =\psi_- (\mu)$ near $x_0=-\infty$ and
continue $g_1
\in\Cal D (H)$ appropriately. Similarly, for $x_2 =+\infty$, take $g_1 =
\psi_- (\mu)
\in\Cal D (H)$ and choose $g_2 =\psi_+  (\tilde\mu)$ in a neighborhood
of $x_2 =
+\infty$ and continue $g_2 \in\Cal D (H)$ appropriately.) Next, one
verifies that
$$
\biggl[H -\frac{1}{2} (\tilde\mu +\mu)\biggr] \eta_j =\frac{1}{2} (\mu
-\tilde\mu)
\tilde\eta_j
$$
and hence for $\eta\in\text{span}\{\eta_j\}$,
$$
\biggl\| \biggl[ H-\frac{1}{2} (\tilde\mu +\mu)\biggr]\eta\biggr\|
=\frac{1}{2}
|\tilde\mu -\mu| \|\eta\|
$$
implying
$$
\dim\,\text{Ran}(P_{[\mu,\tilde\mu]}(H))\geq\dim\,\text{span}\{\eta_j\},
$$
where $P_\Omega (H)$ denotes the spectral projection of $H$ corresponding
to $\Omega
\subseteq\Bbb R$. But $\psi_- (\mu)$ and $\psi_+ (\tilde\mu)$ are linearly
independent
on each interval (since their Wronskian is non-constant) and hence all
$\eta_j$ are
linearly independent. In particular,
$$
\dim\,\text{Ran}(P_{(\mu, \tilde\mu)}(H)) \geq 1,
$$
which contradicts our basic hypothesis that $(E_0, E_1)\subset \Bbb R
\backslash\sigma(H)$. This contradiction shows that (2.24) is impossible,
and hence
$W(x)\neq 0$ for all $x\in\Bbb R$.

Next, consider the case $\tilde\sigma=-\sigma=-$ (and still $\tilde\mu
>\mu$). Define
$$
\widehat W(x)=W(\psi_- (\mu), \psi_- (\tilde\mu))(x), \quad x\in\Bbb R
$$
and suppose
$$
\widehat W (x_1)=0 \quad \text{for some $x_1 \in\Bbb R$}.  \tag 2.27
$$
We introduce
$$
\eta_1 (x) =\cases \psi_- (\mu,x)-\gamma_1 \psi_- (\tilde\mu, x),
&x\leq x_1 \\
0, & x\geq x_1
\endcases
$$
(fixing $\gamma_1$ by demanding $\eta_1 \in\Cal D(H)$) and
$$
\tilde\eta_1 (x) =\cases \psi_- (\mu, x)
+\gamma_1 \psi_- (\tilde\mu, x), &
x\leq x_1 \\
0, & x>x_1
\endcases .
$$
If $\tilde\mu \in\sigma_d(H)$, we introduce in addition
$$
\eta_0 (x) =\psi_- (\tilde\mu, x)=-\tilde\eta_0 (x), \quad x_0 =+\infty
$$
and if $\tilde\mu \in\sigma_d (H)$,
$$
\eta_2 (x)=\psi_- (\mu,x)=\tilde\eta_2 (x), \quad x_0 =+\infty.
$$
The rest of the proof is analogous to the case considered first: The
$\eta_j$'s are linearly
independent by considering their supports and
$$
\int\limits^{x_1}_{-\infty} dx\,\psi_- (\mu, x) \psi_- (\tilde\mu, x)
= (\mu
-\tilde\mu)^{-1} \lim\limits_{c\downarrow -\infty} [\widehat W(x_1)
-\widehat W(c)]=0
$$
since $\widehat W (x_1)=0$ by hypothesis, and both $\psi_- (\mu, x)$ and
$\psi_-
(\tilde\mu, x)$ satisfy the boundary conditions of $H$ at $-\infty$.

Finally, the cases $\psi_+ (\mu, x)$, $\psi_\pm (\tilde\mu, x)$ can be
obtained by
reflection. \qed
\enddemo

Actually, Lemma 2.2 is only the tip of the iceberg. The principal results
of [\gst] relate
the number of zeros of appropriate Wronskians on an arbitrary interval
$(a,b)$ of the
type studied in this section to dimensions of spectral projections of
general Sturm-Liouville
operators on $(a,b)$. For a previous generalization of Sturm's separation
theorem invoking
the sign of Wronskians, see [\lei].

For later reference, we now summarize our basic assumptions on $V$,
$\mu$, and
$\tilde\mu$ in the following hypothesis.

\example{(H.2.3)}  (i) Suppose $V\in L^1_{\text{\rom{loc}}}(\Bbb R)$ to be
real-valued.

(ii)
$$\aligned
&(E_0, E_1)\subset\Bbb R\backslash\sigma(H), \quad E_0, E_1 \in\sigma(H), \\
&\mu\in\sigma_d (H^D), \quad (\mu,\sigma)\in (E_0, E_1)\times \{-,+\}
\text{ or } \mu\in\{E_0, E_1\}\cap \sigma_d (H), \\
&(\tilde\mu,\tilde\sigma)\in (E_0, E_1)\times\{-,+\} \text{ or }
\tilde\mu \in \{E_0, E_1\}\cap\sigma_d (H), \\
&\mu,\tilde\mu \in [E_0, E_1],  \quad \mu\neq\tilde\mu \text{ or }
(\tilde\mu, \tilde\sigma) = (\mu,-\sigma), \quad \mu\in (E_0, E_1).
\endaligned
$$
\endexample

Next, we introduce various solutions of
$(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z)
\tilde\psi (z)=0$ needed in (2.32)--(2.35) to define the self-adjoint
operator
$\tilde H_{(\tilde\mu, \tilde\sigma)}$ in $L^2 (\Bbb R)$ associated with
$\tilde\psi_{(\tilde\mu, \tilde\sigma)}$. Define
$$\align
\tilde\psi_{-\sigma}(\mu,x) &=\psi_{-\tilde\sigma}(\tilde\mu,x) \big/
W_{(\tilde\mu, \tilde\sigma)}(x), \tag 2.28 \\
\tilde\psi_{\tilde\sigma}(\tilde\mu, x) &= \psi_\sigma (\mu,x) \big/
W_{(\tilde\mu,\tilde\sigma)}(x), \quad \tilde\psi_{\tilde\sigma}(\tilde\mu,
x_0)=0.
\tag 2.29
\endalign
$$

Then
$$
(\tilde\tau_{(\tilde\mu,\tilde\sigma)}\tilde\psi_{-\sigma}(\mu))(x)=\mu
\tilde\psi_{-\sigma}(\mu,x), \quad (\tilde\tau_{(\tilde\mu,\tilde\sigma)}
\tilde\psi_{\tilde\sigma}(\tilde\mu))(x)=\tilde\mu \tilde\psi_{\tilde\sigma}
(\tilde\mu, x) \tag 2.30
$$
and
$$
\tilde\psi_{-\sigma}(\mu,x) \tilde\psi_{\tilde\sigma}(\tilde\mu, x) =
[W_{(\tilde\mu,\tilde\sigma)}(x)^{-1}]'. \tag 2.31
$$

The Dirichlet deformation operator $\tilde H_{(\tilde\mu,\tilde\sigma)}$
associated with
$\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ in (2.20) is then defined as follows:
$$\aligned
\tilde H_{(\tilde\mu,\tilde\sigma)}f
&=\tilde\tau_{(\tilde\mu,\tilde\sigma)} f, \\
f\in\Cal D(\tilde H_{(\tilde\mu,\tilde\sigma)}) &= \{g\in L^2(\Bbb R)\mid
g,g'\in AC_{\text{\rom{loc}}}(\Bbb R); \tilde\tau_{(\tilde\mu,
\tilde\sigma)} g\in L^2 (\Bbb R); \\
& \qquad  g \text{ satisfies one of the b.c.'s in Cases I--III if } \\
& \qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)} \text{ is l.c.~at $-\infty$
and/or $+\infty$}\}.
\endaligned \tag 2.32
$$
The boundary conditions (b.c.'s) alluded to in (2.32) are chosen as follows:

\medpagebreak
\flushpar {\bf{Case I:}} Either $\tau$ is l.p.~at $\pm\infty$ or
$\tilde\sigma =\sigma$.
$$\aligned
\lim\limits_{x\to\tilde\sigma \infty}
W(\tilde\psi_{\tilde\sigma}(\tilde\mu), g)(x) &= 0
\text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at $\tilde\sigma
\infty$}, \\
\lim\limits_{x\to -\tilde\sigma\infty}
W(\tilde\psi_{-\sigma}(\mu), g)(x) &=0
\text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at
$-\tilde\sigma\infty$}.
\endaligned \tag 2.33
$$

\flushpar{\bf{Case II:}}  $\tilde\sigma=-\sigma$, $\tau$ is l.c.~at
$-\infty$ or $+\infty$,
and $\mu\in\sigma_d (H)$.
$$
\lim\limits_{x\to\omega\infty}
W(\tilde\psi_{\tilde\sigma}(\tilde\mu), g)(x)=0
\text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at
$\omega\infty$},
\quad \omega\in\{-,+\}. \tag 2.34
$$

\flushpar{\bf{Case III:}} $\tilde\sigma=-\sigma$, $\tau$ is l.c.~at
$-\infty$ or $+\infty$,
and $\tilde\mu\in\sigma_d(H)$.
$$
\lim\limits_{x\to\omega\infty} W(\tilde\psi_{-\sigma}(\mu), g)(x)=0
\text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at
$\omega\infty$}, \quad
\omega\in\{-,+\}. \tag 2.35
$$
(Note that Case II $=$ Case III if $(\tilde\mu, \tilde\sigma) = (\mu,
-\sigma)$.)

As always, there is no boundary condition at $\omega\infty$ in (2.32) if
$\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.p.~at $\omega\infty$,
$\omega\in\{-,+\}$.
Cases I--III, of course, are not exhaustive. We singled them out since they
are the only
situations where the spectra of $H$ and $\tilde
H_{(\tilde\mu,\tilde\sigma)}$ are closely
related (see (3.17) and the discussion at the end of Section 3).

If $\tilde\mu\in\sigma_d(H)$, we will occasionally use the more appropriate
notation
$\tilde V_{\tilde\mu}(x)$, $\tilde\tau_{\tilde\mu}$, and $\tilde
H_{\tilde\mu}$ (instead
of $\tilde V_{(\tilde\mu,\tilde\sigma)}(x)$,
$\tilde\tau_{(\tilde\mu,\tilde\sigma)}$, and
$\tilde H_{(\tilde\mu,\tilde\sigma)}$, cf.~the comments following (2.21)).

We conclude this section by introducing the Dirichlet operator
$\tilde H^D_{(\tilde\mu,\tilde\sigma)}$ associated with $\tilde
H_{(\tilde\mu,\tilde\sigma)}$
and the fixed reference point $x_0\in\Bbb R$,
$$\aligned
\tilde H^D_{(\tilde\mu,\tilde\sigma)} f
&=\tilde\tau_{(\tilde\mu,\tilde\sigma)} f, \\
f\in\Cal D(\tilde H^D_{(\tilde\mu,\tilde\sigma)}) &=
\bigl\{g\in L^2 (\Bbb R) \mid g\in AC_{\text{\rom{loc}}} (\Bbb R), g'\in
AC_{\text{\rom{loc}}}(\Bbb R\backslash \{x_0\});
\lim\limits_{\epsilon\downarrow 0}
g(x_0 \pm\epsilon)=0; \\
& \qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)}g\in L^2 (\Bbb R);
g \text{ satisfies one of the b.c.'s in Cases I--III if }\\
&\qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)} \text{ is l.c.~at $-\infty$
and/or
$+\infty$}\bigr\}.
\endaligned \tag 2.36
$$
In analogy to (2.5), $\tilde H^D_{(\tilde\mu,\tilde\sigma)}$ decomposes into
$$
\tilde H^D_{(\tilde\mu,\tilde\sigma)} = \tilde
H^D_{(\tilde\mu,\tilde\sigma), -}
\oplus \tilde H^D_{(\tilde\mu,\tilde\sigma), +} \tag 2.37
$$
with respect to (2.6).

\vskip 0.3in

\flushpar{\bf \S 3. Half-Line Weyl-Titchmarsh and Spectral Functions}
\vskip 0.1in

In this section we derive the Weyl-Titchmarsh $m$-functions for the
Dirichlet deformation
operator $\tilde H_{(\tilde\mu,\tilde\sigma)}$ and relate them to those of
$H$. Moreover,
we provide a complete spectral characterization of $\tilde
H^D_{(\tilde\mu,\tilde\sigma),
\pm}$ in terms of $H^D_\pm$.

We start by introducing the transformation
$$
U_{(\tilde\mu,\tilde\sigma)}(z): \cases AC_{\text{\rom{loc}}}(\Bbb R)\to
L^1_{\text{\rom{loc}}}(\Bbb R) \\
f(x)\to\tilde f_{(\tilde\mu,\tilde\sigma)}(z,x) =f(x)-(z-\mu)^{-1}
\tilde\psi_{-\sigma}
(\mu, x)W(\psi_\sigma (\mu), f)(x), \, z\in\Bbb C\backslash \{\mu\}
\endcases
\tag 3.1
$$
and note that by inspection,
$$
((\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z)
U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) =0
\text{ if and only if } ((\tau-z)\psi(z))(x)=0, \quad z\in\Bbb C\backslash
\{\mu\}.
\tag 3.2
$$
Moreover, one verifies
$$\aligned
\tilde f_{(\tilde\mu,\tilde\sigma)}(z,x) &=
(U_{(\tilde\mu,\tilde\sigma)}(z)f)(x) \\
&= (z-\mu)^{-1} (z-\tilde\mu) f(x)-(z-\mu)^{-1} \psi_{\tilde\sigma}
(\tilde\mu, x)
W(\psi_{-\tilde\sigma} (\mu), f)(x), \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
f\in AC_{\text{\rom{loc}}}(\Bbb R), \ z\in\Bbb C
\backslash \{\mu\},
\endaligned \tag 3.3
$$
$$\gathered
(U_{(\tilde\mu,\tilde\sigma)}
(\tilde\mu)\psi_{-\tilde\sigma}(\tilde\mu))(x)=0,
\quad
(U_{(\tilde\mu,\tilde\sigma)}
(\tilde\mu)\psi_{\tilde\sigma}(\tilde\mu))(x)=
(\tilde\mu -\mu)^{-1} W(\psi_{\tilde\sigma}(\tilde\mu),
\psi_{-\tilde\sigma}(\tilde\mu)) \tilde\psi_{\tilde\sigma}
(\tilde\mu, x),
\\
\lim\limits_{z\to\mu}(z-\mu) (U_{(\tilde\mu,\tilde\sigma)}(z) \psi_\sigma
(\mu))(x)=0, \\
\lim\limits_{z\to\mu}
(z-\mu)(U_{(\tilde\mu,\tilde\sigma)}(z)\psi_{-\tilde\sigma}
(\mu))(x)= -W(\psi_\sigma (\mu),
\psi_{-\sigma}(\mu))\tilde\psi_{-\sigma}(\mu, x).
\endgathered
\tag 3.4
$$
In addition, if
$$
(\tau -z)\psi(z)=0, \quad (\tau -\widehat z) \widehat\psi (\widehat z)=0,
\quad z, \widehat
z\in\Bbb C\backslash\{\mu\},
$$
then
$$\gather
W(\tilde\psi_{-\sigma}(\mu),
U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) = \frac
{W(\psi_{-\tilde\sigma}(\tilde\mu), \psi (z))(x)}
{W_{(\tilde\mu,\tilde\sigma)}(x)}, \tag 3.5 \\
W(\tilde\psi_{\tilde\sigma}(\tilde\mu),
U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) =
\frac{z-\tilde\mu}{z-\mu} \, \frac{W(\psi_\sigma (\mu), \psi(z))(x)}
{W_{(\tilde\mu,\tilde\sigma)}(x)}, \tag 3.6 \\
W(U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z),
U_{(\tilde\mu,\tilde\sigma)}(\widehat z)
\widehat\psi (\widehat z))(x) = \frac{z-\tilde\mu}{z-\mu} \, W(\psi(z),
\widehat\psi
(\widehat z))(x)  \\
\qquad \qquad  + \ \frac{z-\widehat z}{(z-\mu)(\widehat z-\mu)} \,
\frac{W(\psi_\sigma (\mu), \widehat\psi (\widehat z))(x)
W(\psi_{-\tilde\sigma}(\tilde\mu),
\psi(z))(x)}{W_{(\tilde\mu,\tilde\sigma)}(x)}. \tag 3.7
\endgather
$$
Next, let $\phi(z,x),\theta(z,x)$ be the standard fundamental system of
solutions of
$(\tau -z)\psi(z)=0$, $z\in\Bbb C$ defined by
$$
\phi(z,x_0)=\theta'(z,x_0)=0, \quad \phi'(z,x_0)=\theta(z,x_0)=1, \quad
z\in\Bbb C \tag 3.8
$$
with $x_0 \in\Bbb R$ the reference point used in (2.4), and denote by
$\tilde\theta_{(\tilde\mu,\tilde\sigma)}(z,x),
\tilde\phi_{(\tilde\mu,\tilde\sigma)}(z,x)$
the analogous fundamental system of solutions of
$(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z)
\tilde\psi(z)=0$, $z\in\Bbb C$ satisfying (3.8). Since by definition
(2.15), $\psi_\sigma
(z,\,\cdot\,)\in L^2((R, \sigma\infty))$, $R\in\Bbb R$,
$z\in\Bbb C\backslash
\Bbb R$ satisfy the boundary conditions of $H$ near $\sigma\infty$ (if any)
and, in
particular, $\lim_{x\to\sigma\infty} W(\psi_\sigma (z),g)(x)=0$, $g\in\Cal
D(H)$,
one obtains
$$
\psi'_\sigma (z,x_0)/\psi_\sigma (z,x_0) = m_\sigma (z), \quad
\sigma\in\{-,+\}, \
z\in\Bbb C\backslash\Bbb R, \tag 3.9
$$
where $m_\sigma (z)$ denotes the Weyl-Titchmarsh $m$-function of $H$ with
respect
to the half-line $(x_0, \sigma\infty)$, $\sigma\in\{-,+\}$. Thus,
$$\aligned
(U_{(\tilde\mu,\tilde\sigma)}(z)f)(x_0) &= \frac{z-\tilde\mu}{z-\mu}\,
f(x_0), \\
(U_{(\tilde\mu,\tilde\sigma)}(z) f)' (x_0) &= f'(x_0) -\frac{\tilde\mu
-\mu}{z-\mu}\,
m_{-\tilde\sigma}(\tilde\mu) f(x_0), \quad z,\tilde\mu\in\Bbb
C\backslash\{\mu\}, \
f\in AC_{\text{\rom{loc}}}(\Bbb R),
\endaligned
\tag 3.10
$$
imply
$$\aligned
\tilde\phi_{(\tilde\mu,\tilde\sigma)}(z,x)
&= (U_{(\tilde\mu,\tilde\sigma)}(z)
\phi(z))(x), \\
\tilde\theta_{(\tilde\mu,\tilde\sigma)}(z,x)
&= \frac{z-\mu}{z-\tilde\mu}\,
(U_{(\tilde\mu,\tilde\sigma)}(z)\theta(z))(x) + \frac{\tilde\mu
-\mu}{z-\tilde\mu}\,
m_{-\tilde\sigma}(\tilde\mu) (U_{(\tilde\mu,\tilde\sigma)}(z)\phi(z))(x),
\, z,\tilde\mu \in\Bbb C\backslash\{\mu\}.
\endaligned
\tag 3.11
$$
The case $(\tilde\mu,\tilde\sigma)=(\mu,-\sigma)$ in (3.10) and (3.11) can
be obtained by
a limiting procedure ($\pm m_\pm (z)$ being Herglotz, has at most one
simple pole for
$\lambda\in [E_0, E_1]$ with a negative residue),
$$
\lim\limits_{\tilde\mu\to\mu}(\tilde\mu -\mu)m_\sigma (\tilde\mu) =
\biggl(\,\int\limits^{x_0}_{\sigma\infty}dx \, \phi(\mu,x)^2\biggr)^{-1},
\quad \sigma\in\{-,+\}
\tag 3.12
$$
(see, e.g., [\gstrans], Appendix A for a brief summary on Weyl
$m$-functions).

The following general fact on Weyl $m$-functions, which provides an
effective tool for
computing them in the context of $\tilde H_{(\tilde\mu,\tilde\sigma)}$, may
well be of
independent interest.

\proclaim{Lemma 3.1} Let $\widehat V\in L^1_{\text{\rom{loc}}}(\Bbb R)$ be
real-valued, $\widehat\tau =-\frac{d^2}{dx^2}+\widehat V(x)$,
$x\in\Bbb R$, and
$\widehat\eta_\sigma (\lambda, x)$, $\lambda\in\Bbb R$, $\sigma\in\{-,+\}$
non-zero
real-valued solutions of $(\widehat\tau -\lambda)\widehat\psi(\lambda)=0$.
Define the
self-adjoint operator $\widehat H$ in $L^2 (\Bbb R)$
by
$$\aligned
\widehat Hf &=\widehat\tau f, \\
f\in\Cal D(\widehat H) &= \bigl\{g\in L^2 (\Bbb R) \mid g,g'\in
AC_{\text{\rom{loc}}}
(\Bbb R); \widehat\tau g\in L^2 (\Bbb R); \\
&\qquad \lim\limits_{x\to\pm\infty} W(\widehat\eta_\pm
(\lambda), g)(x) =0 \text{ if $\widehat\tau$ is l.c.~at $\pm\infty$}\bigr\}.
\endaligned
\tag 3.13
$$
\rom(If $\widehat\tau$ is l.p.~at $+\infty$ and/or $-\infty$, the
corresponding boundary
condition in {\rom{(3.13)}} is to be deleted.\rom) Denote by $\widehat\phi
(z,x),
\widehat\theta (z,x)$ the fundamental system of solutions of
$(\widehat\tau -z)
\widehat\psi(z)=0$, $z\in\Bbb C$, with initial values as in {\rom{(3.8)}}.
Then the
limits
$$
\widehat m_\sigma (z) =-\lim\limits_{x\to\sigma\infty}
\frac{W(\widehat\eta_\sigma
(\lambda), \widehat\theta (z))(x)}{W(\widehat\eta_\sigma (\lambda),
\widehat\phi(z))
(x)}, \quad z\in\Bbb C\backslash\Bbb R, \  \sigma\in\{-,+\} \tag 3.14
$$
exist and represent the Weyl-Titchmarsh $m$-functions of $\widehat H$ on
the half-line
$(x_0, \sigma\infty)$.
\endproclaim

\demo{Proof} First suppose that $\widehat\tau$ is l.p.~at
$\sigma\infty$ and
 $z\in
\Bbb C\backslash\Bbb R$. Then
$$
-\frac{W(\widehat\eta_\sigma (\lambda),
\widehat\theta(z))(x)}{W(\widehat\eta_\sigma
(\lambda), \widehat\phi(z))(x)} = -\frac{\widehat\theta(z,x) +
\tan(\beta(x))\widehat\theta'
(z,x)}{\widehat\phi (z,x) + \tan(\beta(x))\widehat\phi' (z,x)}, \quad
\cot(\beta(x))=
-\eta'_\sigma (\lambda, x)/\eta_\sigma (\lambda, x) \tag 3.15
$$
converges to $\widehat m_\sigma (z)$. This does not quite represent the
typical Weyl limit
point consideration in which one usually involves an $x$-independent
boundary condition
parameter $\beta\in [0,\pi)$. However, due to the l.p.~hypothesis
made, the
Weyl disks shrink
to a limit point and the $x$-dependence of $\beta(x)$ in (3.15) becomes
immaterial in the
limit $x\to\sigma\infty$.

Next, assume $\widehat\tau$ is l.c.~at $\sigma\infty$. Then $\widehat\phi
(z,\,\cdot\,),
\widehat\theta (z,\,\cdot\,)\in L^2 ((R, \sigma\infty))$, $R \in\Bbb R$,
and hence the limits
$$\multline
\lim\limits_{x\to\sigma\infty} W(\widehat\eta_\sigma (\lambda),
\widehat\chi(z))(x) =
W(\widehat\eta_\sigma (\lambda), \widehat\chi(z))(x_0)  \\
+ (z-\lambda) \int\limits^{x_0}_{\sigma\infty} dx' \, \widehat\eta_\sigma
(\lambda, x') \widehat\chi (z,x')  \text{ for } \widehat\chi(z,x) =
\cases \widehat\phi (z,x) \\ \widehat\theta (z,x) \endcases
\endmultline  \tag 3.16
$$
exist. Actually, the limits in (3.16) not only exist but they are also
non-zero since otherwise
one could construct self-adjoint operators with boundary conditions at
$\sigma\infty$
induced by $\widehat\phi(z,x)$ or $\widehat\theta(z,x)$ with associated
eigenvalue
$z\in\Bbb C\backslash\Bbb R$.

Next, consider the function
$$
\widehat\psi_{\sigma}(z,x) =\widehat\theta (z,x)+\widehat m_{\sigma} (z)
\widehat\phi (z,x), \quad z\in\Bbb C\backslash\Bbb R,
$$
where $\widehat m_{\sigma}(z)$ denotes the $m$-function associated with
$\widehat H$
on $(x_0, \sigma\infty)$. Since by construction,
$\widehat\psi_{\sigma}(z,x)$ satisfies
the boundary conditions of $\widehat H$ at $\sigma\infty$, one infers
$$\align
0 &=\lim\limits_{x\to\sigma\infty} W(\widehat\eta_{\sigma} (\lambda),
\widehat\psi_{\sigma} (z))(x) \\
&= \bigl[ \lim\limits_{x\to\sigma\infty} W(\widehat\eta_{\sigma}(\lambda),
\widehat\theta (z))(x) \bigr]  +\widehat m_{\sigma}(z)
\bigl[\lim\limits_{x\to\sigma\infty}
W(\widehat\eta_{\sigma}(\lambda), \widehat\phi(z))(x)\bigr],
\endalign
$$
and hence (3.14) again. \qed
\enddemo

Applying Lemma 3.1 to $\tilde H_{(\tilde\mu, \tilde\sigma)}$, we obtain as
our first
major result the following expression for the half-line Weyl-Titchmarsh
$m$-functions
of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ in terms of those of $H$.

\proclaim{Theorem 3.2} Assume {\rom{(H.2.3)}}. Given $H$ and $\tilde
H_{(\tilde\mu,
\tilde\sigma)}$ by {\rom{(2.3)}} and {\rom{(2.32)}}, respectively, denote
by $m_\pm (z)$
and $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ the corresponding
$m$-functions
associated with the half-lines $(x_0, \pm\infty)$. Then
$$\aligned
\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z) &= \frac{z-\mu}{z-\tilde\mu}\,
m_\pm (z)
-\frac{\tilde\mu-\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu), \quad
\tilde\mu\neq\mu, \\
\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z) &= m_\pm (z) -
\biggl(\,\int\limits^{x_0}_{\sigma
\infty}dx\, \phi(\mu,x)^2\biggr)^{-1} \frac{1}{z-\mu}, \quad (\tilde\mu,
\tilde\sigma) =
(\mu, -\sigma), \ z\in\Bbb C\backslash\Bbb R.
\endaligned \tag 3.17
$$
\endproclaim

\demo{Proof} Combining (3.11), (3.5), and (3.6), one computes $\tilde
m_{(\tilde\mu,
\tilde\sigma),\omega}(z)$, $\omega\in\{-,+\}$ either from
$$
-\lim\limits_{x\to\omega\infty}\frac{W(\tilde\psi_{\tilde\sigma}(\tilde\mu),
\tilde\theta_{(\tilde\mu,
\tilde\sigma)}(z))(x)}{W(\tilde\psi_{\tilde\sigma}(\tilde\mu),
\tilde\phi_{(\tilde\mu, \tilde\sigma)}(z))(x)} =
\frac{z-\mu}{z-\tilde\mu}
\biggl[-\lim\limits_{x\to\omega\infty} \frac{W(\psi_\sigma (\mu),
\theta(z))(x)}
{W(\psi_\sigma (\mu), \phi(z))(x)}\biggr] -\frac{\tilde\mu
-\mu}{z-\tilde\mu}\,
m_{-\tilde\sigma}(\tilde\mu) \tag 3.18
$$
or from
$$\split
-\lim\limits_{x\to\omega\infty}& \frac{W(\tilde\psi_{-\sigma}(\mu),
\tilde\theta_{(\tilde\mu, \tilde\sigma)}(z))(x)}{W(\tilde\psi_{-\sigma}(\mu),
\tilde\phi_{(\tilde\mu, \tilde\sigma)}(z))(x)} \\
& = \frac{z-\mu}{z-\tilde\mu}
\biggl[-\lim\limits_{x\to\omega\infty} \frac{W(\psi_{-\tilde\sigma}
(\tilde\mu),
\theta(z))(x)}{W(\psi_{-\tilde\sigma} (\tilde\mu), \phi(z))(x)}\biggr]
- \frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu)
\endsplit\tag 3.19
$$
(or from both), depending on Cases I--III in (2.32)--(2.35) by applying
Lemma 3.1 to
$\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$. \qed
\enddemo

An examination of $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ in (3.17)
then reveals
the following behavior near $\mu$ and $\tilde\mu$.

\proclaim{Corollary 3.3} {\rom{(i)}} Suppose $\mu,\tilde\mu\in (E_0, E_1)$,
$\tilde\mu
\neq\mu$ or $\mu\in\{E_0, E_1\}\cap\sigma_d (H)$, $\tilde\mu\in
(E_0, E_1)$. Then, $\tilde m_{(\tilde\mu, \tilde\sigma),
-\tilde\sigma}(z)$ is
holomorphic in a neighborhood of $\tilde\mu$ whereas $\tilde m
_{(\tilde\mu, \tilde\sigma),
\tilde\sigma}(z)$ has a simple pole at $\tilde\mu$ with residue
$$
\lim\limits_{z\to\tilde\mu} (z-\tilde\mu) \tilde m_{(\tilde\mu,
\tilde\sigma), \tilde\sigma}
(z)= (\tilde\mu -\mu)
[m_{\tilde\sigma}(\tilde\mu)-m_{-\tilde\sigma}(\tilde\mu)] \neq 0.
\tag 3.20
$$
Both $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ are holomorphic in a
neighborhood of
$\mu$.

{\rom{(ii)}} Assume $\mu=\tilde\mu\in (E_0, E_1)$, $\tilde\sigma
=-\sigma$. Then $\tilde m_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}(z)$ is
holomorphic
in a neighborhood of $\tilde\mu$ whereas $\tilde m_{(\tilde\mu,
\tilde\sigma),
\tilde\sigma}(z)$ has a simple pole at $\tilde\mu$ with residue
$$
\lim\limits_{z\to\tilde\mu} (z-\tilde\mu)\tilde m_{(\tilde\mu,
\tilde\sigma), \tilde\sigma}
(z)=-\lim\limits_{\tilde\mu\to\mu} (\tilde\mu-\mu)m_\sigma (\tilde\mu) =
-\biggl( \,
\int\limits^{x_0}_{\sigma\infty} dx\,\phi(\mu,x)^2 \biggr)^{-1}. \tag 3.21
$$

{\rom{(iii)}} Assume $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E_0, E_1\} \cap
\sigma_d(H)$ or $\mu,\tilde\mu\in \{E_0, E_1\} \cap \sigma_d(H)$,
$\mu\neq\tilde\mu$.
Then $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ are both holomorphic in
a neighborhood
of $\mu$ and $\tilde\mu$.
\endproclaim

\demo{Proof} Observing that
$$
m_\sigma (z) \operatornamewithlimits{=}\limits_{z\to\mu}
c_\sigma (z-\mu)^{-1} +
O(1), \  c_\sigma\in\Bbb R\backslash\{0\}, \  m_{-\sigma}(z)
\operatornamewithlimits{=}\limits_{z\to\mu} O(1), \quad \mu\in (\lambda_0,
\lambda_1), \tag 3.22
$$
$$
m_\pm (z) \operatornamewithlimits{=}\limits_{z\to\mu} c_\pm (z-\mu)^{-1}
+O(1), \  c_\pm \in\Bbb R\backslash\{0\}, \  c_\pm\lessgtr 0, \
\mu\in\sigma_d (H^D)\cap\sigma (H), \tag 3.23
$$
cases (i) and (ii) are a straightforward consequence of (3.12),(3.17), and
the fact
that $m_- (\tilde\mu)\neq m_+ (\tilde\mu)$ since
$\tilde\mu\notin\sigma_d (H)$.
For (iii) one observes, in addition, that first,
$$
m_- (\tilde\mu)=m_+ (\tilde\mu) \quad\text{since }\tilde\mu\in\sigma_d (H)
\backslash\sigma_d (H^D)
$$
by hypothesis, and second,
$$
\lim\limits_{z\to\lambda} (z-\lambda) m_\pm (z) \text{ exists for all
$\lambda\in
[E_0, E_1]$} \tag 3.24
$$
since $\pm m_\pm (z)$ are Herglotz functions (and $\sigma_{\text{\rom{ess}}}
(H)\cap [E_0, E_1]=\emptyset$). \qed
\enddemo

As we will explore in more detail in the next section, Corollary 3.3(i) for
$\mu,\tilde\mu
\in(E_0, E_1)$, $\mu\neq\tilde\mu$ just means the Dirichlet datum $(\mu,
\sigma)$ gets changed into $(\tilde\mu, \tilde\sigma)$ and (ii) illustrates
the ``flip" of the
Dirichlet eigenvalue $\mu$ from one half-line $(x_0, \sigma\infty)$ to the
other,
$(x_0, -\sigma\infty)$, changing $(\mu,\sigma)$ into $(\mu, -\sigma)$. The
remaining cases
represent non-isospectral deformations of $H$ where the eigenvalue
$\mu\in\sigma_d (H)$,
respectively, $\tilde\mu\in\sigma_d (H)$, or both,
$\mu,\tilde\mu\in\sigma_d
 (H)$ are
actually ``knocked out" of the spectrum of $H$ (i.e., do not belong to
$\sigma
(\tilde H_{(\tilde\mu, \tilde\sigma)})$, respectively, $\sigma(\tilde
H_{\tilde\mu})$).

\proclaim{Corollary 3.4} Let $z\in\Bbb C\backslash\{\sigma(H)\cup\{\mu\}\}$,
$\tilde\mu\in (E_0, E_1)$. Then $\tilde\psi_{\tilde\sigma}(\tilde\mu, x)$,
$\tilde\psi_{-\sigma}(\mu,x)$, $\mu\neq\tilde\mu$, and $(U_{(\tilde\mu,
\tilde\sigma)}
(z))\psi_\pm (z))(x)$ satisfy the boundary conditions of $\tilde
H_{(\tilde\mu, \tilde\sigma)}$ \rom(if any\rom) at $\tilde\sigma\infty$,
$-\sigma\infty$, and $\pm\infty$, respectively.
In particular,
$$\aligned
\lim\limits_{x\to\tilde\sigma\infty}W(\tilde\psi_\sigma (\tilde\mu), \tilde
g)(x)&=0, \\
\lim\limits_{x\to -\sigma\infty} W(\tilde\psi_{-\sigma}(\mu),\tilde
g)(x)&=0, \quad
\mu\neq\tilde\mu, \\
\lim\limits_{x\to\pm\infty}W(U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z),
\tilde g)(x) &=0
\endaligned \tag 3.25
$$
for all $\tilde g\in\Cal D(\tilde H_{(\tilde\mu, \tilde\sigma)})$. Moreover,
$$\aligned
\psi_{\tilde\sigma}(\tilde\mu, \,\cdot\,)
&\in L^2 ((R,\tilde\sigma\infty)), \\
\tilde\psi_{-\sigma}(\mu,\,\cdot\,) &\in L^2 ((R, -\sigma\infty)), \quad
\mu\neq\tilde\mu, \\
U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z) &\in L^2 ((R,\pm\infty)), \quad
R\in\Bbb R
\endaligned \tag 3.26
$$
\rom(justifying the notation we chose for
$\tilde\psi_{\tilde\sigma}(\tilde\mu, x)$
and $\tilde\psi_{-\sigma}(\mu, x)$\rom).
\endproclaim

\demo{Proof} Let $\tilde H^D_{(\tilde\mu, \tilde\sigma),\omega}$ denote the
Dirichlet
operators (2.37) corresponding to $\tilde\tau_{(\tilde\mu, \tilde\sigma)}$
on the half-line
$(x_0, \omega\infty)$, $\omega\in\{-,+\}$. Since $\tilde m_{(\tilde\mu,
\tilde\sigma),
\tilde\sigma}(z)$ has a pole at $z=\tilde\mu$ by Corollary 3.3, one infers
$\tilde\mu\in
\sigma_d (\tilde H^D_{(\tilde\mu, \tilde\sigma),\tilde\sigma})$. Moreover,
since
$(\tilde\tau_{(\tilde\mu,
\tilde\sigma)}-\tilde\mu)\tilde\psi_{\tilde\sigma}(\tilde\mu)=0$
and $\tilde\psi_{\tilde\sigma}(\tilde\mu, x_0)=0$ (cf.~(2.29)),
$\tilde\psi_{\tilde\sigma}
(\tilde\mu, x)$ is the corresponding eigenfunction of
$\tilde H^D_{(\tilde\mu,
\tilde\sigma), \tilde\sigma}$ and hence
$\tilde\psi_{\tilde\sigma}(\tilde\mu)\in \Cal D
(\tilde H^D_{(\tilde\mu, \tilde\sigma), \tilde\sigma})$ satisfies (3.25)
and (3.26). In the
case of $\tilde\psi_{-\sigma}(\mu, x)$, $\mu\neq\tilde\mu$, one
verifies that
$$
\tilde m_{(\tilde\mu, \tilde\sigma), -\sigma}(\mu)
=m_{-\tilde\sigma}(\tilde\mu) =
\psi'_{-\tilde\sigma} (\tilde\mu, x_0)/\psi_{-\tilde\sigma}
(\tilde\mu, x_0) =
\tilde\psi'_{-\sigma}(\mu, x_0) \big/ \tilde\psi_{-\sigma} (\mu, x_0)
$$
and hence (3.25) and (3.26) are valid for $\tilde\psi_{-\sigma}(\mu)$.
Finally, as a
consequence of (2.15), one infers that
$$
\psi_\pm (z,x)=c_\pm (z) [\theta(z,x)+m_\pm (z)\phi (z,x)], \quad z\in\Bbb
C\backslash
\sigma(H) \tag 3.27
$$
for some constants $c_\pm (z)$. Combining (3.11), (3.17), and applying
$U_{(\tilde\mu, \tilde\sigma)}(z)$ to (3.27) results in
$$\multline
((U_{(\tilde\mu, \tilde\sigma)}(z) \psi_\pm (z))(x)= \\
c_\pm (z) \frac{z-\tilde\mu}{z-\mu}\, [\tilde\theta_{(\tilde\mu,
\tilde\sigma)}(z,x)
+ \tilde m_{(\tilde\mu, \tilde\sigma)}(z) \tilde\phi_{(\tilde\mu,
\tilde\sigma)}(z,x)],
\quad  z\in\Bbb C\backslash\{\sigma(H)\cup\{\mu\}\}.
\endmultline \tag 3.28
$$
Clearly, (3.28) proves that $U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z)$
satisfy
(3.25)  and (3.26). \qed
\enddemo

Given the fundamental relation between $\tilde m_{(\tilde\mu,
\tilde\sigma),\pm}(z)$
and $m_\pm (z)$ in Theorem 3.2, we can readily derive the ensuing relation
between
the corresponding spectral functions $\tilde\rho_{(\tilde\mu,
\tilde\sigma),\pm}(\lambda)$
and $\rho_\pm (\lambda)$ associated with the half-line Dirichlet operators
$\tilde H^D_{(\tilde\mu, \tilde\sigma),\pm}$ and $H^D_\pm$. The
right-continuous
non-decreasing functions $\rho_\pm (\lambda)$ and $\tilde\rho_{(\tilde\mu,
\tilde\sigma),\pm}
(\lambda)$ are defined for $\lambda\in\Bbb R$ by
$$
\rho_\pm (\lambda) -\rho_\pm (\lambda') =
\pm\lim\limits_{\delta\downarrow 0}
\lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda
+\delta}_{\lambda' +
\delta} d\nu \text{ Im}[m_\pm (\nu+i\epsilon)] \tag 3.29
$$
and
$$
\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)
-\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}
(\lambda') =\pm\lim\limits_{\delta\downarrow 0}
\lim\limits_{\epsilon\downarrow 0}
\pi^{-1} \int\limits^{\lambda +\delta}_{\lambda' +\delta} d\nu \text{ Im}
[m_{(\tilde\mu, \tilde\sigma), \pm}(\nu+ i\epsilon)]. \tag 3.30
$$
This sets up the second major result of this section.

\proclaim{Theorem 3.5} Assume {\rom{(H.2.3)}}. Let $H^D_\pm$ and
$\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ denote the Dirichlet operators
{\rom{(2.4)}} and {\rom{(2.37)}}, and $d\rho_\pm (\lambda)$ and $d
\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)$ the corresponding
spectral measures
generated by $\rho_\pm (\lambda)$ and $\tilde\rho_{(\tilde\mu,
\tilde\sigma),\pm}(\lambda)$,
respectively. Then
$$\align
d\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda) &= \frac{\lambda-\mu}
{\lambda-\tilde\mu}\, d\rho_\pm (\lambda) + (\tilde\mu -\mu)
\left\{ \matrix \format\l &\quad\l \\
0, & \tilde\sigma =\mp \\
[m_- (\tilde\mu)-m_+ (\tilde\mu)], & \tilde\sigma=\pm \endmatrix
\right\} d\theta (\lambda-\tilde\mu), \ \mu\neq\tilde\mu, \\
d\tilde\rho_{(\tilde\mu, \tilde\sigma), \pm}(\lambda)
&= d\rho_\pm (\lambda) +
\left\{ \matrix \format\l &\quad\l \\
0, & \sigma=\pm \\
(\pm\int^{x_0}_{\mp\infty} dx\, \phi(\mu, x)^2)^{-1}, & \sigma=\mp
\endmatrix \right \}
d\theta (\lambda-\mu), \tag 3.31 \\
&\qquad \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad
\qquad
{(\tilde\mu, \tilde\sigma) = (\mu, -\sigma), \ \lambda\in\Bbb R.}
\endalign
$$
Here $\theta(x)=\cases 1, & x\geq 0 \\
0, & x<0 \endcases$.
\endproclaim

\demo{Proof} Inserting (3.17), for $\mu\neq\tilde\mu$ (for simplicity) into
(3.30) yields
$$\multline
\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)
-\tilde\rho_{(\tilde\mu, \tilde\sigma) \pm}
(\lambda') = \pm \lim\limits_{\delta\downarrow 0}
\lim\limits_{\epsilon\downarrow 0}
\pi^{-1} \int\limits^{\lambda+\delta}_{\lambda' +\delta} d\nu \biggl\{
\frac{\epsilon^2 +
(\nu-\tilde\mu)(\nu-\mu)}{(\nu-\tilde\mu)^2 +\epsilon^2}\, \text{Im}[m_\pm
(\nu+i\epsilon)]  \\
+ \frac{(\tilde\mu -\mu)\epsilon}{(\nu-\tilde\mu)^2 +\epsilon^2}
\{m_{-\tilde\sigma}
(\tilde\mu) -\text{Re}[m_\pm (\nu+i\epsilon)]\}\biggr\}.
\endmultline \tag 3.32
$$
Since
$$\gather
m_\pm (\nu) \text{ are real-valued for } \nu\in [E_0, E_1]\backslash
\{\mu\},  \\
\text{Im}[m_\pm (\nu+ i0)]\,d\nu \text{ has no support in a sufficiently
small} \\
\text{neighborhood of $\tilde\mu$ (since $\tilde\mu \in [E_0,
E_1]\backslash\{\mu\}$)},
\tag 3.33\\
\pi^{-1} \frac{\epsilon\, d\nu}{(\nu-\tilde\mu)^2 +\epsilon^2}
\underset \epsilon\downarrow 0 \to\longrightarrow  d\theta(\nu-\tilde\mu)
\quad\text{weakly}, \tag 3.34 \\
\pi^{-1} \text{Im}[m_\pm (\nu+i\epsilon)]\, d\nu
\underset \epsilon\downarrow 0 \to\longrightarrow  d\rho_\pm (\nu)
\quad\text
{weakly [49]},  \tag 3.35
\endgather
$$
$$\gathered
\lim\limits_{\epsilon\downarrow 0} (\mp i\epsilon\pi^{-1} m_\pm (\lambda
+i\epsilon))
=\rho_\pm (\lambda) - \rho_\pm (\lambda -0), \\
\lim\limits_{\epsilon\downarrow 0} (\mp i\epsilon\pi^{-1} m_\pm (\lambda
+i\epsilon))
\leq \rho_\pm (E_1)- \rho_\pm (E_0 -0), \quad \lambda\in [E_0, E_1], \\
|\epsilon\pi^{-1} m_\pm (\lambda + i\epsilon)| \leq C, \quad\epsilon\in [0,
\epsilon_0], \
\lambda \in [E_0, E_1]\text{ for some } \epsilon_0 >0, \ C>0,
\endgathered \tag 3.36
$$
(3.32) implies (3.31) for $\mu\neq\tilde\mu$ by splitting the integral in
(3.32) into a
sufficiently small interval around $\tilde\mu$ (if $\tilde\mu\in [\lambda',
\lambda]$)
and the remaining intervals (applying the dominated convergence theorem).
The case
$(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ is treated analogously. \qed
\enddemo

\remark{Remark 3.6} If $\tilde\mu\neq\mu$, the factor
$(\lambda-\mu)/(\lambda-\tilde\mu)
\neq 1$ in (3.31) shows that the half-line Dirichlet deformation method
$H^D_\pm \to
\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ affects all remaining norming
 constants
corresponding to eigenvalues of $H^D_\pm$. More precisely, denote by
$$
c^2_{\pm, n} = \biggl( \mp \int\limits^{x_0}_{\pm\infty} dx\,
\phi(\lambda_{\pm, n},
x)^2\biggr)^{-1} =\rho_\pm (\lambda_{\pm, n}) -\rho_\pm (\lambda_{\pm, n}-0)
$$
the norming constant associated with $\lambda_{\pm, n}\in\sigma_p (H^D_\pm)$,
$\lambda_{\pm, n}\neq\mu$ and denote by $\tilde c^2_{(\tilde\mu,
\tilde\sigma), \pm, n}$
the one associated with $\lambda_{\pm, n}\in\sigma_p (\tilde H^D_{(\tilde\mu,
\tilde\sigma), \pm})$. Then
$$
\tilde c^2_{(\tilde\mu, \tilde\sigma), \pm, n} =\frac{\lambda_{\pm, n}-\mu}
{\lambda_{\pm, n}-\tilde\mu}\, c^2_{\pm, n}. \tag 3.37
$$
Only in the case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$, the remaining
norming constants
stay invariant,
$$
\tilde c^2_{(\mu, -\sigma), \pm, n}=c^2_{\pm, n}. \tag 3.38
$$
In fact, the deformation $(\mu, \sigma)\to (\mu, -\sigma)$ coincides with
the isospectral
case of the double commutation method considered in Appendix B (cf.~Remark
B.3(i)).
The corresponding invariance in (3.38) was originally proven in [\gjfa].
\endremark

Theorem 3.5 implies the following half-line deformation result.

\proclaim{Theorem 3.7} Assume {\rom{(H.2.3)}} and denote by $H^D_\pm$ and
$\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ the half-line Dirichlet
operators {\rom{(2.4)}}
and {\rom{(2.37)}}.
\roster
\item"\rom{(i)}" Suppose $\mu, \tilde\mu \in (E_0, E_1)$. Then
$$\aligned
\sigma_{(p)} (\tilde H^D_{(\tilde\mu, \tilde\sigma), \tilde\sigma}) &=
\cases \{\sigma_{(p)} (H^D_\sigma )\backslash \{\mu\}\}\cup \{\tilde\mu\},
&\sigma
=\tilde\sigma \\
\sigma_{(p)} (H^D_{-\sigma})\cup \{\tilde\mu\}, & \sigma=-\tilde\sigma
\endcases , \\
\sigma_{(p)} (\tilde H^D_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}) & =
\cases \sigma_{(p)}(H^D_\sigma)\backslash \{\mu\},
& \sigma=-\tilde\sigma \\
\sigma_{(p)}(H^D_{-\sigma}), &\sigma=\tilde\sigma
\endcases ,
\quad \tilde\mu\neq\sigma (H^D_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}).
\endaligned \tag 3.39
$$
\item"\rom{(ii)}" Assume $\mu\in\{E_0, E_1\}\cap\sigma_d(H)$,
$\tilde\mu\in (E_0, E_1)$. Then
$$\aligned
\sigma_{(p)}(\tilde H^D_{(\tilde\mu,\tilde\sigma), \tilde\sigma}) &=
\{\sigma_{(p)}
(H^D_{\tilde\sigma})\backslash \{\mu\}\}\cup\{\tilde\mu\}, \\
\sigma_{(p)} (H^D_{(\tilde\mu,\tilde\sigma), -\tilde\sigma}) &=\sigma_{(p)}
(H^D_{-\tilde\sigma})\backslash\{\mu\}, \quad \tilde\mu\notin\sigma
(H^D_{(\tilde\mu,\tilde\sigma), -\tilde\sigma}).
\endaligned \tag 3.40
$$
\item"\rom{(iii)}" Suppose $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E _0, E_1\}
\cap\sigma_d (H)$. Then
$$\aligned
\sigma_{(p)}(\tilde H^D_{\tilde\mu,\sigma}) &=
\sigma_{(p)}(H^D_\sigma)\backslash
\{\mu\}, \\
\sigma_{(p)}(H^D_{\tilde\mu,-\sigma}) &=\sigma_{(p)}(H^D_{-\sigma}), \quad
\tilde\mu\notin\sigma (H^D_{\tilde\mu,\pm}).
\endaligned \tag 3.41
$$
\item"\rom{(iv)}" Assume $\mu,\tilde\mu\in \{E_0, E_1\}\cap\sigma_d (H)$,
$\mu\neq\tilde\mu$. Then
$$
\sigma_{(p)}(\tilde H^D_{\tilde\mu, \pm}) =\sigma_{(p)} (H^D_\pm)\backslash
\{\mu\}, \quad \tilde\mu\notin\sigma  (\tilde H^D_{\tilde\mu, \pm}).
\tag 3.42
$$
\rom(Here $\sigma_{(p)}(\,\cdot\,)$ denotes $\sigma (\,\cdot\,)$ or
$\sigma_p (\,\cdot\,)$
\rom(the point spectrum, i.e., the set of eigenvalues\rom) and we recall
our occasional use
of the notation of $\tilde H^D_{\tilde\mu,\pm}$ instead of $\tilde
H^D_{(\tilde\mu,
\tilde\sigma), \pm}$ if $\tilde\mu\in\sigma_d (H)$, cf.~the paragraph
preceding Lemma
2.2.\rom)
\item"\rom{(v)}"
$$
\sigma_{\text{\rom{ess, ac, sc}}}
(\tilde H^D_{(\tilde\mu,\tilde\sigma), \pm})
=\sigma_{\text{\rom{ess, ac, sc}}} (H^D_\pm). \tag 3.43
$$
Moreover, $\tilde H^D_{(\tilde\mu,\tilde\sigma),\pm}$ and $H^D_\pm$,
restricted to
the orthogonal complements of the \rom(at most one-dimensional, possibly
equaling
$\{0\}$\rom) eigenspaces corresponding to $\tilde\mu$ and $\mu$, are
unitarily equivalent.
\endroster
\endproclaim

\demo{Proof} This is a direct consequence of Corollary 3.3, Theorem 3.5,
and the
fact that
half-line spectra corresponding to separated boundary conditions are
simple. In
 particular,
we note that by Corollary 3.4(i) and (iii), $\tilde
m_{(\tilde\mu,\tilde\sigma), \pm} (z)$ are
holomorphic in a sufficiently small neighborhood of $\mu$ and/or
$\tilde\mu$ whenever
they belong to $\sigma_d (H)$. \qed
\enddemo

As long as $\mu,\tilde\mu\in (E_0, E_1)$ and hence $\mu,\tilde\mu\notin
\sigma_d(H)$, (3.39) just says that the Dirichlet datum $(\mu,\sigma)$
associated with
$H^D=H^D_- \oplus H^D_+$ got changed into the Dirichlet datum
$(\tilde\mu,\tilde\sigma)$ associated with $\tilde
H^D_{(\tilde\mu,\tilde\sigma)}=\tilde H^D_{(\tilde\mu,
\tilde\sigma), -} \oplus \tilde H^D_{(\tilde\mu,\tilde\sigma), +}$. The
cases (ii)--(iv) examine
all remaining possibilities where $\mu$ and/or $\tilde\mu$ belong to
$\sigma_d (H)$ and
possibly $\sigma_d (H^D_\pm)$ in which case, however, they no longer belong
to $\sigma_d
(\tilde H^D_{(\tilde\mu,\tilde\sigma)})$.

We have yet to show that our choices I--III of boundary conditions of
$\tilde H_{(\tilde\mu,
\tilde\sigma)}$ in (2.32)--(2.35) are indeed the only ones that lead to our
fundamental
formula (3.17) as claimed after (2.35). We only need to focus on l.c.~cases
and hence
assume that $\tilde\tau$ is l.c.~at $\pm\infty$. By Lemma 3.1, the
$m$-functions
$\tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z)$ of $\tilde
H_{(\tilde\mu,\tilde\sigma)}$ can be
computed as follows,
$$
\tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z)
=-\lim\limits_{x\to\pm\infty} \frac
{W(\tilde f(\lambda), \tilde\theta_{(\tilde\mu,\tilde\sigma)}(z))(x)}
{W(\tilde f(\lambda), \tilde\phi_{(\tilde\mu,\tilde\sigma)}(z))(x)},
\tag 3.44
$$
where $(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-\lambda)\tilde f(\lambda)=0$
for some
$\lambda\in\Bbb R$. Consider a corresponding $f(\lambda, x)$ satisfying
$(\tau - \lambda)
f(\lambda)=0$ and $\tilde f(\lambda)=U_{(\tilde\mu,\tilde\sigma)}(\lambda)
f(\lambda)$.
Then (3.44) becomes
$$
\tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z) =\frac{z-\mu}{z-\tilde\mu}
\biggl[
-\lim\limits_{x\to\pm\infty} \frac{W(U_{(\tilde\mu,\tilde\sigma)}(\lambda)
f(\lambda),
U_{(\tilde\mu,\tilde\sigma)}(z)
\theta(z))(x)}{W(U_{(\tilde\mu,\tilde\sigma)}(\lambda)
f(\lambda), U_{(\tilde\mu,\tilde\sigma)}(z)\phi(z))(x)}\biggr] -
\frac{\tilde\mu -\mu}
{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu). \tag 3.45
$$
Applying (3.7) to (3.45) then yields
$$\multline
\frac{z-\tilde\mu}{z-\mu} \biggl\{ \tilde m_{(\tilde\mu,\tilde\sigma),
\pm}(z) +
\frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu)\biggr\} =
-\lim\limits_{x\to\pm\infty} \biggl\{(\lambda-\tilde\mu) W(f(\lambda),
\phi(z))(x) \\
- \frac{(\lambda-z)(\tilde\mu -\mu)}{z-\mu} \frac{W(\psi_\sigma (\mu),
\phi(z))(x)
W(f(\lambda), \psi_{-\sigma}(\tilde\mu))(x)}{W(\psi_\sigma(\mu),
\psi_{-\tilde\sigma}
(\tilde\mu))(x)}\biggr\}
\biggl\{(\lambda-\tilde\mu) W(f(\lambda), \theta (z))(x) \\
-\frac{(\lambda-z) (\tilde\mu-\mu)}{z-\mu}
 \frac{W(\psi_\sigma (\mu),\theta(z))(x) W(f(\lambda),
\psi_{-\sigma}(\tilde\mu))(x)}{W(\psi_\sigma (\mu), \psi_{-\tilde\sigma}
(\tilde\mu))(x)}\biggr\}.
\endmultline \tag 3.46
$$

In order to reproduce (3.17), the right-hand side of (3.46) would have
to equal
$$
-\lim\limits_{x\to\pm\infty} \frac{W(\eta(\lambda),
\theta(z))(x)}{W(\eta(\lambda),
\phi(z))(x)} =m_\pm (z) \tag 3.47
$$
for some real-valued solution $\eta(z,x)$ of
$(\tau-\lambda)\psi(\lambda)=0$ which
satisfies the boundary conditions of $H$ at $\pm\infty$. Clearly,
$f(\lambda, x) =
\psi_\sigma (\mu,x)$ and $f(\lambda, x)=\psi_{-\tilde\sigma}
(\tilde\mu, x)$ are
distinguished in (3.46) and these were precisely the cases we singled out
in (2.32)--(2.35).
No other choice of $f(\lambda, x)$ in (3.46) is compatible with (3.47).

\vskip 0.3in

\flushpar{\bf \S 4. Spectral and Weyl-Titchmarsh Matrices, Isospectral
Deformations}
\vskip 0.1in

In this section we prove our principal results including explicit
computations of the
Weyl-Titchmarsh and spectral matrices of $\tilde H_{(\tilde\mu,
\tilde\sigma)}$ in terms
of those of $H$. Moreover, we provide a complete spectral
characterization of
$\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $\tilde H^D_{(\tilde\mu,
 \tilde\sigma)}$
in terms of $H$ and $H^D$.

We start with the Weyl-Titchmarsh matrices for $H$ and
$\tilde H_{(\tilde\mu,
\tilde\sigma)}$. To fix notation, we introduce the Weyl-Titchmarsh
$M$-matrix in
$\Bbb C^2$ associated with $H$ by
$$\align
M(z) &= (M_{p, q}(z))_{1\leq p, q\leq 2} \\
&= [m_- (z) -m_+ (z)]^{-1} \pmatrix m_- (z) m_+(z)
& [m_- (z)+m_+ (z)]/2 \\
[m_- (z)+m_+ (z)]/2 & 1 \endpmatrix , \quad z\in\Bbb C\backslash\Bbb R
\tag 4.1
\endalign
$$
and similarly, in connection with $\tilde H_{(\tilde\mu,
\tilde\sigma)}$, by
$$\align
\tilde M_{(\tilde\mu, \tilde\sigma)}(z) &= (\tilde M_{(\tilde\mu,
\tilde\sigma), p,q}
(z))_{1\leq p, q\leq 2} \\
&= [\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) -\tilde m_{(\tilde\mu,
\tilde\sigma), +}(z)]^{-1} \\
& \ \times
\pmatrix \tilde m_{(\tilde\mu, \tilde\sigma), -}(z) \tilde m_{(\tilde\mu,
\tilde\sigma), +}(z) &
[\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) + \tilde m_{(\tilde\mu,
\tilde\sigma), +}(z)]/2 \\
[\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) + \tilde m_{(\tilde\mu,
\tilde\sigma), +}(z)]/2 & 1
\endpmatrix, \  z\in\Bbb C\backslash\Bbb R. \tag 4.2
\endalign
$$
An application of Theorem 3.2 then yields

\proclaim{Theorem 4.1} Assume {\rom{(H.2.3)}} and
$z\in\Bbb C\backslash\Bbb R$.
Given $H$ and $\tilde H_{(\tilde\mu, \tilde\sigma)}$ by {\rom{(2.3)}} and
{\rom{(2.32)}},
respectively, their Weyl-Titchmarsh $M$-matrices are related by
$$\align
\tilde M_{(\tilde\mu, \tilde\sigma), 1,1}(z) &= \frac{z-\mu}{z-\tilde\mu}\,
M_{1,1}(z)
- 2\frac{\tilde\mu-\mu}{z-\tilde\mu}\,
m_{-\tilde\sigma}(\tilde\mu)M_{1,2}(z) \\
&\qquad + \frac {(\tilde\mu-\mu)^2}{(z-\mu)(z-\tilde\mu)}\, m_{-\tilde\sigma}
(\tilde\mu)^2 M_{2,2}(z), \tag 4.3 \\
\tilde M_{(\tilde\mu, \tilde\sigma), 1,2}(z) &= M_{1,2}(z)
-\frac{\tilde\mu-\mu}{z-\mu}\,
m_{-\tilde\sigma}(\tilde\mu) M_{2,2}(z), \tag 4.4 \\
\tilde M_{(\tilde\mu, \tilde\sigma),2,2}(z) &= \frac{z-\tilde\mu}{z-\mu}\,
M_{2,2}
(z), \quad \tilde\mu\neq\mu. \tag 4.5
\endalign
$$
Equivalently,
$$\split
\tilde M_{(\tilde\mu, \tilde\sigma)}(z) &=(z-\mu)^{-1} (z-\tilde\mu)^{-1}
\pmatrix z-\mu & -(\tilde\mu-\mu)m_{-\tilde\sigma}(\tilde\mu) \\
0 & z-\tilde\mu \endpmatrix  \\
&\qquad \times M(z) \pmatrix z-\mu & 0 \\
-(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) & z-\tilde\mu \endpmatrix ,
\quad z\in\Bbb C\backslash\Bbb R.
\endsplit \tag 4.6
$$
\endproclaim

The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ follows by a
straightforward limiting
argument (see (3.12), (3.17), and (3.31)).

\demo{Proof} This is just a combination of (3.17), (4.1), and (4.2). \qed
\enddemo

We note that (4.3) can be written as
$$
\tilde M_{(\tilde\mu, \tilde\sigma), 1,1}(z)=\frac{z-\mu}{z-\tilde\mu}\,
M_{2,2}(z)
\biggl[m_+ (z) -\frac{\tilde\mu-\mu}{z-\mu}\,
m_{-\tilde\sigma}(\tilde\mu)\biggr]
\biggl[m_- (z) -\frac{\tilde\mu -\mu}{z-\mu}\, m_{-\tilde\sigma}(\tilde\mu)
\biggr]. \tag 4.7
$$
A close examination of $\tilde M_{(\tilde\mu, \tilde\sigma)}(z)$ then
reveals the following
behavior near $\mu$ and $\tilde\mu$.

\proclaim{Corollary 4.2} $\tilde M_{(\tilde\mu, \tilde\sigma)}(z)$ is
holomorphic in a
neighborhood of $\mu$ and $\tilde\mu$ \rom(for all values of $\mu$ and
$\tilde\mu$
admitted by {\rom{(H.2.3)}}\rom).
\endproclaim

\demo{Proof} It suffices to examine the pole structure (or better, the lack
thereof) of
$\tilde M_{(\tilde\mu, \tilde\sigma),p,p}(z)$, $p=1,2$ since $\det[\tilde
M(z)]=-\frac{1}
{4}$ then controls that one of $\tilde M_{(\tilde\mu,
\tilde\sigma),1,2}(z)$ as well. The
proof then proceeds along a case-by-case study depending on whether $\mu$,
respectively
$\tilde\mu$, lie in $(E_0, E_1)$ or in $\{E_0, E_1\}\cap
\sigma_d (H)$. More specifically, one uses (3.26)--(3.28),
$$\aligned
&m_- (\tilde\mu) =m_+ (\tilde\mu), \ M_{2,2}(z) {\underset {z\to\tilde\mu}
\to =}
c(z-\tilde\mu)^{-1}+O(1), \quad c\in\Bbb R\backslash \{0\} \\
& \text{ if and only if }\tilde\mu\in \sigma_d(H)\backslash \sigma_d (H^D)
\endaligned \tag 4.8
$$
and
$$
M_{2,2}(z) \underset {z\to\mu} \to = c(z-\mu) +O((z-\mu)^2),
\quad c\in\Bbb R
\backslash \{0\}\text{ for } \mu\in [E_0, E_1]\cap\sigma_d (H^D). \tag 4.9
$$
The holomorphy assertion then follows directly from (4.5) and (4.7). \qed
\enddemo

Given the basic connection between $\tilde M_{(\tilde\mu,
\tilde\sigma)}(z)$ and $M(z)$
in Theorem 4.1, we can now proceed to derive the analogous relations
between the spectral
matrices $\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ and
$\rho(\lambda)$ associated with
$\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$, respectively. The
right-continuous
non-decreasing matrices $\rho(\lambda)$ and $\tilde \rho_{(\tilde\mu,
\tilde\sigma)}(\lambda)$
in $\Bbb C^2$ are defined for $\lambda\in\Bbb R$ by
$$\gather
\rho(\lambda) = (\rho_{p,q}(\lambda))_{1\leq p, q\leq 2}, \quad
\tilde\rho_{(\tilde\mu,
\tilde\sigma)}(\lambda) =(\tilde\rho_{(\tilde\mu,
\tilde\sigma),p,q}(\lambda))_{1\leq p,
q\leq 2}, \\
\rho(\lambda)- \rho(\lambda') =\lim\limits_{\delta\downarrow 0}
\lim\limits_{\epsilon\downarrow 0} \pi^{-1}
\int\limits^{\lambda+\delta}_{\lambda'+\delta}
d\nu \, \text{Im}[M(\nu+i\epsilon)],
\tag 4.10 \\
\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda) - \tilde\rho_{(\tilde\mu,
\tilde\sigma)}(\lambda')
=\lim\limits_{\delta\downarrow 0}
\lim\limits_{\epsilon\downarrow 0} \pi^{-1}
\int\limits^{\lambda+\delta}_{\lambda'+\delta} d\nu\, \text{Im}[\tilde
M_{(\tilde\nu,
\tilde\sigma)}(\nu+i\epsilon)]. \tag 4.11
\endgather
$$
The result for $\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ in terms
of that of
$\rho(\lambda)$ then reads as follows.

\proclaim{Theorem 4.3} Assume {\rom{(H.2.3)}}. Given $H$ and $\tilde
H_{(\tilde\mu, \tilde\sigma)}$ by {\rom{(2.3)}} and {\rom{(2.32)}}, let
$d\rho(\lambda)$ and
$d\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ be the corresponding
$\Bbb C^2$-valued
spectral measures generated by $\rho(\lambda)$ and $\tilde\rho_{(\tilde\mu,
\tilde\sigma)}
(\lambda)$, respectively. Then
$$\align
d\tilde\rho_{(\tilde\mu, \tilde\sigma),1,1}(\lambda) &= \frac{\lambda-\mu}
{\lambda -\tilde\mu} \, d\rho_{1,1}(\lambda) -2
\frac{\tilde\mu-\mu}{\lambda-\tilde\mu}
\, m_{-\tilde\sigma}(\tilde\mu)\, d\rho_{1,2}(\lambda)  \\
&\qquad + \frac{(\tilde\mu-\mu)^2}{(\lambda-\mu)(\lambda-\tilde\mu)} \,
m_{-\tilde\sigma}(\tilde\mu)^2 \, d\rho_{2,2}(\lambda), \tag 4.12 \\
d\tilde\rho_{(\tilde\mu, \tilde\sigma),1,2}(\lambda)
&= d\rho_{1,2}(\lambda) -
\frac{\tilde\mu -\mu}{\lambda-\mu}\, m_{-\tilde\sigma}(\tilde\mu)\,
d\rho_{2,2}(\lambda), \tag 4.13 \\
d\tilde\rho_{(\tilde\mu, \tilde\sigma),2,2}(\lambda)
&= \frac{\lambda-\tilde\mu}
{\lambda -\mu}\, d\rho_{2,2}(\lambda), \quad \tilde\mu\neq\mu. \tag 4.14
\endalign
$$
Equivalently,
$$\split
d\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda) & = (\lambda-\mu)^{-1}
(\lambda-\tilde\mu)^{-1}
\pmatrix \lambda -\mu & -(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) \\
0 & \lambda - \tilde\mu \endpmatrix  \\
&\qquad \times d\rho(\lambda) \pmatrix \lambda-\mu & 0 \\
-(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) & \lambda-\tilde\mu
\endpmatrix,
\quad \tilde\mu\neq\mu.
\endsplit \tag 4.15
$$
\endproclaim

The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ follows by a
straightforward limiting
argument (cf.~(3.12), (3.17), and (3.31)).

\demo{Proof} It suffices to consider $\tilde\rho_{(\tilde\mu,
\tilde\sigma),2,2}(\lambda)$,
the remaining cases being analogous. Equation (4.5) and
$$
\text{Im}[\tilde M_{(\tilde\mu, \tilde\sigma),2,2}(\nu+i\epsilon)]
=\frac{\epsilon^2
+ (\nu-\mu)(\nu-\tilde\mu)}{(\nu-\mu)^2 +\epsilon^2} \text{Im}[M_{2,2}(\nu
+i\epsilon)]
+ \frac{(\tilde\mu -\mu)\epsilon}{(\nu-\mu)^2 +\epsilon^2}
\text{Re}[M_{2,2}(\nu +i\epsilon)]
$$
show that one can follow the proof of Theorem 3.5 step by step involving
(3.34)--(3.36)
(replacing $m_\pm (z), \rho_\pm(\lambda)$ by $M_{2,2}(z),
\rho_{2,2}(\lambda)$, etc.).
\qed
\enddemo

This finally leads to the principal spectral deformation result of
this paper.

\proclaim{Theorem 4.4} Assume {\rom{(H.2.3)}} and let $H$, $\tilde
H_{(\tilde\mu,
\tilde\sigma)}$, $H^D$, and $\tilde H^D_{(\tilde\mu, \tilde\sigma)}$ be
defined by
{\rom{(2.3)}}, {\rom{(2.32)}}, {\rom{(2.4)}}, and {\rom{(2.36)}},
respectively.

{\rom{(i)}} Suppose $\mu,\tilde\mu\in (E_0, E_1)$. Then $\tilde
H_{(\tilde\mu, \tilde\sigma)}$
and $H$ are unitarily equivalent. Moreover, $\tilde H^D_{(\tilde\mu,
\tilde\sigma)}$ and
$H^D$, restricted to the orthogonal complements of the one-dimensional
eigenspaces
corresponding to $\tilde\mu$ and $\mu$, are unitarily equivalent.

{\rom{(ii)}} Assume $\mu\in\{E_0, E_1\}\cap\sigma_d (H)$, $\tilde\mu\in
(E_0, E_1)$. Then
$$\align
\sigma_{(p)}(\tilde H_{(\tilde\mu, \tilde\sigma)})
&=\sigma_{(p)}(H)\backslash
\{\mu\}, \tag 4.16 \\
\sigma_{(p)}( \tilde H^D_{(\tilde\mu, \tilde\sigma)})
&= \{\sigma_{(p)}(H^D)
\backslash \{\mu\}\}\cup\{\tilde\mu\}. \tag 4.17
\endalign
$$

{\rom{(iii)}} Suppose $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E_0, E_1\}\cap
\sigma_d (H)$.
Then
$$\align
\sigma_{(p)}(\tilde H_{\tilde\mu}) &= \sigma_{(p)}(H)\backslash
\{\tilde\mu\},
\tag 4.18 \\
\sigma_{(p)}(\tilde H^D_{\tilde\mu}) &=\sigma_{(p)}(H^D)\backslash\{\mu\},
\quad \tilde\mu\notin\sigma(\tilde H^D_{\tilde\mu}). \tag 4.19
\endalign
$$

{\rom{(iv)}} Assume $\mu, \tilde\mu\in \{E_0, E_1\}\cap
\sigma_d (H)$, $\mu
\neq\tilde\mu$. Then
$$\align
\sigma_{(p)}(\tilde H_{\tilde\mu}) &= \sigma_{(p)}(H)\backslash\{E_0, E_1\},
\tag 4.20 \\
\sigma_{(p)}(\tilde H^D_{\tilde\mu}) &= \sigma_{(p)}(H^D)\backslash \{\mu\},
\quad \tilde\mu\notin\sigma (\tilde H^D_{\tilde\mu}). \tag 4.21
\endalign
$$
In cases {\rom{(ii)--(iv)}}, the corresponding pairs of operators,
restricted to the obvious
orthogonal complements corresponding to $\mu$ and/or $\tilde\mu$, are
unitarily
equivalent. In particular,
$$
\sigma_{\text{\rom{ess, ac, sc}}}(\tilde H_{(\tilde\mu, \tilde\sigma)}) =
\sigma_{\text{\rom{ess, ac, sc}}}(\tilde H^D_{(\tilde\mu,
\tilde\sigma)}) =
\sigma_{\text{\rom{ess, ac, sc}}}(H^D) =
\sigma_{\text{\rom{ess, ac, sc}}}(H). \tag 4.22
$$
\endproclaim

\demo{Proof} This is a direct consequence of Corollary 3.3,
Theorems 3.5, 3.7, and
4.4, and the
orthogonal decompositions of $H^D =H^D_- \oplus H^D_+$, $\tilde
H^D_{(\tilde\mu,
\tilde\sigma)}= \tilde H^D_{(\tilde\mu, \tilde\sigma),-}\oplus \tilde
H^D_{(\tilde\mu,
\tilde\sigma),+}$. Moreover, in connection with case (iv), one observes
that $\mu\in
\sigma_d (H)\cap \sigma_d (H^D)$ necessarily implies that $\tilde\mu\in
\{\{E_0, E_1\}
\cap\sigma_d (H)\}\backslash\{\mu\}$ cannot lie in $\sigma_d (H^D)$
(i.e., two
consecutive discrete eigenvalues of $H$ cannot both belong to the spectrum
of $H^D$).
\qed
\enddemo

\remark{Remark 4.5} Perhaps the most spectacular consequence of
Theorem 4.4(i),
from an inverse spectral point of view, is the fact that {\it{any}}
finite
number of
deformations of Dirichlet data within spectral gaps of $V$ yields a
potential $\tilde V$
in the isospectral class of $V$\!. No further constraints on $(\mu_j,
\sigma_j),
(\tilde\mu_j, \tilde\sigma_j)$, other than $(\mu_j, \sigma_j),
(\tilde\mu_j, \tilde\sigma_j)
\in (E_{j-1}, E_j) \times \{-,+\}$, $(E_{j-1}, E_j)\in\Bbb R\backslash
\sigma (H)$,
$j=1, \dots, N$, $N\in\Bbb N$ are involved.
\endremark

\smallpagebreak

On an intuitive level, the Dirichlet deformation method amounts to the
following two-step
procedure. In the first commutation step, effected by $\psi_\sigma (\mu,
x)$ in (2.19),
the Dirichlet eigenvalue $\mu\in (E_0, E_1)$ associated with
$H=-\frac{d^2}{dx^2}
+V$ on the interval $(x_0, \sigma\infty)$ for some $x_0 =x_0 (\mu)\in\Bbb
R$ is moved
to $\infty$, thereby producing a singular intermediate potential
deformation of $V(x)$ in
the process. The second commutation step, effected by $\psi_{-\tilde\sigma}
(\tilde\mu, x)$
in (2.19), then moves back this ``Dirichlet eigenvalue" from $\infty$ to
$\tilde\mu \in
(E_0, E_1)$ associated with the interval $(x_0, \tilde\sigma \infty)$. In
the latter process,
the resulting deformation $\tilde V_{(\tilde\mu, \tilde\sigma)}(x)$
becomes
regular again
(i.e., $W_{(\tilde\mu, \tilde\sigma)}(x)\neq 0$, $x\in\Bbb R$) and
isospectral to the original
base potential $V(x)$.

We conclude this section with a series of remarks. A variety of additional
results and
possible extensions in connection with the Dirichlet deformation
method will be
presented in Section 5.

\remark{Remark 4.6} (i) The isospectral property (i) in Theorem 4.4,
in the
special case
of periodic potentials $V(x)$, has first been proven by Finkel, Isaacson,
and Trubowitz
[\fit]. Further results can be found in Buys and Finkel [\bufi] and
Iwasaki
[\iwa] (see also [\mckcpam]). Similar constructions in connection with
Schr\"odinger operators on a compact interval can be found in P\"oschel
and
Trubowitz [\potr] and Ralston and Trubowitz [\ratr] (see
our discussion in the introduction).

 (ii) By inspection, Dirichlet deformations produce the commuting diagram
\vskip 0.1in

$$
\matrix
& (\mu_2, \sigma_2) &\\
\arrowne& &\arrowse\\
\llap{$(\mu_1$}, \sigma_1) & \arrowl & (\mu_3, \rlap{$\sigma_3)$}
\endmatrix
$$

\vskip 0.1in
\flushpar for $(\mu_j, \sigma_j)\in [E_0, E_1] \times \{-, +\}$, $1\leq
j\leq 3$ according
to (H.2.3).

(iii) Let $\mu\in (E_0, E_1)$. Then the (isospectral) Dirichlet deformation
$(\mu,\sigma)
\to (\mu, -\sigma)$ is precisely the isospectral case of the double
commutation method
considered in Appendix B (see Remark B.3(i)). It simply flips the Dirichlet
eigenvalue
$\mu$ on the half-line $(x_0, \sigma\infty)$ to the other half-line $(x_0,
-\sigma\infty)$.
In the special case where $V(x)$ is periodic, this procedure has first been
used by McKean
and van Moerbeke [\mckmoe].

(iv) In analogy to Remark 3.6, the Dirichlet deformation method as
displayed in
(4.12)--(4.14) changes magnitudes of discontinuities of $\rho(\lambda)$ at
all eigenvalues
$\lambda_n \in\sigma_p (H)$ as long as $\tilde\mu\neq\mu$. Even in the
special case
$(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ discussed in item (iii)
above, one
obtains invariance of the magnitudes of jumps at $\lambda_n$ only for the
spectral
matrix element $\rho_{2,2}(\lambda)$.

(v) In the non-isospectral cases (ii)--(iv), a combination of the present
Dirichlet deformation
method with the double commutation method in Appendix B can restore
isospectrality by
inserting an eigenvalue at $\mu$, $\tilde\mu$, or both.
\endremark

\remark{Remark 4.7} In certain cases where the base (background) potential
$V$ is
reflectionless (cf.~(5.6)) and $H$ is bounded from below and has no
singularly continuous
spectrum, the isospectral class $\text{Iso}(V)$ of $V$ (the set of all
$\tilde V$'s such that
$\sigma(\tilde H)=\sigma(H)$) is completely characterized by the
distribution of Dirichlet
(initial) data $(\mu_{j+1}(x_0), \sigma_{j+1}(x_0))\in [E_j, E_{j+1}]
\times \{-,+\}$,
$j\in J$ in non-trivial spectral gaps of $H$. Here $x_0\in\Bbb R$ is a
fixed reference point
and $J=\{0,1,\dots, N-1\}$, $N\in\Bbb N$ or $j\in J=\Bbb N_0$ ($=\Bbb
N\cup\{0\}$)
parametrizes these non-trivial spectral gaps $(E_j, E_{j+1})$ of $H$ (the
trivial one being
$(-\infty, \inf \sigma(H))$). Prime examples of this type are periodic
potentials,
algebro-geometric quasi-periodic finite-gap potentials,  and certain
limiting cases thereof
(e.g., soliton potentials). In these cases, an iteration of the Dirichlet
deformation method,
in the sense that $(\mu_{j+1}(x_0), \sigma_{j+1} (x_0))\to
(\tilde\mu_{j+1}(x_0),
\tilde\sigma_{j+1}(x_0))$ within $[E_j, E_{j+1}]\times \{-, +\}$ for each
$j\in J$,
independently of each other (cf.~(5.4), (5.5)) yields an explicit
realization of the underlying
isospectral class $\text{Iso}(V)$ of potentials with base $V$\!.  In the
periodic case, this has
first been proved by Finkel, Isaacson, and Trubowitz [\fit] (see also
[\bufi], [\iwa]). More
precisely, the inclusion of limiting cases $\mu_{j+1}(x_0)\in \{E _j,
E_{j+1}\} \cap \sigma_{\text{\rom{ess}}}(H)$ requires a special argument
(since it is excluded by (H.2.3))
but this can be provided in the special cases at hand.
\endremark

\remark{Remark 4.8} Another case of primary interest concerns potentials
$V$ with
purely discrete spectra bounded from below, that is,
$$\gather
\sigma(H)=\sigma_d(H) = \{E_j\}_{j\in\Bbb N_0}, \quad -\infty< E_0,  \
E_j <E _{j+1}, \  j\in\Bbb N_0,  \\
\sigma_{\text{\rom{ess}}}(H)=\emptyset.
\endgather
$$
(For simplicity, one may think in terms of the harmonic oscillator
$V(x)=x^2$, [\levsbor],
[\mcktr].) In this case, either
$$
(\mu_{j+1}(x_0), \sigma_{j+1}(x_0)) \in (E_j, E_{j+1}) \times \{-, +\}
$$
or
$$
\mu_{j+1}(x_0) =E _{j+1} =\mu_{j+2} (x_0),
$$
that is, Dirichlet eigenvalues necessarily meet in pairs whenever they hit
an eigenvalue
of $H$. The following trace formula for $V(x)$ in terms of $\sigma(H)=
\{E _j\}_{j\in\Bbb N_0}$ and $\sigma (H^D_x)=\{\mu_j (x)\}_{j\in\Bbb N}$
($H^D_y$ the Dirichlet operator associated with $\tau=-\frac{d^2}{dx^2}
+V(x)$ and an
additional Dirichlet boundary condition at $x=y$), proven in [\gsacta],
$$
V(x)=E _0 +\lim\limits_{\alpha\downarrow 0} \alpha^{-1} \sum^{\infty}_{j=1}
(2e^{-\alpha\mu_j (x)} - e^{-\alpha E _j} - e^{-\alpha E _{j+1}}),
\tag 4.23
$$
then shows one crucial difference to the periodic-type cases mentioned
previously. Unlike
in the periodic case, though, the initial Dirichlet eigenvalues
$\mu_{j+1}(x_0)$ {\it{cannot}}
be prescribed arbitrarily in the spectral gaps $(E _j, E_{j+1})$ of $H$.
Indeed, the fact
that the Abelian regularization in the trace formula (4.23) for $V(x)$
converges to a limit
restricts the asymptotic distribution of $\mu_{j+1}(x)\in [E_j, E_{j+1}]$
as $j\to\infty$.
However, as stressed in Remark 4.5, one of the fundamental consequences of
this paper
concerns the fact that there is no such restriction for any finite number
of spectral gaps of
$H$ (see (5.4), (5.5)). In other words, only the tail end of the Dirichlet
eigenvalues
$\mu_{j+1}(x_0)$ as $j\to\infty$ is restricted (the precise nature of this
restriction being
unknown at this point), any finite number of them can be placed arbitrarily
in the spectral
gaps $(E _j, E_{j+1})$ (with the obvious ``crossing" restrictions at the
common boundary
$E _{j+1}$ of $(E _j, E_{j+1})$ and $(E _{j+1}, E_{j+2})$). This fact
served as one of
our prime motivations for this paper.The only other known restriction
to date on Dirichlet initial data $(\mu_j
(x_0), \sigma_j
(x_0))$ is that $\sigma_j (x_0) =-$ and $\sigma_j (x_0)=+$ infinitely
often, that is,
both half-lines $(-\infty, x_0)$ and $(x_0, \infty)$ support (naturally)
infinitely many
Dirichlet eigenvalues.
\endremark

The general characterization of the full isospectral class of operators
with purely discrete
spectra remains a (very interesting) open problem. It is quite surprising
that more than sixty
years after the founding of quantum mechanics, the isospectral class
of the
one-dimensional
harmonic oscillator remains shrouded in mystery.

Finally, it seems appropriate to comment on the map from $V$ to Dirichlet
data alluded to in the introduction and describe the role played by the
additional parameter needed in the Dirichlet data in the special case
where eigenvalues of $H^D$ and $H$ coincide.

\remark{Remark 4.9} Suppose $H$ (and hence $H^D$) has empty essential
spectrum and is
bounded from below. In order to show that the map from $V$ to Dirichlet
data (suitable interpreted to allow for eigenvalue coincidences of $H^D$
and $H$) is one-one when defined on the isospectral set of $V$, one can
use results in [\gstrans] and [\gsacta] as follows. Since the spectra of
both $H$ and $H^D$ are purely discrete they determine the diagonal Green's
function $G(z,x_0,x_0)= [m_-(z) - m_+(z)]^{-1}$ by formula (6.7)
of [\gsacta].
Moreover, since the Weyl $m$-functions are meromorphic, we only need to
know whether the
pole of $G(z,x_0,x_0)^{-1}$ at each $z=\mu$ belongs to $m_+(z)$ or
$m_-(z)$
in order to recover $m_\pm(z)$, that is, we need $\sigma$ as in (2.7). If
$\mu$ is an eigenvalue of $H^D$ and $H$, and hence a pole of $m_-(z)$
and $m_+(z)$,  $\sigma$ is not merely a sign but needs to
contain the information about how the residue of $G(z,x_0,x_0)^{-1}$ at
$z=\mu$ is
distributed between $m_+(z)$ and $m_-(z)$ as discussed in Theorem~3.6 of
[\gstrans]. A convenient choice for this additional parameter (see, e.g.,
[\gkt]) would be
$\sigma = (\gamma_+ - \gamma_-)/(\gamma_+ + \gamma_-) \in (-1,1)$,
where $\gamma_\pm$
denote the respective residues of $m_\pm(z)$ at $z=\mu$. In this extended
sense (when compared to (2.7)) the spectrum of $H$ and the Dirichlet data
${(\mu,\sigma)}$ uniquely determine $V(x)$ for a.e. $x\in\Bbb R$. These
considerations are not confined to operators with purely discrete
spectra but also apply to situations where $H$ is reflectionless and
has no
singularly continuous spectrum. This has been discussed in the
context of Jacobi operators in [\gkt] but analogous arguments work in the
Schr\"odinger operator case.
\endremark

\remark{Remark 4.10} The additional parameter $\sigma_0$ introduced
in Remark
4.9 in the case where $E_0$ is an eigenvalue of $H$ and $H^D$ (and both
have
purely discrete spectra) can be tuned to produce all corresponding
isospectral
potentials in $Iso(V)$. In fact, the double commutation procedure (see
Appendix B) allows to add/subtract $\tilde\gamma_1$ to the residues
of the
Weyl $m$-functions (see (B.27)) as long as the term under the
logarithm in
(B.14) does not become zero.

In addition, one can use the following three-step procedure to generate a
prescribed degeneracy at an eigenvalue $E_0$ of $H$:

(i) Use the Dirichlet deformation method to move $\mu$ to a discrete
eigenvalue $E_0$ of $H$. (This removes both the discrete eigenvalue $E_0$
of $H$ and the (Dirichlet) eigenvalue $\mu$ of $H^D$).

(ii) As a consequence of step (i), there is now another eigenvalue
$\tilde\mu$ of $H^D$ in the resulting larger spectral gap of $H$. Move
$\tilde\mu$ to $E_0$ using the Dirichlet deformation method.

(iii) Use the double commutation method to insert an eigenvalue of $H$ at
$E_0$.

Finally, use the method at the beginning of this remark to change
$\sigma_0$
into any allowed value.

Theorems 3.7 and B.2 then show that the resulting operator is unitarily
equivalent to the original operator $H$, and (3.17) and (B.27) then prove
that the remaining Dirichlet eigenvalues remain invariant.
\endremark


\vskip 0.3in

\flushpar {\bf \S 5. Various Additional Results and Possible Extensions}
\vskip 0.1in

In our final section we discuss a variety of additional results and sketch
possible extensions,
including limit point/limit circle considerations, iterations of the
Dirichlet deformation
procedure, scattering theory, and general Sturm-Liouville operators on
arbitrary intervals.

We start with two limit point results. The first, although trivial from a
technical point of
view, nevertheless will apply in a great variety of situations.

\proclaim{Lemma 5.1} Assume {\rom{(H.2.3)}} and define $H$ and
$H^D_\pm$ as in
{\rom{(2.3)}} and {\rom{(2.4)}}. Let $\sigma\in\{-,+\}$ and suppose that
one of the
following conditions {\rom{(i)--(iii)}} holds.
\roster
\item"\rom{(i)}" $\sigma_{\text{\rom{ess}}}(H^D_\sigma )\neq\emptyset$.
\item"\rom{(ii)}" $\sigma_{\text{\rom{ess}}}(H^D_\sigma ) =\emptyset$ and
$H^D_\sigma$ is bounded from below.
\item"\rom{(iii)}" $\sigma(H^D_\sigma)= \sigma_d (H^D_\sigma) =
\{E_{\sigma, n}\}_{n\in\Bbb Z}$ with $\sum_{n\in\Bbb Z}
(1+E^2_{\sigma, n})^{-1}
=\infty$.
\endroster
Then, both $\tau$ and $\tilde\tau_{(\tilde\mu, \tilde\sigma)}$ are
in the
limit point case
at $\sigma\infty$.
\endproclaim

\demo{Proof} Clearly, $\tau$ is l.p.~ at $\sigma\infty$ if condition (i)
holds since
differential expressions being regular at $x_0$ and l.c.~at
$\sigma\infty$
can only generate
self-adjoint operators in $L^2 ((x_0, \sigma\infty))$ with purely discrete
spectra. (Indeed,
all solutions of $\tau\psi =z\psi$, $z\in\Bbb C$ being in $L^2 ((x_0,
\sigma\infty))$ yield
a compact, in fact, Hilbert-Schmidt resolvent). These spectra necessarily
accumulate at
$+\infty$ and $-\infty$ (see, e.g., Lemma C.1 in [\gjfa] for a short
argument based on
principal solutions in Hartman's terminology.) Finally, the Hilbert-Schmidt
argument for
the resolvent would lead to $\sum_{n\in\Bbb Z}(1+ E^2_n)^{-1} <\infty$
for the
corresponding eigenvalues $\{E_n\}_{n\in\Bbb Z}$ in the l.c.~case at
$\sigma\infty$.
Theorem 3.7 then shows that $\tilde H^D_{(\tilde\mu, \tilde\sigma),
\sigma}$ shares the
corresponding property (i), (ii), or (iii), rendering
$\tilde\tau_{(\tilde\mu, \tilde\sigma)}$
l.p.~at $\sigma\infty$ as well. \qed
\enddemo

Our second limit point result is a bit more refined and tailored
toward the
Dirichlet
deformation method (denoted as DDM for brevity in the remainder of this
section).

\proclaim{Lemma 5.2} In addition to {\rom{(H.2.3)}}, assume that $\mu,
\tilde\mu\in
(E_0, E_1)$, $\mu\neq\tilde\mu$ and $\tilde\sigma=\sigma$. Then
$\tilde\tau_{(\tilde\mu,
\sigma)}$ is in the limit point \rom(resp., limit circle\rom) case at
$\omega\infty$ if and
only if $\tau$ is limit point \rom(resp., limit circle\rom) at
$\omega\infty$, $\omega\in
\{-, +\}$.
\endproclaim

\demo{Proof} Assume that $\tau$ is l.p.~at $\omega\infty$ and suppose the
contrary
for $\tilde\tau_{(\tilde\mu, \sigma)}$, that is, suppose
$\tilde\tau_{(\tilde\mu, \sigma)}$
is l.c.~at $\omega\infty$ and hence
$$
\tilde\psi_\sigma (\tilde\mu, \,\cdot\,), \tilde\psi_{-\sigma}(\mu,
\,\cdot\,)\in
L^2 ((x_0, \omega\infty)). \tag 5.1
$$
Since by hypothesis, $\tau$ is l.p.~at $\omega\infty$, both functions in
(5.1) generate
the same $m$-function $\tilde m_{(\tilde\mu, \tilde\sigma), \omega}(z)$
associated with
$\tilde H_{(\tilde\mu, \tilde\sigma)}$ on $(x_0, \omega\infty)$. This
follows directly
from (3.18) and (3.19). In particular, both
$\tilde\psi_\sigma (\tilde\mu,
x)$ and
$\psi_{-\sigma}(\mu, x)$ fulfill the boundary conditions of $\tilde
H_{(\tilde\mu,
\tilde\sigma)}$ and the analog of (3.25) at $\omega\infty$. As a
consequence of (5.1),
we obtain existence, in fact, vanishing of the limit
$$
\lim\limits_{x\to\omega\infty} W(\tilde\psi_\sigma (\tilde\mu),
\tilde\psi_{-\sigma}
(\mu))(x)=0.
$$
Since by Corollary 3.4 $\tilde\psi_\sigma (\tilde\mu, \,\cdot\,)\in L^2
((x_0, \sigma\infty))$
and $\tilde\psi_{-\sigma}(\mu, \,\cdot\,)\in L^2 ((x_0,
-\sigma\infty))$ satisfy
$$
\lim\limits_{x\to\sigma\infty} W(\tilde\psi_\sigma (\tilde\mu), \tilde g)
(x)=0,
\quad \lim\limits_{x\to -\sigma\infty} W(\tilde\psi_{-\sigma} (\tilde\mu),
\tilde g) (x)=0
$$
for all $\tilde g\in \Cal D (\tilde H_{(\tilde\mu, \sigma)})$, we arrive at
the following
case distinction. Either

(i)  $\omega =\sigma$. Then $\tilde\psi_{-\sigma}(\mu)\in\Cal D
(\tilde H_{(\tilde\mu, \sigma)})$ and hence $\mu\in\sigma_p (\tilde
H_{(\tilde\mu,
\sigma)})$,

\flushpar or

(ii) $\omega=-\sigma$. Then $\tilde\psi_\sigma (\tilde\mu)\in \Cal D
(\tilde H_{(\tilde\mu, \sigma)})$ and hence $\tilde\mu\in\sigma_p (\tilde
H_{(\tilde\mu,
\sigma)})$.

\flushpar Either way, since $\sigma (\tilde H_{(\tilde\mu,
\sigma)})=\sigma(H)$ by
Theorem 4.4(i), we get a contradiction since by hypothesis $\mu,
\tilde\mu\in (E_0, E_1)
\subset \Bbb R\backslash \sigma (H)$. If $\tau$ is l.c.~at $\omega\infty$,
suppose
$\tilde\tau_{(\tilde\mu, \tilde\sigma)}$ is l.p.~ at $\omega\infty$. By
studying the reverse
deformation $(\tilde\mu, \sigma)\to (\mu,\sigma)$, $\tau$ would have to be
l.p.~at
$\omega\infty$ by our previous argument. This contradiction shows
$\tilde\tau_{(\tilde\mu, \sigma)}$ is l.c.~at $\omega\infty$. By symmetry
in $\tau$ and
$\tilde\tau_{(\tilde\mu, \sigma)}$, the proof is complete. \qed
\enddemo

After these encouraging results, we shall take a chance (and possibly
disappoint the reader)
by describing a construction showing that DDM in general neither respects
the l.c.~ nor
the l.p.~case if $\tilde\sigma=-\sigma$. More precisely, we will construct
an example
where we ``hop" from l.c.~to l.p.~ and then back to a l.c.~case. This
illustrates our
warning raised in the paragraph following (2.3).

\proclaim{Lemma 5.3} If $\tilde\sigma=-\sigma$, the Dirichlet deformation
method
\rom(as presented by {\rom{(H.2.3)}}, {\rom{(2.3)}}, and
{\rom{(2.32)}}\rom), in
general, neither preserves the limit point nor limit circle case.
\endproclaim

\demo{Proof} Let $\mu\in (E_0, E_1)$ and choose $H$ in such a way that
$\tau$ is
l.c.~at $\sigma\infty$ but l.p.~at $-\sigma\infty$ by assuming
$\sigma_{\text{\rom{ess}}}
(H^D_{-\sigma})\neq\emptyset$. Now consider the (sign flip) deformation
$(\mu,\sigma)
\to (\mu, -\sigma)$. Clearly, $\tilde\tau_{(\mu, -\sigma)}$ is l.p.~at
$-\sigma\infty$
since $\sigma_{\text{\rom{ess}}}(\tilde H^D_{(\mu, \sigma), -\sigma}) =
\sigma_{\text{\rom{ess}}}(H^D_{ -\sigma})\neq\emptyset$. However, we
claim that
$\tilde\tau_{(\mu, -\sigma)}$ is l.p.~at $\sigma\infty$ as well. To prove
this assertion,
we suppose the contrary, that is, we assume $\tilde\tau_{(\mu,-\sigma)}$ to
be l.c.~at
$\sigma\infty$. Then the left-hand side of the following identity
(cf.~(2.17))
$$
\tilde\psi_{-\sigma}(\mu, x) \tilde\psi_{-\sigma}(\mu, x) = -\frac{d}{dx}
\biggl(\, \int\limits^x_{\sigma\infty} dx' \, \psi_\sigma (\mu,
x')^2\biggr)^{-1}
$$
is in $L^1 ((x_0, \sigma\infty))$. However, the right-hand side is clearly
not integrable
near $\sigma\infty$, providing the desired contradiction. Hence,
$\tilde\tau_{(\mu,
-\sigma)}$ is indeed l.p.~at $\pm\infty$. A further sign flip, that is,
$(\mu, -\sigma)\to
(\mu, \sigma)$, restores the original differential expression $\tau$ which
was l.c.~at
$\sigma\infty$ (see Remark 4.6(ii)). Summarizing,
$$
\tau\longrightarrow \tilde\tau_{(\mu, -\sigma)} \longrightarrow
\widetilde{(\tilde\tau_{(\mu, -\sigma)})}_{(\mu, \sigma)} =\tau, \tag 5.2
$$
that is, in obvious notation,
$$
\underset {\text{l.c.}} \to {(\mu, \sigma)} \longrightarrow
\underset {\text{l.p.}} \to {(\mu, -\sigma)} \longrightarrow
\underset {\text{l.c.}} \to {(\mu, \sigma)}, \tag 5.3
$$
displays the required deformations. \qed
\enddemo

By Remark 4.6(ii) again, (5.3) can be modified to an example of the type
$$
\underset {\text{l.p.}} \to {(\mu, \sigma)} \longrightarrow
\underset {\text{l.c.}} \to {(\tilde\mu, -\sigma)}, \quad \mu,
\tilde\mu \in
(E_0, E_1), \  \mu\neq\tilde\mu
$$
using the chain
$$
\underset {\text{l.p.}} \to {(\mu, \sigma)} \longrightarrow
\underset {\text{l.p.}} \to {(\tilde\mu, \sigma)} \longrightarrow
\underset {\text{l.c.}} \to {(\tilde\mu, -\sigma)}
$$
(relying on Lemma 5.2 in the first step).

Next, we briefly comment on how to iterate DDM (see [\bufi], [\fit]).
Suppose
$V\in L^1_{\text{\rom{loc}}}(\Bbb R)$ and
$$
(E_n, E_{n+1}), \quad \mu_{n+1}, \tilde\mu_{n+1}\in [E_n, E_{n+1}],
\quad \sigma_{n+1}, \tilde\sigma_{n+1} \in \{-,+\}
$$
satisfy (H.2.3) for each $n=0,1,\dots, N-1$, $N\in\Bbb N$. Then the
Dirichlet
deformation result after $N$ steps, denoted by $\tilde\tau_{(\tilde\mu_1,
\tilde\sigma_1),
\dots, (\tilde\mu_N, \tilde\sigma_N)}$, reads as follows.
$$\align
\tilde\tau_{(\tilde\mu_1, \tilde\sigma_1), \dots, (\tilde\mu_N,
\tilde\sigma_N)} &=
-\frac{d^2}{dx^2} + \tilde V_{(\tilde\mu_1, \tilde\sigma_1), \dots,
(\tilde\mu_N,
\tilde\sigma_N)}(x), \\
\tilde V_{(\tilde\mu_1, \tilde\sigma_1), \dots, (\tilde\mu_N,
\tilde\sigma_N)}(x) &=
V(x) -2 \bigl\{ \ln [W_{(\tilde\mu_1, \tilde\sigma_1), \dots,
(\tilde\mu_N,
\tilde\sigma_N)}
(x)]\bigr\}'', \quad x\in\Bbb R, \tag 5.4 \\
W_{(\tilde\mu_1, \tilde\sigma_1), \dots, (\tilde\mu_N,
\tilde\sigma_N)}(x) &=
\frac{W(\psi_{\sigma_1} (\mu_1), \psi_{-\tilde\sigma} (\tilde\mu_1),
\dots,
\psi_{\sigma_N}(\mu_N), \psi_{-\tilde\sigma_N}(\tilde\mu_N))(x)}
{(\tilde\mu_1 -\mu_1)  \dots (\tilde\mu_N - \mu_N)}, \\
&\qquad \mu_{n+1}, \tilde\mu_{n+1} \in [E_n, E_{n+1}], \ \mu_n \neq
\tilde\mu_n, \
0\leq n\leq N-1. \tag 5.5
\endalign
$$
In case $(\tilde\mu_{n_0}, \tilde\sigma_{n_0})=(\mu_{n_0},
-\sigma_{n_0})$
for some
$0\leq n_0 \leq N-1$, one amends (5.5) by replacing the pair
$(\tilde\mu_{n_0}-
\mu_{n_0})^{-1} (\psi_{\sigma_{n_0}} (\tilde\mu_{n_0}),
 \psi_{-\tilde\sigma_{n_0}}
(\tilde\mu_{n_0}))$ by $(\psi_{\sigma_{n_0}}(\mu_{n_0}),
\dot\psi_{\sigma_{n_0}}(\mu_{n_0}))$ (where ``$\,\cdot\,$" abbreviates
$d/d\lambda$).  It should perhaps be pointed out that
$W(\psi_{\sigma_1}(\mu_1),
\dots, \psi_{-\tilde\sigma_N}(\tilde\mu_N))(x)$ in (5.5) denotes a
slightly
modified
$2N\times 2N$ Wronskian of solutions of $\tau\psi(z)=z\psi(z)$. In
particular, it is
understood that $\psi''(z,x)$ must be replaced by $(V(x)-z)\psi(z,x)$
with $-z\psi(z,x)$ remaining, and similarly for higher
derivatives of $\psi$. At the end of this process only $\psi$, $\psi'$,
and
$z$ enter
(5.5) and no additional smoothness on $V$ is required.  At this point
each
of our
previous results has an obvious counterpart in connection with (5.4),
(5.5).

Next, we will show that DDM leaves reflectionless potentials invariant.
We
recall that
$H$ (resp.~$V$) is called {\it{reflectionless}} if and only if
$$
\text{for all }x\in\Bbb R, \quad \arg (G(\lambda+ i0, x,x))=\pi/2 \quad
\text{for (Lebesgue) a.e. } \lambda\in\sigma_{\text{\rom{ess}}}(H).
\tag 5.6
$$
Here $G(z,x,x')$ denotes the Green's function of $H$ (i.e., the integral
kernel of
$(H-z)^{-1}$) and $G(\lambda+i0, x,x)=\lim_{\epsilon\downarrow 0}
G(\lambda+
i\epsilon, x,x)$ in obvious notation. As discussed in [\gnow], (5.6) is
equivalent to
$$
m_+ (\lambda+i0) =\overline{m_- (\lambda+i0)}
\quad \text{for a.e. }\lambda
\in\sigma_{\text{\rom{ess}}}(H). \tag 5.7
$$
This then implies

\proclaim{Lemma 5.4} Assume {\rom{(H.2.3)}}. Then $\tilde H_{(\tilde\mu,
\tilde\sigma)}$,
is reflectionless if and only if $H$ is.
\endproclaim

\demo{Proof} By (3.17) and Theorem 4.4, one observes that (5.7) holds if
and only if
$$
\tilde m_{(\tilde\mu, \tilde\sigma), +}(\lambda +i0) =\overline{\tilde
m_{(\tilde\mu,
\tilde\sigma), -}(\lambda +i0)} \quad \text{for a.e. }
\lambda\in\sigma_{\text{\rom{ess}}}
(\tilde H_{(\tilde\mu, \tilde\sigma)})=\sigma_{\text{\rom{ess}}}(H). \qed
$$
 \enddemo

Since
$$
G(z, x_0, x_0) = [m_- (z) -m_+ (z)]^{-1}, \tag 5.8
$$
we might add the fact that by (4.2) and (4.5),
$$
\tilde G_{(\tilde\mu, \tilde\sigma)}(z, x_0, x_0)
=\frac{z-\tilde\mu}{z-\mu}\,
G(z, x_0, x_0), \tag 5.9
$$
where $\tilde G_{(\tilde\mu, \tilde\sigma)}(z, x,x')$ denotes the Green's
function of
$\tilde H_{(\tilde\mu, \tilde\sigma)}$. (5.9) again underscores the change
$\mu\to\tilde
\mu$ (but it stops short of indicating $\sigma\to\tilde\sigma$).

In the following we describe how to define DDM for general Sturm-Liouville
operators.
To keep this section within a reasonable length, we only point out the
major changes required
in this context.
$$\gather
{\align
V\in L^1_{\text{\rom{loc}}}(\Bbb R) \text{ real-valued}, \  x_0\in\Bbb R \to
p^{-1}, q, & k\in L^1_{\text{\rom{loc}}} ((a,b)), \quad kp\in
AC_{\text{\rom{loc}}}((a,b)) \\
&\quad p>0, \ k>0,  \  q \text{ real-valued},  \\
&\quad -\infty\leq a<b\leq\infty, \  x_0\in (a,b),
\endalign} \\
f'\to pf', \\
W(f,g)(x)\to \widehat W(f,g)(x) =f(x) p(x)g'(x)-p(x)f'(x) g(x), \\
L^2(\Bbb R) \to L^2 ((a,b); kdx), \\
\tau\to\widehat\tau =\frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x)
\frac{d}{dx} +
q(x)\biggr), \quad x\in (a,b), \tag 5.10\\
\tilde\tau_{(\tilde\mu, \tilde\sigma)}\to
\widetilde{\widehat\tau}_{(\tilde\mu, \tilde\sigma)} =
\frac{1}{k(x)}\biggl( -\frac{d}{dx}\, p(x)\frac{d}{dx}+\tilde
q_{(\tilde\mu, \tilde\sigma)}
(x)\biggr), \\
\tilde q_{(\tilde\mu, \tilde\sigma)}(x) =q(x)+ \biggl(\frac{1}{k(x)}\,
(k(x)p(x))' - 2
\frac{d}{dx}\, p(x)\biggr) \frac{d}{dx}\, \ln [\widehat W_{(\tilde\mu,
\tilde\sigma)}
(x)], \\
\widehat W_{(\tilde\mu, \tilde\sigma)}(x) =\cases (\tilde\mu -\mu)^{-1}
\widehat W
(\psi_\sigma (\mu), \psi_{-\tilde\sigma}(\tilde\mu))(x), & \mu,
\tilde\mu\in [E_0, E_1], \
\mu\neq\tilde\mu \\
-\sigma \int^x_{\sigma\infty} k(x')\, dx' \psi_\sigma (\mu, x')^2,
&(\tilde\mu, \tilde\sigma)
=(\mu, -\sigma), \ \mu\in (E_0, E_1)
\endcases, \\
\tau\psi_\sigma (\mu) =\mu\psi_\sigma(\mu), \quad
\tau\psi_{-\tilde\sigma}(\tilde\mu)
=\tilde\mu \psi_{-\tilde\sigma}(\tilde\mu).
\endgather
$$

Since (a generalization of ) Lemma 2.2 is actually proven in [\gst]
for the
 general
Sturm-Liouville case on $(a,b)$, every result in this paper extends to the
general setting
in (5.10). In particular, the fundamental Theorems 3.2, 3.5, 3.7, 4.1, 4.3,
and 4.4 (replacing $\phi(\mu, x')^2$ by $k(x')\phi(\mu, x')^2$ if
$(\tilde\mu,
\tilde\sigma) =
(\mu, -\sigma)$) do not change at all.

Next, we turn to short-range scattering. Assuming temporarily
$$
V\in L^1 (\Bbb R, (1+|x|)\,dx) \text{ to be real-valued}, \tag 5.11
$$
the Jost solutions $f_\pm (z,x)$ associated with $V$ are defined as
usual by
$$
f_\pm (z,x) =e^{\pm iz^{1/2}x} +\int\limits^x_{\pm\infty} dx'\, z^{-1/2}
\sin [z^{1/2}
(x-x')] V(x') f_\pm (z,x'), \  z\in\Bbb C\backslash \{0\},
\text{Im}(z^{1/2})
\geq 0. \tag 5.12
$$
Denoting
$$
f_\pm (\lambda, x) =\lim\limits_{\epsilon\downarrow 0} f_\pm (\lambda
+i\epsilon, x),
\quad \lambda >0 ,\tag 5.13
$$
one obtains
$$
f_\pm (\lambda, x) =\frac{1}{T(\lambda)}\,
\overline{f_\mp (\lambda, x)} \, +
\frac{R\Sp\ell \\ r\endSp (\lambda)}{T(\lambda)}\, f_\mp (\lambda, x),
\quad \lambda >0
\tag 5.14
$$
and
$$\aligned
T(\lambda) &= \frac{2i\lambda^{1/2}}
{W(f_- (\lambda), f_+ (\lambda))}\, , \\
R^\ell (\lambda) &= -\frac{W(\overline{f_- (\lambda)}, f_+ (\lambda))}
{W(f_- (\lambda), f_+ (\lambda))}\, , \quad R^r (\lambda) = -\frac
{W(f_- (\lambda), \overline{f_+ (\lambda)})}{W(f_- (\lambda), f_+
(\lambda))}\, ,
\quad \lambda >0
\endaligned \tag 5.15
$$
define the scattering matrix $S(\lambda)$ in $\Bbb C^2$ associated
with the
pair
$(H, H_0)$, where $H_0 =-\frac{d^2}{dx^2}$, $\Cal D(H_0)
=H^{2,2}(\Bbb R)$,
$$
S(\lambda) =\pmatrix T(\lambda) & R^r (\lambda) \\
R^\ell (\lambda) & T(\lambda) \endpmatrix , \quad \lambda >0. \tag 5.16
$$
(5.12) and (5.14) then yield (see, e.g. [\detr], Section 2)
$$\aligned
f_\pm (\lambda, x) &= \cases e^{\pm i\lambda^{1/2}x} + o(1),
& x\to\pm\infty \\
\frac {1}{T(\lambda)} e^{\pm i\lambda^{1/2}x} +
\frac{ R\Sp \scriptscriptstyle \ell \\
\scriptscriptstyle r\endSp (\lambda)}
{T(\lambda)} e^{\mp i\lambda^{1/2}x} +o(1), & x\to\mp\infty,
\endcases , \\
f'_\pm (\lambda, x) &= \cases
\pm i\lambda^{1/2} e^{\pm i\lambda^{1/2}x} +
o(1), & x\to\pm\infty \\
\frac{\pm i\lambda^{1/2}}{T(\lambda)} e^{\pm i\lambda^{1/2}x} \mp
i\lambda^{1/2}
\frac{R\Sp \scriptscriptstyle \ell \\ \scriptscriptstyle r\endSp
(\lambda)}{T(\lambda)}
e^{\mp i\lambda^{1/2}x} +o(1), & x\to\mp\infty
\endcases,
\quad \lambda > 0.
\endaligned \tag 5.17
$$

The following result proves that DDM preserves the class of $L^1 (\Bbb R;
(1+|x|)\,dx)$ potentials and computes the scattering matrix $\tilde
S_{(\tilde\mu,
\tilde\sigma)}(\lambda)$,
$$
\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)= \pmatrix
\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda) & \tilde R^r_{(\tilde\mu,
\tilde\sigma)}
(\lambda)\\
\tilde R^\ell_{(\tilde\mu, \tilde\sigma)}(\lambda) & \tilde T_{(\tilde\mu,
\tilde\sigma)}
(\lambda)\endpmatrix, \quad \lambda >0, \tag 5.18
$$
associated with the pair $(\tilde H_{(\tilde\mu, \tilde\sigma)}, H_0)$ in
terms of
$S(\lambda)$ in (5.16) associated with $(H, H_0)$.

\proclaim{Lemma 5.5} In addition to {\rom{(H.2.3)}}, assume $\mu,
\tilde\mu\in
(E_0, E_1)\subset (-\infty,0)$. Then $\tilde V_{(\tilde\mu, \tilde\sigma)}
\in L^1
(\Bbb R; (1+|x|)\,dx)$ if and only if $V\in  L^1 (\Bbb R; (1+|x|)\,dx)$ and
the scattering
matrices $\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)$ and $S(\lambda)$
in {\rom{(5.18)}}
and {\rom{(5.16)}} associated with $(\tilde H_{(\tilde\mu, \tilde\sigma)},
H_0)$ and
$(H, H_0)$, respectively, are related via
$$\aligned
\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda) &= T(\lambda), \\
\tilde R_{(\tilde\mu, \tilde\sigma)}\Sp\ell \\ r\endSp (\lambda) &=
\frac{\lambda^{1/2}
\mp i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\tilde\sigma
(-\tilde\mu)^{1/2}}
\, \frac{\lambda^{1/2} \pm i\sigma (-\mu)^{1/2}}{\lambda^{1/2}\mp i\sigma
(-\mu)^{1/2}}
\, R\Sp\ell \\ r\endSp (\lambda), \quad \lambda >0.
\endaligned \tag 5.19
$$
\endproclaim

\demo{Proof} First, we prove that $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and
only if
$\tilde V_{(\tilde\mu, \tilde\sigma)}\in L^1 (\Bbb R; (1+|x|)\, dx)$ for
 $\mu\neq \tilde\mu$.
We adopt the strategy of Deift and Trubowitz [\detr] in their proof of
Theorem 3.2 (which
treats the analog of Lemma 5.5 in the single commutation context; see
Appendix A and
especially the paragraph preceding (A.32), (A.33)). Introduce
$$
g_\sigma (\mu, x) =\psi_\sigma (\mu, x) e^{\sigma(-\mu)^{1/2}x},
\quad g_{-\tilde\sigma} (\tilde\mu, x) =\psi_{-\tilde\sigma} (\tilde\mu, x)
e^{-\tilde\sigma(-\mu)^{1/2}x}, \quad \mu\neq\tilde\mu. \tag 5.20
$$
Then Lemmas 2.1 and 2.6 of [\detr] yield
$$
g_\sigma (\mu, x) =C_{\sigma,\pm} (\mu) (1+o(1)), \quad x\to\pm\infty,
\tag 5.21
$$
with $C_{\sigma, \pm}(\mu)>0$ and
$$\gathered
g'_\sigma (\mu, x) = o(1), \quad |x|\to \infty, \\
g'_\sigma (\mu, \,\cdot\, )\in L^\infty (\Bbb R)
\cap L^1 (\Bbb R; (1+|x|)\,dx)
\endgathered \tag 5.22
$$
and similarly for $g_{-\tilde\sigma}(\tilde\mu, x)$. Next,
abbreviating $W(x)=
W(\psi_\sigma (\mu), \psi_{-\tilde\sigma}(\tilde\mu))(x)$, one computes
using (5.21),
$$\align
\tilde V_{(\tilde\mu, \tilde\sigma)} -V &=2W^{-2} [W^{\prime 2}-WW''] \\
&= \{[\tilde\sigma (-\tilde\mu)^{1/2} +\sigma (-\mu)^{1/2}] g_\sigma
g_{-\tilde\sigma}
 + g_\sigma g'_{-\tilde\sigma}-g_{-\tilde\sigma}g'_\sigma\}^{-2}
\tag 5.23 \\
&\quad \times 2(\tilde\mu-\mu) \{2\tilde\sigma (-\tilde\mu)^{1/2}
g^2_\sigma g_{-\tilde\sigma}g'_{-\tilde\sigma}
+2 \sigma(-\mu)^{1/2} g_\sigma
g^2_{-\tilde\sigma} g'_\sigma +g^2_\sigma g^{\prime 2}_{-\tilde\sigma}
-g^2_{-\tilde\sigma} g^{\prime 2}_\sigma\}, \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \mu\neq\tilde\mu.
\endalign
$$
By (5.21) and (5.22), the right-hand side of (5.23) is clearly in $L^1
(\Bbb R; (1+|x|)\,dx)$
at least as long as $(\tilde\mu, \tilde\sigma)\neq(\mu, -\sigma)$. The case
$(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ leads to a rather cumbersome
$0/0$ expression in (5.23).
Fortunately, this is quite irrelevant since we can simply apply DDM twice,
that is, use
the deformation sequence $(\mu, \sigma)\to (\tilde\mu, -\sigma)\to (\mu,
-\sigma)$ with
$\tilde\mu\neq\mu$ (instead of $(\mu, \sigma)\to (\mu, -\sigma)$ in one
step) by appealing
to Remark 4.6(ii). This then proves $\tilde V_{(\tilde\mu, \tilde\sigma)}
\in L^1 (\Bbb R;
(1+|x|)\,dx)$ if and only if $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ in all cases.

Next, define
$$
\tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x) = (U_{(\tilde\mu,
\tilde\sigma), \pm}
(\lambda) f_\pm (\lambda))(x), \quad \lambda >0, \ \tilde\mu\neq\mu
$$
(cf.~(3.1)). Then (5.17) yields
$$\aligned
\tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x) &=
\frac{\lambda^{1/2} \pm
i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\sigma (-\mu)^{1/2}}
[e^{\pm i\lambda^{1/2}x} + o(1)], \quad x\to\pm\infty, \\
\tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x)
&=\frac{\lambda^{1/2}
\pm
i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\sigma (-\mu)^{1/2}}
\biggl[\frac{1}{T(\lambda)}\, e^{\pm i\lambda^{1/2}x}
+ \frac{R\Sp\ell \\
r\endSp (\lambda)}{T(\lambda)} \\
&\qquad  \times \frac{\lambda^{1/2} \mp i\tilde\sigma
(-\tilde\mu)^{1/2}}{\lambda^{1/2}
\pm i\tilde\sigma (-\tilde\mu)^{1/2}} \, \frac{\lambda^{1/2} \pm i\sigma
(-\mu)^{1/2}}{\lambda^{1/2}\mp i\sigma (-\mu)^{1/2}} \,
e^{\mp i\lambda^{1/2}x} +o(1)\biggr], \ x\to\mp\infty \\
&= \frac{\lambda^{1/2}
\pm i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm
i\sigma (-\mu)^{1/2}}  \biggl[ \frac{1}{\tilde T_{(\tilde\mu,
\tilde\sigma)}(\lambda)}
\, e^{\pm i\lambda^{1/2}x} + \frac{\tilde R_{(\tilde\mu,
\tilde\sigma)}\Sp\ell \\ r\endSp
(\lambda)} {\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda)}\, e^{\mp
i\lambda^{1/2}x}
 + o(1)\biggr ], \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
x\to\mp\infty, \ \lambda >0,  \ \tilde\mu\neq \mu,
\endaligned \tag 5.24
$$
with $\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda)$,
$\tilde R_{(\tilde\mu, \tilde\sigma)}\Sp\ell \\ r\endSp (\lambda)$ as given
by (5.19).  Applying
this two-step procedure to $\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)$
then proves (5.19)
also in the remaining (sign flip) case $(\mu, \sigma)
\to (\mu, -\sigma)$. \qed
\enddemo

For simplicity we only considered the case  $\mu, \tilde\mu\in (E_0, E_1)$
in Lemma 5.5
There is no problem in moving $\tilde\mu$ to the boundary of the interval
$(E_0, E_1)$ as
long as the boundary point in question is an eigenvalue of $H$ (i.e.,
different from
$E_1 =0$). Indeed, in the case $(\mu,\sigma)\to \tilde\mu\in \{E_0,
E_1\}\cap \sigma_d
(H)$, $\mu\in (E_0, E_1)$, the analog of (5.19) then reads
$$\aligned
\tilde T_{\tilde\mu}(\lambda) &= \frac{\lambda^{1/2} -i(-\tilde\mu)^{1/2}}
{\lambda^{1/2} + i(-\tilde\mu)^{1/2}}\, T(\lambda), \\
\tilde R_{\tilde\mu}\Sp \ell \\ r\endSp (\lambda) &= \frac{\lambda^{1/2}
-i(-\tilde\mu)^{1/2}}
{\lambda^{1/2} +i(-\tilde\mu)^{1/2}}\, \frac{\lambda^{1/2}\pm
i\sigma(-\tilde\mu)^{1/2}}
{\lambda^{1/2}\mp -i\sigma (-\tilde\mu)^{1/2}}\, R\Sp \ell \\ r\endSp
(\lambda), \\
&\qquad \lambda>0, \ (\mu,\sigma)\in (E_0, E_1) \times \{-,+\}, \
\tilde\mu\in\{E_0,
E_1\}\cap\sigma_d(H).
\endaligned \tag 5.25
$$
One observes that the transmission coefficient in (5.19) stays invariant
with respect to
DDM (since Lemma 5.5 describes the isospectral case $\sigma_{(p)}(\tilde
H_{(\tilde\mu, \tilde\sigma)})=\sigma_{(p)}(H)$ as $\mu,\tilde\mu\in (E_0,
E_1)$) whereas (5.25)
exhibits a change of $T(\lambda)$ (as it must since now $\tilde\mu\in
\{E_0, E_1\}
\cap\sigma_d(H)$ got knocked out of the spectrum of $\tilde H_{(\tilde\mu,
\tilde\sigma)}$,
$\sigma_{(p)} (\tilde H_{(\tilde\mu,
\tilde\sigma)})=\sigma_{(p)}(H)\backslash \{\tilde\mu
\}$). In this context we invite the reader to compare with the
corresponding single and
double commutation results in Appendices A and B (see (A.32), (A.33), and
(B.35), (B.36)).

\remark{Remark 5.6} It should be pointed out at the end that the material
in this
paper is not at all confined to Schr\"odinger or Sturm-Liouville operators.
In fact, (a
generalization of) Lemma 2.2 for general second-order finite-difference
(Jacobi) operators
appeared in [\te]. Moreover, the discrete analog of DDM was used in
[\gtdiff] to construct
an explicit realization of the isospectral torus for algebro-geometric
quasi-periodic
Jacobi operators. A detailed analysis of the discrete version of DDM will
be given in [\tpre].
\endremark

\vskip 0.3in

\flushpar {\bf Acknowledgments.} F.G.~is indebted to C.W.~Peck and
A.~Kechris for
kind invitations to Caltech during the summers of 1994 and 1996 where some
of this
work was done. The extraordinary hospitality and support by the
Department of
Mathematics at Caltech are gratefully acknowledged.

\vskip 0.3in

\flushpar{\bf Appendix A. The Single Commutation or Crum-Darboux Method}
\vskip 0.1in

We briefly summarize the main spectral characteristics of the single
commutation\linebreak
method (abbreviated occasionally as SCM in this appendix). The principal
source for the
following material is the fundamental paper by Deift [\deift] (see also
[\detr], [\gsvi],
[\schm]). Further hints to the literature and to applications of this
method in spectral theory
and completely integrable systems are collected at the end of this appendix.

Suppose
$$
V\in L^1_{\text{\rom{loc}}}(\Bbb R)\text{ is real valued} \tag A.1
$$
and assume that the differential expression
$$
\tau =-\frac{d^2}{dx^2}+V(x), \quad x\in\Bbb R\text{ is non-oscillatory at
both $\pm\infty$}.
\tag A.2
$$
Denote by $H$ the uniquely associated self-adjoint operator in $L^2(\Bbb
R)$, maximally
defined, that is,
$$\aligned
Hf &=\tau f, \\
f\in\Cal D(H) &= \{g\in L^2(\Bbb R)\mid g,
g'\in AC_{\text{\rom{loc}}}(\Bbb R);
\tau g\in L^2 (\Bbb R)\}.
\endaligned \tag A.3
$$
Because of (A.2), $H$ is bounded from below,
$$
H\geq \Sigma_0, \quad \Sigma_0 =\inf\sigma(H)>-\infty. \tag A.4
$$
Next, let $\psi_\pm (z,x)$, $z\in\Bbb
C\backslash\sigma_{\text{\rom{ess}}}(H)$ be the
unique (up to constant multiples) solutions of
$$
\tau\psi (z) =z\psi(z) \tag A.5
$$
satisfying for all $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$,
$R\in\Bbb R$,
$$
\psi_\pm (z, ,\cdot\,) \in L^2 ((R, \pm\infty)) \text{ and } \psi_\pm
(\lambda, x) >0
\text{ for } \lambda < \Sigma_0. \tag A.6
$$
(The latter condition in (A.6) can always be achieved since $\psi_\pm
(\lambda, x)\neq 0$
for $\lambda < \Sigma_0$.) One defines
$$
\psi_{\nu_1} (\lambda_1,x) =\frac{1}{2} (1-\nu_1)\psi_- (\lambda_1,x) +
\frac{1}{2}
(1+\nu_1)\psi_+ (\lambda_1,x), \quad \nu_1 \in [-1, 1],
\ \lambda_1 <\Sigma_0
\tag A.7
$$
(we identify $\psi_{\pm 1} =\psi_\pm$ for notational convenience) and
$$\align
\phi_{\nu_1}(\lambda_1, x) &=\psi'_{\nu_1} (\lambda_1, x) / \psi_{\nu_1}
(\lambda_1, x), \tag A.8 \\
\alpha_{\nu_1}(\lambda_1) &=\frac{d}{dx} +\phi_{\nu_1} (\lambda_1, x),
\quad \alpha_{\nu_1}(\lambda_1)^+ = -\frac{d}{dx} + \phi_{\nu_1}
(\lambda_1, x). \tag A.9
\endalign
$$
(We note that $\psi_{\nu_1} (\lambda_1, x)\neq 0$ for $\lambda_1
<\Sigma_0$). One
infers that
$$
\tau=\alpha_{\nu_1}(\lambda_1) \alpha_{\nu_1} (\lambda_1)^+ + \lambda_1
=-\frac{d^2}{dx^2}+V(x) \tag A.10
$$
is independent of $\nu_1\in [-1,1]$ and $\lambda_1 <\Sigma_0$ and
introduces a
commutation of $\alpha_{\nu_1}(\lambda_1)$ and $\alpha_{\nu_1}
(\lambda_1)^+$,
$$\aligned
\tilde\tau_{\nu_1}(\lambda_1) &= \alpha_{\nu_1}(\lambda_1)^+
\alpha_{\nu_1}(\lambda_1)+\lambda_1 = -\frac{d^2}{dx^2} + \tilde V_{\nu_1}
(\lambda_1, x),  \quad x\in\Bbb R,\\
\tilde V_{\nu_1}(\lambda_1, x) &= V(x) -2\{\ln [\psi_{\nu_1} (\lambda_1,
x)]\}'',
\quad \lambda_1 <\Sigma_0, \  \nu_1\in [-1, 1].
\endaligned \tag A.11
$$
One verifies
$$\gathered
\alpha_{\nu_1}(\lambda_1)^+ \psi_{\nu_1}(\lambda_1) =0,  \quad
\alpha_{\nu_1}(\lambda_1) \psi_{\nu_1}(\lambda_1)^{-1} =0, \\
\psi_\pm (\lambda_1, \,\cdot\,)\in L^2 ((R, \pm\infty)), \quad \psi_\pm
(\lambda_1,
\,\cdot\,)\notin L^2 ((R, \mp\infty)),\quad R\in\Bbb R ,\\
\psi_{\nu_1}(\lambda_1, \,\cdot\,)\notin L^2 ((-\infty, R))\cup L^2 ((R,
\infty)),
\quad R\in \Bbb R, \\
\psi_{\nu_1}(\lambda_1, \,\cdot\,)^{-1}\in L^2 (\Bbb R), \quad \nu_1 \in
(-1, 1).
\endgathered \tag A.12
$$
Next, let $\overline{A_{\nu_1}(\lambda_1)}$,
$\overline{A_{\nu_1}(\lambda_1)}^*$,
and $\tilde H_{\nu_1}(\lambda_1)$ be the following closed linear
operators in
$L^2 (\Bbb R)$ associated with $\alpha_{\nu_1}(\lambda_1)$, $\alpha_{\nu_1}
(\lambda_1)^+$, and $\tilde\tau_{\nu_1}(\lambda_1)$, respectively. Consider
$$\gather
\Cal D_0 = \{g\in\Cal D(H)\mid \text{ supp$(g)$ compact}\}, \tag A.13 \\
A_{\nu_1}(\lambda_1) = \left. \alpha_{\nu_1}(\lambda_1)\right|_{\Cal D_0}.
\tag A.14
\endgather
$$
Then,
$$
\left. A_{\nu_1}(\lambda_1)^*\right|_{\Cal D_0} = \left. \alpha_{\nu_1}
(\lambda_1)^+ \right|_{\Cal D_0} \tag A.15
$$
and (cf.~[\gzh])
$$
H=\overline{A_{\nu_1}(\lambda_1)} \  \overline{A_{\nu_1}(\lambda_1)}^*
+\lambda_1, \quad \tilde H_{\nu_1}(\lambda_1) =
\overline{A_{\nu_1}(\lambda_1)}^* \, \overline{A_{\nu_1}(\lambda_1)} +
\lambda_1.
\tag A.16
$$
In particular, by Lemma 5.1(i), $\tilde\tau_{\nu_1}(\lambda_1)$ (and, of
course,
$\tau$) is l.p.~at $\pm\infty$ since
$\tilde H_{\nu_1}(\lambda_1)\geq\lambda_1$
($H\geq\Sigma_0$). Let
$$\gathered
\overline{A_{\nu_1}(\lambda_1)}^* = S_{\nu_1}(\lambda_1)
|\overline{A_{\nu_1}
(\lambda_1)}^*| = |\overline{A_{\nu_1}(\lambda_1)}| \,
S_{\nu_1} (\lambda_1), \\
|\overline{A_{\nu_1}(\lambda_1)}^*| = [\overline{A_{\nu_1}(\lambda_1)} \
\overline{A_{\nu_1}(\lambda_1)}^*]^{1/2}, \quad
|\overline{A_{\nu_1}(\lambda_1)}| =
[\overline{A_{\nu_1}(\lambda_1)}^* \
\overline{A_{\nu_1}(\lambda_1)}]^{1/2}
\endgathered \tag A.17
$$
denote the polar decomposition of $\overline{A_{\nu_1}(\lambda_1)}^*$,
where
$$
S_{\nu_1}(\lambda_1): L^2 (\Bbb R) \to \text{Ker}
(\,\overline{A_{\nu_1}(\lambda_1)}\,)^\perp \tag A.18
$$
is unitary. At this point we used the fact that
$\text{Ker}(\overline{A_{\nu_1}(\lambda_1)}^*)=
\{0\}$ by the hypothesis $\lambda_1 <\Sigma_0 =\inf \sigma(H)$.
Moreover, we
introduce the orthogonal projection $\tilde P_{\nu_1}(\lambda_1)$ onto
$\text{Ker}
(\overline{A_{\nu_1}(\lambda_1)})$, that is,
$$
\tilde P_{\nu_1}(\lambda_1) =\cases 0, & \nu_1 \in \{-1, 1\} \\
\| \psi_{\nu_1}(\lambda_1)^{-1}\|^{-2}_2 (\psi_{\nu_1}(\lambda_1)^{-1},
\,\cdot\,)\psi_{\nu_1}(\lambda_1)^{-1}, & \nu_1 \in (-1, 1)
\endcases \tag A.19
$$
(cf.~(A.12)).

The fundamental result regarding the spectra of $H$ and $\tilde H_{\nu_1}
(\lambda_1)$ then follows as a special case of the unitary equivalence of
$\overline A \,
\overline A^*$ and $\overline A^* \, \overline A$, restricted to the
orthogonal complements
of their respective null spaces (see Deift [\deift]).

\proclaim{Theorem A.1 (Deift [\deift], see also [\gsvi], [\schm])}
Let $H,
\tilde H_{\nu_1}(\lambda_1)$, $\nu_1\in [-1,1]$, $\lambda_1 <\Sigma_0
=\inf\sigma(H)$, and $S_{\nu_1}(\lambda_1)$, $\tilde P_{\nu_1}(\lambda_1)$
be given as in {\rom{(A.3)}}, {\rom{(A.16)}}, and {\rom{(A.18)}},
{\rom{(A.19)}}.
Then
$$
\tilde H_{\nu_1}(\lambda_1) \restriction \text{\rom{Ran}}(1 -\tilde
P_{\nu_1}(\lambda_1))
= S_{\nu_1}(\lambda_1) H S _{\nu_1}(\lambda_1)^{-1}, \tag A.20
$$
that is, $\tilde H_{\pm 1}(\lambda_1)$ and $H$, and $\tilde
H_{\nu_1}(\lambda_1)$
and $H$, $\nu_1 \in (-1, 1)$, restricted to the orthogonal complement
of the
\rom(one-dimensional\rom) eigenspace associated with the eigenvalue
$\lambda_1$ of
$\tilde H_{\nu_1}(\lambda_1)$, are unitarily equivalent. In particular,
$$\align
\sigma_{(p)} (\tilde H_{\nu_1}(\lambda_1)) &= \cases
\sigma_{(p)}(H), & \nu_1 \in \{-1, 1\} \\
\sigma_{(p)}(H)\cup \{\lambda_1\}, &\nu_1 \in (-1, 1) \endcases , \\
\text{\rom{Ker}}(\tilde H_{\nu_1}(\lambda_1)-\lambda_1) &=
\cases \{0\}, & \nu_1 \in \{-1, 1\} \\
{\text{\rom{span}}}\{\psi_{\nu_1}(\lambda_1)^{-1}\}, & \nu_1 \in (-1, 1)
\endcases ,
\tag A.21 \\
\sigma_{\text{\rom{ess, ac, sc}}} (\tilde H_{\nu_1}(\lambda_1)) &=
\sigma_{\text{\rom{ess, ac, sc}}}(H).
\endalign
$$
\endproclaim

Next, we describe a variety of additional results and possible extensions
paralleling Section 5.
This is intended to facilitate comparisons with DDM as well as the double
commutation
method in Appendix B.

One verifies that
$$
\tilde\psi_{\nu_1, \pm}(z,\lambda_1,x) = W(\psi_\pm (z),
\psi_{\nu_1}(\lambda_1))
(x) / \psi_{\nu_1}(\lambda_1, x) \tag A.22
$$
satisfies
$$\gathered
\tilde\tau_{\nu_1}(\lambda_1)\tilde\psi_{\nu_1, \pm}(z,\lambda_1) =
z\tilde\psi_{\nu_1, \pm}(z,\lambda_1), \\
\tilde\psi_{\nu_1, \pm}(z, \lambda_1, \,\cdot\,) \in L^2 ((R, \pm\infty)),
\quad z\in\Bbb C\backslash \sigma(H), \ R\in\Bbb R.
\endgathered \tag A.23
$$
(The latter fact is proven in [\gsvi], Appendix A for $z\leq \lambda_1$;
one can use (2.16)
to extend it to $z\in\Bbb C\backslash\sigma(H)$.) Hence, normalizing
$\psi_\pm (z,x)$
temporarily (and without loss of generality) by
$$
\psi_\pm (\lambda_1, x_0) =1, \quad x_0 \in\Bbb R\text{ fixed}, \tag A.24
$$
the Weyl-Titchmarsh $m$-function $\tilde m_{\nu_1, \pm}(z, \lambda_1)$
associated with $\tilde H_{\nu_1}(\lambda_1)$ and the reference point
$x_0$ is
given by (cf.~[\gnow])
$$\gathered
\tilde m_{\nu_1, \pm}(z, \lambda_1) = \frac{z-\lambda_1}
{\cot(\alpha_{\nu_1}(\lambda_1)) - m_\pm (z)} - \cot(\alpha_{\nu_1}
(\lambda_1)), \\
\cot(\alpha_{\nu_1}(\lambda_1)) = \frac{1}{2} (1-\nu_1) m_- (\lambda_1)
+\frac{1}{2} (1+\nu_1) m_+ (\lambda_1), \quad \nu_1 \in [-1,1], \
z\in\Bbb C\backslash\Bbb R.
\endgathered \tag A.25
$$
Here use has been made of
$$
\tilde m_{\nu_1, \pm}(z, \lambda_1) = \tilde\psi'_{\nu_1, \pm}
(z, \lambda_1, x_0),
$$
(A.22), (A.24), and the fact that $\tilde\tau_{\nu_1}(\lambda_1)$
is l.p.~at
$\pm\infty$.

Given (A.25) one can compute $\tilde M_{\nu_1}(z,\lambda_1)$, the
Weyl-Titchmarsh
$M$-matrix in $\Bbb C^2$ associated with $\tilde H_{\nu_1}(\lambda_1)$ in
terms of $M(z)$, the $M$-matrix of $H$ (see (4.1)). One obtains
$$\align
\tilde M_{\nu_1,1,1}(z, \lambda_1)&= \frac{\cot^2 (\alpha_{\nu_1}
(\lambda_1))}
{z-\lambda_1} M_{1,1}(z) + 2\cot (\alpha_{\nu_1}(\lambda_1)) \biggl[ 1-
\frac{\cot^2(\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1}\biggr] M_{1,2}
(z) \\
&\qquad + \biggl[ (z-\lambda_1) -2\cot^2 (\alpha_{\nu_1}(\lambda_1)) +
\frac{\cot^4 (\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1}\biggr]
M_{2,2}(z),  \tag A.26 \\
\tilde M_{\nu_1,1,2}(z,\lambda_1) &= -\frac{\cot(\alpha_{\nu_1}
(\lambda_1))}
{z-\lambda_1} M_{1,1}(z) + \biggl[ \frac{2\cot^2 (\alpha_{\nu_1}
(\lambda_1))}
{z-\lambda_1} -1\biggr] M_{1,2}(z), \tag A.27 \\
\tilde M_{\nu_1, 2,2}(z, \lambda_1) &= \frac{1}{z-\lambda}  M_{1,1} (z) -
\frac{2\cot(\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1} M_{1,2}(z)
+ \frac{\cot^2 (\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1}  M_{2,2}(z), \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad
z\in\Bbb C\backslash\Bbb R.
\tag A.28
\endalign
$$
One readily confirms that all matrix elements $\tilde M_{\nu_1,p,q}
(z,\lambda_1)$, $1\leq p, q\leq 2$ have a pole at $z=\lambda_1$ if and
only
if $\nu_1
\in (-1, 1)$ (i.e., if and only if $\cot (\alpha_{\nu_1}(\lambda_1))\neq
m_\pm (\lambda_1$)
in agreement with Theorem A.1. Moreover, we might note that $m_-
(\lambda_1) \neq
m_+ (\lambda_1)$ since $\lambda_1 <\Sigma_0 =\inf\sigma(H)$.

One could use (A.26)--(A.28) to compute the corresponding $\Bbb
C^2$-valued
spectral
matrix $\tilde\rho_{\nu_1}(\lambda, \lambda_1)$ of $\tilde
H_{\nu_1}(\lambda_1)$
in terms of $\rho(\lambda)$, the one associated with $H$. The resulting
formulas (although
providing an alternative proof of Theorem A.1), however, are rather
complex and
hence omitted. (A.26)--(A.28) become simpler if the Dirichlet boundary
condition $\psi
(x_0\pm 0)=0$, used to compute $\tilde m_{\nu_1, \pm}(z,\lambda_1)$,
$\tilde M_{\nu_1}(z,\lambda_1)$, is replaced by $\sin (\alpha_{\nu_1}
(\lambda_1)) \psi' (x_0\pm 0) +\cos (\alpha_{\nu_1}(\lambda_1))
\psi (x_0 \pm 0)
=0$. We will not pursue this now but return to this approach in
Appendix B.

Iterations of SCM can be handled as follows. Assume $V\in
L^1_{\text{\rom{loc}}}
(\Bbb R)$ to be real-valued and pick $\lambda_1
<\lambda_2 <\cdots <\lambda_N
<\Sigma_0=\inf\sigma(H)$, $\nu_j \in [-1, 1]$, $C_{\pm, j} >0$, $1\leq
j\leq N$,
$N\in\Bbb N$. Then the SCM result after $N$ iteration steps, denoted by
$\tilde\tau_{\nu_1,\dots,\nu_N}(\lambda_1, \dots, \lambda_N)$, reads as
follows.
$$\gather
\tilde\tau_{\nu_1, \dots, \nu_N}(\lambda_1,\dots, \lambda_N) = -
\frac{d^2}{dx^2} +\tilde V_{\nu_1,\dots, \nu_N}(\lambda_1, \dots,
\lambda_N, x), \quad x\in\Bbb R, \tag A.29 \\
\tilde V_{\nu_1, \dots, \nu_N}(\lambda_1, \dots, \lambda_N,x) = V(x) -2
\{\ln [W(\psi_{\nu_1}(\lambda_1), \dots,
\psi_{\nu_N}(\lambda_N))(x)]\}'', \\
\psi_{\nu_j}(\lambda_j, x) =\frac{1}{2} (1-\nu_j)C_{-,j}
\psi_- (\lambda_j, x)
+\frac{1}{2} (1+\nu_j) C_{+, j} \psi_+ (\lambda_j, x),\quad
1\leq j\leq N.
\endgather
$$
The obvious analog of Theorem A.1 (distinguishing between
$\nu_j \in (-1,
1)$ and
$\nu_j \in \{-1, 1\}$) then applies to $\tilde H_{\nu_1, \dots, \nu_N}
(\lambda_1, \dots, \lambda_N)$, the unique semi-bounded self-adjoint
operator
associated with $\tilde\tau_{\nu_1,\dots,\nu_N}(\lambda_1, \dots,
\lambda_N)$
(see [\gsvi], Appendix A for more details).

In analogy to Lemma 5.4, one infers from (A.25) that $\tilde
H_{\nu_1}(\lambda_1)$
is reflectionless if and only if $H$ is (as observed in [\gnow]).

In order to obtain the Sturm-Liouville generalization of
(A.1)--(A.16) (see
[\gtproc],
[\schm]), one assumes
$$\gathered
p,p', k,k'\in AC_{\text{\rom{loc}}} ((a,b),
\quad q\in L^1_{\text{\rom{loc}}}
((a,b)) \text{ real-valued}, \\
p>0, \ k>0, -\infty\leq a < b \leq \infty,
\endgathered \tag A.30
$$
and makes the changes
$$\gathered
\alpha_{\nu_1}(\lambda_1) \to \widehat\alpha_{\nu_1}(\lambda_1) =
\frac{1}{k(x)} \biggl(\sqrt{k(x)p(x)} \frac{d}{dx}
+ \phi_{\nu_1}(\lambda_1, x)
\biggr), \\
\alpha_{\nu_1}(\lambda_1)^+ \to \widehat\alpha_{\nu_1}(\lambda_1)^+ =
\frac{1}{k(x)} \biggl(-\frac{d}{dx} \sqrt{k(x)p(x)}\,
+\phi_{\nu_1}(\lambda_1,x)
\biggr), \\
\tau\to\widehat\tau = \widehat\alpha_{\nu_1}(\lambda_1)
\widehat\alpha_{\nu_1}(\lambda_1)^+ +\lambda_1 =\frac{1}{k(x)}
\biggl( -
\frac{d}{dx}\, p(x) \frac{d}{dx}+ q(x)\biggr), \\
\tilde\tau_{\nu_1}(\lambda_1) \to
\widetilde{\widehat\tau}_{\nu_1}(\lambda_1)
=\widehat\alpha_{\nu_1}(\lambda_1)^+ \widehat\alpha_{\nu_1}(\lambda_1)
+\lambda_1 =\frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x) \frac{d}{dx} +
\tilde q_{\nu_1}(\lambda_1, x)\biggr), \\
\tilde q_{\nu_1}(\lambda_1,x) = q(x) -\frac{p''(x)}{2}
+ \frac{p'(x)^2}{4p(x)}
+ \frac{3k'(x)^2 p(x)}{4k(x)^2} - \frac{k''(x)p(x)}{2k(x)} \\
\qquad \qquad \qquad + \biggl( \frac{1}{k(x)} (k(x)p(x))'
-2 \frac{d}{dx}\,
p(x)\biggr)
\frac{d}{dx} \, \ln [\psi_{\nu_1}(\lambda_1, x)], \\
\widehat\tau \psi_{\nu_1}(\lambda_1)
= \lambda_1 \psi_{\nu_1}(\lambda_1),
\quad \phi_{\nu_1}(\lambda_1,x) = \bigl(\sqrt{k(x)p(x)}\,
\psi_{\nu_1}(\lambda_1, x)\bigr)' \big/ \psi_{\nu_1}(\lambda_1, x).
\endgathered \tag A.31
$$

It remains to sketch the scattering theory formulas analogous to (5.19),
assuming $V\in
L^1 (\Bbb R; (1+|x|)\,dx)$ to be real-valued. (We use the conventions
established in
(5.11)--(5.17).) It was proved by Deift and Trubowitz ([\detr], Theorem
3.2) that
$\tilde V_{\nu_1}(\lambda_1)\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if
$V$ is,
and also the scattering matrix $\tilde S_{\nu_1}(\lambda, \lambda_1)$
associated
with the pair $(\tilde H_{\nu_1}(\lambda_1), H_0)$ in terms of
$S(\lambda)$ in
(5.16), the one corresponding to $(H, H_0)$, was determined as follows.
$$\aligned
\tilde T_{\nu_1}(\lambda, \lambda_1) &=
\frac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2} -
i(-\lambda_1)^{1/2}}\,
T(\lambda), \\
\tilde R_{\nu_1}\Sp \ell \\ r\endSp (\lambda, \lambda_1) &=-
\frac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2} -
i(-\lambda_1)^{1/2}}\,
R\Sp \ell \\ r\endSp (\lambda), \quad \lambda >0, \
\lambda_1 < \Sigma_0, \
\nu_1 \in (-1, 1),
\endaligned \tag A.32
$$
$$\aligned
\tilde T_{\nu_1}(\lambda, \lambda_1) &= T(\lambda), \\
\tilde R_{\nu_1}\Sp\ell \\ r\endSp (\lambda, \lambda_1) &=-
\frac{\lambda^{1/2} \pm i\nu_1 (-\lambda_1)^{1/2}}
{\lambda^{1/2} \mp i\nu_1 (-\lambda_1)^{1/2}}\, R\Sp \ell
\\ r\endSp (\lambda),
\quad \lambda >0, \  \lambda_1 <\Sigma_0, \ \nu_1\in \{-1, 1\}.
\endaligned \tag A.33
$$
Further generalizations of (A.32), (A.33) in the context of
supersymmetric
quantum
mechanics can be found in [\bggss] and [\gss].

The discrete analog of SCM for general second-order finite-difference
(Jacobi) operators
has been developed in detail in [\gtdiff].

Finally we provide a brief historical account and hints to some
applications
of SCM.
SCM, or as it is
often called, the Crum-Darboux method, (A.7)--(A.11) goes back at
least to
Jacobi
[\jac] and Darboux [\dar]. Important later contributions are due
to Crum
[\crum],
Schmincke [\schm], and especially, Deift [\deift] (see also
[\detr]). In
particular, the
spectral deformation results of this method, as summarized in
Theorem A.1,
are due to
Deift [\deift].

In recent years, this method has been applied to a description of the
isospectral manifold of
periodic and algebro-geometric quasi-periodic finite-gap solutions
of the
(m)KdV
hierarchy (see, e.g, [\ehkn], [\erfl], [\gdekker], [\gcam], [\gss],
[\gwei], [\mckcpam],
[\mckrev], [\mckjsp], [\mckspm], [\mcknew], and the references therein)
and
the construction
of soliton solutions (resp., reflectionless potentials) of the (m)KdV
hierarchy relative to given
background (base) solutions (resp., potentials) by means of B\"acklund
transformations
(cf., e.g., [\deift], [\detr], [\flmcl], [\gdekker], [\gsvi], [\gss],
[\mckjsp], and the literature
cited therein).

As is obvious from (A.11), $\psi_{\nu_1}(\lambda_1, x)$ had to be
chosen
zero-free
(and hence $\lambda_1 <\Sigma_0$) in order to guarantee
$V_{\nu_1}(\lambda_1)
\in L^1_{\text{\rom{loc}}}(\Bbb R)$. This considerable restriction
on the range of $\lambda_1$ will be overcome in the following appendix.

\vskip 0.3in

\flushpar {\bf Appendix B. The Double Commutation Method}
\vskip 0.1in

We review the double commutation method (occasionally abbreviated
as DCM)
to insert
eigenvalues into spectral gaps of general background (base)
Schr\"odinger
operators
following [\gjfa] and [\gtproc]. Applications of this method and
pertinent
references to the
literature are collected at the end of this appendix.

Assuming $V$ satisfies
$$
V\in L^1_{\text{\rom{loc}}}(\Bbb R), \quad V \text{ real-valued}
\tag B.1
$$
and introducing the differential expression $\tau=-\frac{d^2}{dx^2} +
V(x)$, $x\in
\Bbb R$, we pick $\lambda_0\in\Bbb R$ and $\eta_\pm (\lambda_0, x)$
satisfying
$$\gathered
\tau\psi(\lambda_0) =\lambda_0 \psi(\lambda_0), \\
\eta_\pm (\lambda_0, \,\cdot\,) \in L^2 ((R, \pm\infty)),
\quad R\in\Bbb R, \
\eta_\pm (\lambda_0, x)\text{ real-valued}.
\endgathered \tag B.2
$$
Given $\eta_\pm (\lambda_0, x)$, we define the self-adjoint background
(base) operator
$H_\pm$ in $L^2(\Bbb R)$ via
$$\aligned
H_\pm f &=\tau f, \\
f\in\Cal D(H_\pm ) &= \{g\in L^2 (\Bbb R)\mid g,g'\in
AC_{\text{\rom{loc}}}(\Bbb R);
\tau g\in L^2 (\Bbb R); \\
& \qquad \lim\limits_{x\to\omega\infty} W(\eta_\pm (\lambda_0), g) (x)
=0 \text{ if $\tau$ is l.c.~at $\omega \infty, \ \omega \in\{-,+\}$}\}.
\endaligned \tag B.3
$$
Our choice of notation purposely stresses a possible dependence of
$H_\pm$ on
$\eta_\pm (\lambda_0, x)$. If $\tau$ is in the l.p.~case at $\omega
\infty$, $\omega \in
\{-,+\}$, the corresponding boundary condition in (B.3) is
superfluous at
$\omega \infty$
and hence to be deleted from (B.3). In particular, if $\tau$ is
l.p.~at
$\pm\infty$,
then $H_- = H_+ =H$ is independent of the choice of
$\eta_\pm (\lambda_0, x)$.

Next, denote
$$
L^2_{\text{\rom{loc}}} ([\mp\infty, \pm\infty))
= \{g\in L^2_{\text{\rom{loc}}}
(\Bbb R)\mid g\in L^2 ((\mp\infty, c))\}
$$
for some $c\in\Bbb R$ and pick $\gamma_1 \in (0,\infty)$,
$\lambda_1\in\Bbb
R$ and
$\psi_\pm (z,x)$ satisfying
$$\gathered
\tau\psi(z) =z\psi(z), \\
\psi_\pm (z, \,\cdot\,) \in L^2 ((R,\pm\infty)),
\quad z\in\Bbb C\backslash
\sigma_{\text{\rom{ess}}} (H_\pm), \ R\in\Bbb R, \\
\psi_\pm (\lambda, x) \text{ real-valued for }\lambda\in\Bbb R, \\
\psi_\pm (\lambda_1, \,\cdot\,) \in L^2 ((R,\pm\infty)),
\quad R\in\Bbb R, \\
\lim\limits_{x\to\omega\infty} W(\eta_\pm (\lambda_0), \psi_\pm
(\lambda_1))(x) = 0
\text{ if $\tau$ is l.c.~at $\omega\infty, \ \omega\in \{-, +\}$}.
\endgathered \tag B.4
$$
As in (B.3), the last condition in (B.4) is superfluous at
$\omega\infty$
if $\tau$ is
l.p.~at $\omega\infty$, $\omega\in\{-,+\}$. Given $\gamma_1 >0$
and $\psi_\pm
(\lambda_1, x)$, we define the linear transformation
$$
\widehat U_{\pm, \gamma_1}(\lambda_1) : \cases
L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty))
\to L^2_{\text{\rom{loc}}}
([\pm\infty, \mp\infty)) \\
f(x) \to \tilde f_{\pm, \gamma_1} (\lambda_1, x)
= f(x) \pm \gamma_1
\tilde\psi_{\pm, \gamma_1} (\lambda_1, x) \int^x_{\pm\infty} dx'
\psi_\pm
(\lambda_1, x') f(x') \endcases ,
\tag B.5
$$
where
$$
\tilde\psi_{\pm, \gamma_1} (\lambda_1, x) = (\widehat U_{\pm,
\gamma_1}
(\lambda_1)
\psi_\pm (\lambda_1))(x) = \biggl[ 1\mp\gamma_1
\int\limits^x_{\pm\infty}
dx' \,
\psi_{\pm} (\lambda_1, x')^2\biggr]^{-1} \psi_{\pm} (\lambda_1, x).
\tag B.6
$$
By inspection, one infers for the inverse transformation
$$
\widehat U_{\pm, \gamma_1} (\lambda_1)^{-1} : \cases
L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty))
\to L^2_{\text{\rom{loc}}}
([\pm\infty, \mp\infty)) \\
h(x) \to h(x) \mp \gamma_1 \psi_\pm (\lambda_1, x) \int^x_{\pm\infty}dx'
\tilde\psi_{\pm, \gamma_1}(\lambda_1, x') h(x')
\endcases . \tag B.7
$$
We list a few more facts (cf.~[\gtproc]) which explain Lemma B.1 and
Theorem B.2 below.
$$
1\pm \gamma_1 \int\limits^x_{\pm\infty} dx' \, \tilde\psi_{\pm, \gamma_1}
(\lambda_1, x')^2 = \biggl[1\mp\gamma_1 \int\limits^x_{\pm\infty} dx' \,
\psi_\pm
(\lambda_1, x')^2\biggr]^{-1}, \tag B.8
$$
$$
\tilde\psi_{\pm, \gamma_1}(\lambda_1) \in L^2 (\Bbb R), \
\|\tilde\psi_{\pm, \gamma_1} (\lambda_1)\|^2_2 = \gamma^{-1}_1 \biggl\{
1-\lim\limits_{x\to\mp\infty} \biggl[ 1\mp
\gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1,
x')^2\biggr]^{-1}\biggr\},
\tag B.9
$$
$$\multline
\mp\int\limits^x_{\pm\infty} dx' \, \overline{\tilde f_{\pm,
\gamma_1}(\lambda_1, x')} \,
\tilde g_{\pm, \gamma_1}(\lambda_1, x')
=\mp\int\limits^x_{\pm\infty} dx' \,
\overline{f(x')}\, g(x') \\ \pm\gamma_1 \biggl[1\mp\gamma_1
 \int\limits^x_{\pm\infty} dx' \,
\psi_\pm (\lambda_1, x')^2\biggr]^{-1}
\int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')
\overline{f(x')}
\int\limits^x_{\pm\infty} dx'' \, \psi_\pm (\lambda_1, x'')g(x''),
\endmultline \tag B.10
$$
$$\multline
\mp\int\limits^x_{\pm\infty} dx'\, \overline{f(x')}\, g(x') =
\mp\int\limits^x_{\pm\infty}
dx' \, \overline{\tilde f_{\pm, \gamma_1}(\lambda_1, x')}\,
\tilde g_{\pm, \gamma_1}(\lambda_1, x')  \\
\mp \gamma_1 \biggl[ 1\pm\gamma_1 \int\limits^x_{\pm\infty} dx' \,
\tilde\psi_{\pm, \gamma_1}(\lambda_1, x')^2\biggr]^{-1}
\int\limits^x_{\pm\infty} dx' \, \tilde\psi_{\pm, \gamma_1}(\lambda_1,
x')\, \overline{\tilde\psi_{\pm, \gamma_1} (\lambda_1, x')} \\
\times \int\limits^x_{\pm\infty} dx''\, \tilde\psi_{\pm, \gamma_1}
(\lambda_1, x'')
\tilde g_{\pm, \gamma_1}(\lambda_1, x'').
\endmultline
\tag B.11
$$
Next, we denote the restriction of $\widehat U_{\pm, \gamma_1}
(\lambda_1)$
to $L^2
(\Bbb R)$ by
$$
\left. U_{\pm, \gamma_1}(\lambda_1)= \widehat U_{\pm, \gamma_1}
(\lambda_1)
\right|_{L^2 (\Bbb R)}, \tag B.12
$$
define the orthogonal projections
$$\aligned
P_\pm (\lambda_1) &=\cases 0, &\psi_\pm (\lambda_1)\notin
L^2 (\Bbb R) \\
\|\psi_\pm (\lambda_1)\|^{-2}_2 (\psi_\pm (\lambda_1), \,\cdot\,)
\psi_\pm
(\lambda_1),
& \psi_\pm (\lambda_1)\in L^2 (\Bbb R) \endcases , \\
\tilde P_{\pm, \gamma_1}(\lambda_1) &= \|\tilde\psi_{\pm, \gamma_1}
(\lambda_1)\|^{-2}_2 (\tilde\psi_{\pm, \gamma_1}(\lambda_1), \,\cdot\,)
\tilde\psi_{\pm, \gamma_1}(\lambda_1),
\endaligned \tag B.13
$$
and introduce the double commutation differential expression
$$\aligned
\tilde\tau_{\pm, \gamma_1}(\lambda_1) &= -\frac{d^2}{dx^2}+
\tilde V_{\pm,
\gamma_1}
(\lambda_1,x), \quad x\in\Bbb R, \\
\tilde V_{\pm, \gamma_1}(\lambda_1,x) &= V(x) -2\biggl\{ \ln \biggl[
1\mp\gamma_1
\int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2
\biggr]\biggr\}''.
\endaligned \tag B.14
$$
Relations (B.5)--(B.11) then yield

\proclaim{Lemma B.1 [\gtproc]}
\roster
\item"\rom{(i)}" $U_{\pm, \gamma_1}(\lambda_1): (1-P_\pm (\lambda_1))
L^2 (\Bbb R) \to (1-\tilde P_{\pm,
\gamma_1}(\lambda_1))L^2 (\Bbb R)$ is
unitary.
\item"\rom{(ii)}" $\tilde\tau_{\pm, \gamma_1}(\lambda_1)
(\widehat U_{\pm,
\gamma_1}
(\lambda_1)f) = \widehat U_{\pm, \gamma_1}(\lambda_1) (\tau f)$.
\endroster
\endproclaim

Lemma B.1(ii) shows that $U_{\pm, \gamma_1}(\lambda_1)$ are
transformation
operators
for the pairs \linebreak $(\tilde H_{\pm, \gamma_1}(\lambda_1),
H_\pm )$ in
the terminology
of [\levbook], Chapter 1, [\mar], Chapter 1, that is, they map
solutions of
$\tau\psi(z)=z\psi
(z)$, $z\in\Bbb C\backslash \{\lambda_1\}$ into those of
$\tilde\tau_{\pm,
\gamma_1} (\lambda_1)\tilde\psi (z) =z\tilde\psi (z)$.

The self-adjoint operator $\tilde H_{\pm, \gamma_1}(\lambda_1)$
corresponding to
$\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is then defined by
$$\align
\tilde H_{\pm, \gamma_1}(\lambda_1)f &= \tilde\tau_{\pm,
\gamma_1}(\lambda_1)f, \\
f\in\Cal D(\tilde H_{\pm, \gamma_1}(\lambda_1)) &=
\{g\in L^2 (\Bbb R) \mid g,g'
\in AC_{\text{\rom{loc}}} (\Bbb R); \tilde\tau_{\pm,
\gamma_1}(\lambda_1) g\in
L^2 (\Bbb R); \tag B.15 \\
&\ \lim\limits_{x\to\omega\infty} W(\tilde\psi_{\pm,
\gamma_1}(\lambda_1),g)
(x)=0 \text{ if $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.c.~at
$\omega\infty, \
\omega\in \{-,+\}$}\}.
\endalign
$$
As usual, the last boundary condition at $\omega\infty$ in (B.15)
is to be
deleted if
$\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.p.~at $\omega\infty$.

The principal result concerning the spectra of $\tilde H_{\pm,
\gamma_1}(\lambda_1)$
and $H_\pm$ then reads as follows.

\proclaim{Theorem B.2 [\gtproc]} Let $H_\pm$, $\tilde H_{\pm,
\gamma_1}(\lambda_1)$,
$\gamma_1 >0$, $\psi_\pm (\lambda_1, x)$, $\tilde\psi_{\pm,
\gamma_1}(\lambda_1,x)$,
$\lambda_1 \in\Bbb R, $ and $U_{\pm, \gamma_1}(\lambda_1)$, $P_\pm
(\lambda_1)$,
$\tilde P_{\pm, \gamma_1}(\lambda_1)$ be given as in {\rom{(B.3)}},
{\rom{(B.15)}},
{\rom{(B.4)}}, {\rom{(B.6)}}, and {\rom{(B.12)}}, {\rom{(B.13)}}. Then

{\rom{(i)}} $\lambda_1 \in \sigma_p (\tilde H_{\pm,
\gamma_1}(\lambda_1))$ and
$$
\text{\rom{Ker}}(\tilde H_{\pm, \gamma_1}(\lambda_1)-\lambda_1) =
{\text{\rom{span}}}\{
\tilde\psi_{\pm, \gamma_1}(\lambda_1)\}. \tag B.16
$$

{\rom{(ii)}} If $\psi_\pm (\lambda_1) \notin L^2 (\Bbb R)$ and
\rom(and
hence $\tau$
is l.p.~at $\pm\infty$\rom), one obtains
$$
\tilde H_{\pm, \gamma_1}(\lambda_1)(1-\tilde P_{\pm, \gamma_1}
(\lambda_1)) =
U_{\pm, \gamma_1}(\lambda_1)HU_{\pm, \gamma_1}(1-\tilde P_{\pm,
\gamma_1}
(\lambda_1)), \tag B.17
$$
that is, $\tilde H_{\pm, \gamma_1}(\lambda_1)$ and $H$,
restricted to the
orthogonal
complement of the \rom(one-dimensional\rom) eigenspace associated
with the
eigenvalue
$\lambda_1$ of $\tilde H_{\pm, \gamma_1}(\lambda_1)$, are unitarily
equivalent. In
particular,
$$
\sigma_{(p)}(\tilde H_{\pm, \gamma_1}(\lambda_1)) = \sigma_{(p)}(H)\cup
\{\lambda_1\}.
\tag B.18
$$

{\rom{(iii)}} If $\psi_\pm (\lambda_1) \in L^2 (\Bbb R)$, then
there exists
a unitary
operator $\tilde U_{\pm, \gamma_1}(\lambda_1)= \tilde U_{\pm,
\gamma_1}(\lambda_1)
\oplus \sqrt{1+\gamma_1 \|\psi_\pm (\lambda_1)\|^2_2}\, 1$ on
$(1-P_\pm
(\lambda_1))
L^2 (\Bbb R) \oplus P_\pm (\lambda_1) L^2 (\Bbb R)$ such that
$$
\tilde H_{\pm, \gamma_1}(\lambda_1) = \tilde U_{\pm, \gamma_1}
(\lambda_1)H_\pm
\tilde U_{\pm, \gamma_1}(\lambda_1)^{-1}. \tag B.19
$$
Moreover,
$$
\sigma_{\text{\rom{ess, ac, sc}}} (\tilde H_{\pm, \gamma_1}
(\lambda_1)) =
\sigma_{\text{\rom{ess, ac, sc}}} (H_\pm ) \tag B.20
$$
in cases {\rom{(ii)}} and {\rom{(iii)}}.
\endproclaim

\remark{Remark B.3}  (i) Thus far we considered the case
$0<\gamma_\pm
<\infty$. The limit
$\gamma_\pm \to\infty$ in (B.14) (implying $\tilde\psi_{\pm,
\gamma_1}(\lambda_1,x)
\underset {\gamma_1\to\infty} \to\longrightarrow 0$ and hence
$\tilde
P_{\pm, \gamma_1} (\lambda_1)=0$) formally seems to yield an
isospectral
deformation of $H_\pm$ when compared
with Theorem B.2. One computes
$$\aligned
\tilde\tau_{\pm, \infty }(\lambda_1) &= -\frac{d^2}{dx^2} +
\tilde V_{\pm,
\infty}
(\lambda_1,x), \quad x\in \Bbb R, \\
\tilde V_{\pm, \infty }(\lambda_1, x) &= V(x) -2\biggl\{\ln
\biggl[\mp
\int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2
\biggr]\biggr\}''.
\endaligned \tag B.21
$$
A quick look at (2.19) and (2.20) then shows that (B.21) is
precisely the
(sign flip) Dirichlet
deformation (cf.~Remark 4.6(iii)) where
$$
(\lambda_1, \pm) \to (\lambda_1, \mp), \quad \tilde\tau_{\pm,
\infty}(\lambda_1)
= \tilde\tau_{(\lambda_1, \mp)}, \ \mu=\tilde\mu=\lambda_1, \
\tilde\sigma
=-\sigma
=\mp.  \tag B.22
$$
As a consequence of (B.22), Cases II and III in (2.34) and (2.35)
coincide
and the boundary
conditions in $\tilde H_{(\lambda_1, \mp)}$ (if any) are identical
to those
in (B.15), upon
replacing $\tilde\psi_{\pm, \gamma_1} (\lambda_1, x)$ by $\gamma_1
\tilde\psi_{\pm, \gamma_1} (\lambda_1, x)$ and formally letting
$\gamma_1
\to \infty$.
Thus
$$
\tilde H_{\omega, \infty}(\lambda_1) = \tilde H_{(\lambda_1, -\omega)},
\quad
(\lambda_1, \omega) = (\mu, \omega) = (\tilde\mu, -\omega), \ \omega\in
\{-, +\}
\tag B.23
$$
is the right definition for the self-adjoint operator associated with
$\tilde\tau_{\omega, \infty}
(\lambda_1)$ in (B.21). Hence the case $\gamma_1 =\infty$ is fully
covered
by Sections
3--5, and (B.21) indeed gives rise to an isospectral deformation of
$H_\pm$. The isospectral
nature of (B.21) has been systematically exploited in the context of
B\"acklund
transformations for the (m)KdV equation in [\gsvi].

(ii) If $H_\pm$ has an eigenfunction $\psi_\pm (\lambda_1)
\in\Cal D(H_\pm)$
associated
with the eigenvalue $\lambda_1$, one can reverse DCM and remove
$\lambda_1$
upon
choosing $\gamma_1= -\|\psi_\pm (\lambda_1)\|^{-2}_2$. In this case,
$\tilde\psi_{\pm, \gamma_1} (\lambda_1) \notin L^2 (\Bbb R)$ and hence
$\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.p.~at $\pm\infty$
(cf.~[\gjfa], [\gtproc]).

(iii) Similarly to DDM (and in contrast to SCM in Appendix A where we
exploited that $\tau$
was non-oscillatory and hence l.p.~at $\pm\infty$), DCM does not
necessarily produce a
l.p.~differential expression $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ at
 $\mp\infty$
even if $\tau$ was l.p.~at $\mp\infty$. In fact, one can use (ii)
above to
construct an example
where $\tau$ is l.p.~at $\mp\infty$ but $\tilde\tau_{\pm,
\gamma_1}(\lambda_1)$ is
l.c.~at $\mp\infty$ ([\gjfa], [\gtproc]). However, $\tilde\tau_{\pm,
\gamma_1}(\lambda_1)$
is l.p.~at $\pm\infty$ if and only if $\tau$ is and $\tilde\tau_{\pm,
\gamma_1}(\lambda_1)$
is l.c.~at $\mp\infty$ if $\tau$ is ([\gjfa], [\gtproc]). Of course,
Lemma
5.1 immediately
covers the present situation upon entering the obvious changes in
notation.
\endremark

Next, we turn to a computation of Weyl-Titchmarsh functions associated
with
$\tilde H_{\pm, \gamma_1}(\lambda_1)$ in terms of those of $H_\pm$.
Since
some of the
following results (such as (B.26)) are new, we provide a bit more
details.
First, some necessary
notation. Let $x_0 \in\Bbb R$ be a fixed reference point and assume
temporarily the notation
used in Lemma 3.1. Given $\widehat H$, $\widehat V$, and
$\widehat m_\pm
(z)$, the
corresponding $m$-functions associated with the half-line $(x_0,
\pm\infty)$, we define
the usual fractional linear transformation of $\widehat m_\pm (z)$,
$$\aligned
\widehat m^\alpha_\pm (z) &= \frac{-1 +\cot (\alpha) \widehat m_\pm (z)}
{\cot(\alpha) +\widehat m_\pm (z)}\, , \quad \alpha\in (0,\pi), \\
\widehat m^0_\pm (z) &= \widehat m_\pm (z),
\quad z\in\Bbb C\backslash\Bbb R
\endaligned  \tag B.24
$$
and
$$\aligned
\widehat M^\alpha (z) &= \pmatrix \format \l&\quad \r\\
\cos(\alpha) & -\sin(\alpha) \\
\sin(\alpha) & \cos(\alpha) \endpmatrix
\widehat M(z) \pmatrix \format \l&\quad \r\\
\cos(\alpha) & -\sin(\alpha) \\
\sin(\alpha) & \cos(\alpha) \endpmatrix^{-1} \\
&= [\widehat m^\alpha_- (z) -\widehat m^\alpha_+ (z)]^{-1}
\pmatrix
\widehat m^\alpha_- (z) \widehat m^\alpha_+ (z) & \tfrac{1}{2} [\widehat
m^\alpha_- (z)
+\widehat m^\alpha_+ (z) ] \\
\tfrac{1}{2} [\widehat m^\alpha_- (z) +\widehat m^\alpha_+ (z)] & 1
\endpmatrix , \\
\widehat M^0 (z) &= \widehat M(z), \quad z\in\Bbb R,
\endaligned \tag B.25
$$
with $\widehat M(z)$ defined in terms of $\widehat m_\pm (z)$ as in (4.1).
(B.24) and (B.25)
are associated with the boundary condition $\sin(\alpha)\psi' (x_0\pm
0)+\cos(\alpha)
\psi (x_0 \pm 0)=0$, $\alpha\in (0,\pi)$ as opposed to the Dirichlet
boundary condition $\alpha
=0$, $\psi (x_0 \pm 0)=0$ in connection with $\widehat m_\pm (z)$ and
$\widehat M(z)$.

\proclaim{Lemma B.4} Denote by
$\tilde m^\beta_{\omega, \gamma_1, \pm}(z, \lambda_1)$ and
$m^\alpha_\pm (z)$
the corresponding $m$-functions for \linebreak
$\tilde H_{\omega, \gamma_1}
(\lambda_1)$,
$\omega\in \{-,+\}$ and $H$ \rom($=H_\omega$\rom) associated with
the half-line
$(x_0, \pm\infty)$.

\rom{(i)} Suppose $m^0_\omega (\lambda_1)\ne\infty$ (i.e.,
$\psi_\omega (\lambda_1, x_0)\ne0$), $\omega \in \{-,+\}$. Then
$$\align
&\tilde m^{\beta_\omega (\lambda_1)}_{\omega, \gamma_1, \pm}(z,
\lambda_1) =
\cot (\beta_\omega (\lambda_1)) \\
&\quad + \frac{\sin^2 (\alpha_\omega (\lambda_1))}{\sin^2 (\beta_\omega
(\lambda_1))}
\biggl[ m^{\alpha_\omega (\lambda_1)}_\pm (z) - \cot (\alpha_\omega
(\lambda_1)) +
\frac{\omega \tilde\gamma_1 \big( \psi_\omega (\lambda_1, x_0)^2
+ \psi'_\omega(\lambda_1, x_0)^2\big) }
{z-\lambda_1} \biggr], \\
\qquad\tilde\gamma_1 &=\gamma_1 \biggl[ 1-\omega\gamma_1
\int\limits^{x_0}_{\omega\infty}  dx \, \psi_\omega (\lambda_1,
x)^2\biggr]^{-1}, \tag
B.26 \\ &\cot(\alpha_\omega (\lambda_1))
=-m^0_\omega (\lambda_1), \ \cot(\beta_\omega (\lambda_1))
= \cot(\alpha_\omega
(\lambda_1)) -\omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2.
\endalign
$$

\rom{(ii)} Suppose $m^0_\omega (\lambda_1)=\infty$ (i.e.,
$\psi_\omega (\lambda_1, x_0)=0$), $\omega \in \{-,+\}$. Then
$$
\tilde m^0_{\omega, \gamma_1, \pm}(z, \lambda_1) =
m^0_\pm (z) +
\frac{\omega \tilde\gamma_1 \psi'_\omega(\lambda_1, x_0)^2
}{z-\lambda_1} ,
\tag B.27
$$
with $\tilde\gamma_1$ as in (B.26).
\endproclaim

\demo{Proof}
We recall that
$$
\tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1,x) =
\psi_\pm (z,x) - \omega \gamma_1
\frac{ \tilde\psi_{\omega, \gamma_1}(\lambda_1, x)}{z-\lambda_1} W(
\psi_\omega (\lambda_1), \psi_\pm (z)) (x) \tag B.28
$$
satisfies
$$\gathered
\tilde\tau_{\omega, \gamma_1}(\lambda_1) \tilde\psi_{\omega,
\gamma_1,\pm}
(z, \lambda_1) = z\tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1) , \\
\tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1,\,\cdot\,)\in L^2
((R,\pm\infty)),
\quad z\in\Bbb C\backslash \sigma (\tilde H_{\omega, \gamma_1}
(\lambda_1)), \
R\in\Bbb R
\endgathered \tag B.29
$$
and also note that
$$\gathered
\tilde\psi_{\omega, \gamma_1,\omega}(\lambda_1, \lambda_1, x) =
\tilde\psi_{\omega, \gamma_1}(\lambda_1, x) = \biggl[ 1-\omega\gamma_1
\int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_1, x')^2
\biggr]^{-1}
\psi_\omega (\lambda_1, x), \\
\left. \frac{d}{dx} W(\psi_{\omega_1} (z_1), \psi_{\omega_2}(z_2))(x)
\right|_{x=x_0} = z_1 - z_2 , \\
m^0_\omega (z) =\psi'_\omega (z, x_0) / \psi_\omega (z, x_0) , \\
W(\psi_{\omega_1}(z_1), \psi_{\omega_2}(z_2))(x_0)
= m^0_{\omega_2} (z_2) -
m^0_{\omega_1}(z_1).
\endgathered \tag B.30
$$
(B.29) and (B.30) then yield
$$\align
& \tilde m^0_{\omega, \gamma_1, \pm}(z, \lambda_1)
= \tilde\psi'_{\omega,
\gamma_1, \pm}
(z, \lambda_1, x_0) / \tilde\psi_{\omega, \gamma_1, \pm}
(z, \lambda_1, x_0) \\
&\quad = \{1-\omega \tilde\gamma_1 \psi_\omega
(\lambda_1, x_0)^2 (m^0_\pm
(z) - m^0_\omega
(\lambda_1))  (z-\lambda_1)^{-1}\}^{-1} \tag B.31 \\
&\quad\times \{m^0_\pm (z) + \omega\tilde\gamma_1 \psi_\omega
(\lambda_1, x_0)^2
-\omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2\\
&\quad\times [(m^0_\omega (\lambda_1) + \omega\tilde\gamma_1
\psi_\omega
(\lambda_1,
x_0)^2) (m^0_\pm (z) - m^0_\omega (\lambda_1))]
(z-\lambda_1)^{-1}\} \quad
\text{if }
m^0_\omega (\lambda_1)\ne\infty
\endalign
$$
and (B.27) by a limiting procedure if $m^0_\omega (\lambda_1)=
\infty$.
Applying (B.24) to
$m^0_\pm (z)$ and $\tilde m^0_{\omega, \gamma_1, \pm}
(z,\lambda_1)$ with
the choices
$\cot(\alpha_\omega (\lambda_1)) =- m^0_\omega  (\lambda_1)$ and
$\cot(\beta_\omega
(\lambda_1)) =\cot(\alpha_\omega (\lambda_1))
- \omega\tilde\gamma_1 \psi_\omega (\lambda_1,x_0)^2$ then
shows by a
straightforward (but somewhat painful)  computation that (B.31) is
equivalent to (B.26).
\qed
\enddemo

The singularity structure of (B.26) and (B.27) near $z=\lambda_1$
then
leads to a
corresponding pole behavior of $\tilde M^{\beta_\omega
(\lambda_1)}_{\omega, \gamma_1}
(z, \lambda_1)$, $\tilde M^0_{\omega, \gamma_1}(z, \lambda_1)$
(the $\Bbb
C^2$-valued
$M$-matrices of $\tilde H_{\omega, \gamma_1}(\lambda_1)$) when
compared to
$M^{\alpha_\omega (\lambda_1)}(z)$, $M^0 (z)$ (the $M$-matrices
of $H$). The
actual expressions for the $M$-matrices, the half-line spectral
functions,
and the
$\Bbb C^2$-valued spectral matrix of $\tilde H_{\omega,
\gamma_1}(\lambda_1)$ for
$\alpha_\omega \neq 0$, $\beta_\omega \neq 0$ in terms of those of
$H$ are
similar
to the special case $\alpha_\omega = \beta_\omega =0$ and
$m_{\omega_0}(\lambda_1)
=\infty$ described in detail in [\gjfa]. While they provide an
alternative
proof of Theorem
B.2, we resist the temptation of providing detailed formulas at
this point.
(B.26) appears to
be a new result.

Iterations of DCM can now be performed as follows. Assume $V\in
L^1_{\text{\rom{loc}}}
(\Bbb R)$ to be real-valued and pick $\omega\in \{-,+\}$,
$\gamma_j >0$,
$\lambda_j
\in\Bbb R$, $1\leq j\leq N$, $N\in\Bbb N$. Then the DCM result
after $N$
iteration steps,
denoted by $\tilde\tau_{\omega, \gamma_1, \dots, \gamma_N}
(\lambda_1, \dots,
\lambda_N)$, reads as follows.
$$\gathered
\tilde\tau_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots,
\lambda_N)
=-\frac{d^2}{dx^2} + \tilde V_{\omega, \gamma_1, \dots,
\gamma_N}(\lambda_1, \dots, \lambda_N, x), \quad x\in\Bbb R, \\
\tilde V_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots,
\lambda_N, x) =
V(x) -2\{\ln [\det (1+C_{\omega, N}(x))]\}'', \\
C_{\omega, N}(x) = \biggl(-\omega\gamma^{1/2}_k \gamma^{1/2}_\ell
\int\limits^x_{\omega\infty} dx' \,
\psi_\omega (\lambda_{k}, x')\psi_\omega
(\lambda_{\ell}, x')\biggr)_{1\leq k, \ell\leq N}.
\endgathered \tag B.32
$$
The analog of Theorem B.2 then applies to $\tilde H_{\omega, \gamma_1,
\dots, \gamma_N}(\lambda_1, \dots, \lambda_N)$, the self-adjoint
operator
associated with
$\tilde\tau_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots,
\lambda_N)$
(defined similarly to (B.15)) as discussed in detail in [\gtproc]
(see also
[\gss],
[\gsvi]).

As in Lemma 5.4 one infers from (B.31) that $\tilde H_{\omega,
\gamma_1}(\lambda_1)$
is reflectionless if and only if $H$ is (as observed in [\gnow]).

The Sturm-Liouville generalization of (B.1)--(B.14) then leads to the
following results.
One assumes
$$\gather
p^{-1}, q, k\in L^1_{\text{\rom{loc}}} ((a,b)), \quad kp\in
AC_{\text{\rom{loc}}}
((a,b)), \quad q \text{ real-valued}, \\
p>0, \ k>0, \quad -\infty\leq a < b \leq \infty
\endgather
$$
and makes the substitutions (see [\gtproc])
$$\gathered
\tau\to \widehat\tau =\frac{1}{k(x)} \biggl( -\frac{d}{dx}\,
p(x) \frac{d}{dx} +
q(x) \biggr), \\
\tilde\tau_{\omega, \gamma_1}(\lambda_1) \to
\widetilde{\widehat\tau}_{\omega,
\gamma_1}(\lambda_1) = \frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x)
\frac{d}{dx}
+ \tilde q_{\omega, \gamma_1}(\lambda_1, x)\biggr), \\
\tilde q_{\omega, \gamma_1}(\lambda_1, x) =q(x) +
\biggl(\frac{1}{k(x)} \,
(k(x)p(x))' -2\frac{d}{dx}\, p(x)\biggr) \\
\qquad \qquad \times \frac{d}{dx}\, \ln \biggl[ 1-\omega\gamma_1
\int\limits^x_{\omega\infty} k(x')dx'\,
\psi_\omega (\lambda_1, x')^2 \biggr],
\quad \omega\in \{-, +\}.
\endgathered \tag B.33
$$

It remains to sketch scattering theory for real-valued potentials
$V\in L^1
(\Bbb R;
(1+|x|)\,dx)$ similar to DDM and SCM in Section 5 and Appendix A.
(We again
use the
conventions established in (5.11)--(5.17).) Suppose $\lambda_1 <0$,
$\gamma_1 >0$.
Then we claim that DCM leaves $L^1 (\Bbb R; (1+|x|)\,dx)$ potentials
invariant as in
DDM and SCM. More precisely, we assert that $\tilde V_{\omega,
\gamma_1}
(\lambda_1)\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if
$V\in L^1 (\Bbb R;
(1+|x|)\,dx)$. Indeed, since
$$\multline
\tilde V_{\omega, \gamma_1}(\lambda_1,x) -V(x)
= \biggl[ 1-\omega\gamma_1
\int\limits^x_{\omega\infty} dx' \,
\psi_\omega (\lambda_1, x')^2\biggr] ^{-2}
2\gamma^2_1 \psi_\omega (\lambda_1, x)^4 \\
+ \bigg[ 1-\omega\gamma_1
\int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_1, x')^2
\biggr]^{-1}
4\omega\gamma_1 \psi_\omega (\lambda_1, x) \psi'_\omega (\lambda_1, x),
\quad \omega\in \{-, +\},
\endmultline \tag B.34
$$
the right-hand side of (B.34) is exponentially decreasing near
$\omega\infty$ and hence
in \linebreak $L^1 ((R, \omega\infty); (1+|x|)\,dx)$ for all
$R\in\Bbb R$.
In order to treat
(B.34) near $-\omega\infty$, one expands
$$\split
\biggl[ \gamma^{-1}_1 -\omega \int\limits^x_{\omega\infty} dx'
\psi_\omega
(\lambda_1, x')^2 \biggr]^{-1} &\underset{x\to -\omega\infty} \to =
\biggr(-\omega \int\limits^x_{\omega\infty} dx'
\psi_\omega (\lambda_1, x')^2
\biggr)^{-1}  \\
&\qquad \quad \times
\biggl[ 1+O \biggl(\biggl(-\omega \int\limits^x_{\omega\infty} dx'
\psi_\omega (\lambda_1, x')^2 \biggr)^{-1}\biggr)\biggr]
\endsplit
$$
in (B.34) and notices that $O((\dots)^{-1})$ is exponentially
decreasing as
$x\to
-\omega\infty$. The proof of our assertion is then finished by
observing
that the leading
order term in (B.34) is precisely the isospectral double commutation
deformation
corresponding to $\gamma_1 =\infty$ (cf.~Remark B.4(i)), which in
turn
corresponds to
the (sign flip) Dirichlet deformation $(\lambda_1, +\omega) \to
(\lambda_1,
-\omega)$.
The latter has been dealt with in Lemma 5.5.

The fact that DCM leaves $L^1 (\Bbb R; (1+|x|)\,dx)$ potentials
invariant
was proved
by Levitan [\levbook], Section 6.6 using a different strategy (which
yields
exponential decay
of $[\tilde V_{\omega, \gamma_1}(\lambda_1, x)-V(x)]$ also as
$x\to-\omega\infty$).

Finally, we compute the scattering matrix in this context.
Following the
arguments in
(5.11)--(5.20), one readily verifies the following expressions for the
 scattering matrix
$\tilde S_{\omega, \gamma_1}(\lambda, \lambda_1)$ of the pair $(\tilde
H_{\omega, \gamma_1}(\lambda_1), H_0)$ in terms of $S(\lambda)$
corresponding to $(H, H_0)$.
$$\align
\tilde T_{\omega, \gamma_1}(\lambda, \lambda_1) &= \frac{\lambda^{1/2}
+ i(-\lambda_1)^{1/2}}{\lambda^{1/2} - i(-\lambda_1)^{1/2}} \,
T(\lambda),
\quad
\omega\in \{-,+\}, \\
\tilde R^\ell_{\omega, \gamma_1}(\lambda, \lambda_1) &=
\cases R^\ell (\lambda), & \omega =- \\
\bigl(\tfrac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2}-
i(-\lambda_1)^{1/2}}
\bigr)^2 R^\ell (\lambda), & \omega =+ \endcases , \tag B.35 \\
\tilde R^r_{\omega, \gamma_1}(\lambda, \lambda_1) &= \cases
\bigl(\tfrac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2}-
i(-\lambda_1)^{1/2}}
\bigr)^2 R^r (\lambda), & \omega =- \\
R^r (\lambda), & \omega=+ \endcases , \ \lambda >0, \ \lambda_1\in
(-\infty, 0)\backslash
\sigma_d (H), \  \gamma_1 >0, \\
\tilde T_{\omega, \gamma_1}(\lambda, \lambda_1) &= T(\lambda),
\quad
\tilde R_{\omega, \gamma_1}\Sp \ell \\ r\endSp (\lambda,
\lambda_1) = R\Sp
\ell \\ r\endSp
(\lambda), \quad \omega\in \{-, +\}, \ \lambda >0, \,
\lambda_1\in\sigma_d
(H), \
\gamma_1 > 0. \tag B.36
\endalign
$$
The case $\gamma_1 =\infty$ is a special case of (5.19) with
$\mu=\tilde\mu=
\lambda_1$, $\sigma =-\tilde\sigma =\omega$.

The discrete analog of DCM for general second-order finite-difference
(Jacobi) operators
can be found in [\gtdiff].

As in Appendix A, we conclude with a brief account of the history of
 DCM
and references
to further applications of it. The seminal work by Gel'fand and Levitan
[\gellev] in 1955
on a solution of the inverse spectral problem seems to mark the first
appearance of DCM
where it has been used in connection with Wigner-von Neumann examples on
the half-line
$(0, \infty)$. (For a more recent treatment of the half-line case $(0,
\infty)$, see [\deift].)
Shortly afterward, the construction of reflectionless potentials in the
particular case of
$\tau=-\frac{d^2}{dx^2}$ (i.e., $V\equiv 0$), using double commutation
formulas as
a result of applying the inverse scattering approach, was derived by Kay
and Moses
[\kaymo]. Their result regained prominence when Gardner, Greene, Kruskal,
and Miura
[\ggkm] used this formalism to solve the initial value problem for
the KdV
equation
and derived the KdV $N$-soliton solutions. The case of background (base)
potentials
$V\in L^1 (\Bbb R; (1+|x|)\,dx)$ is considered at length in [\detr] and
[\levbook],
Section 6.6. The case of periodic finite-gap background potentials is
treated in
[\fir], [\flmcl], [\gsvi], and [\kumi]. Close connections between the
double commutation
technique and the inverse spectral method based on Marchenko's
approach can be
inferred, for instance, from [\fir], [\flmcl], [\ggkm], [\gss], [\kaymo],
and [\kumi].
General backgrounds were first treated in [\gsvi] (see also [\eastkalf],
Chapter 4). In
particular, the construction of KdV and mKdV soliton solutions
relative to
general
(m)KdV background solutions on the basis of (single and double)
commutation
techniques
has been systematically studied in [\gsvi]. In spite of the
widespread use
of the double
commutation method, its spectral characterization, as summarized in
Theorem
B.2, under
slightly stronger assumptions on $\tau$, was first proven only
recently in
[\gjfa].

\vskip 0.3in
\Refs
\endRefs
\vskip 0.2in

\item{\baum.}\ref{B.~Baumgartner}{Level comparison theorems}{Ann.~Phys.}
{168}{1986}{484--526}
\gap
\item{\belok.} E.D.~Belokolos, A.I.~Bobenko, V.Z.~Enol'skii,
A.R.~Its, and
V.B.~Matveev, {\it{Algebro-geometric Approach to Nonlinear Integrable
Equations}},
Springer, Berlin, 1994.
\gap
\item{\bggss.}\ref{D.~Bolle, F.~Gesztesy, H.~Grosse, W.~Schweiger,
and B.~Simon}
{Witten index, axial anomaly, and Krein's spectral shift function in
supersymmetric
quantum mechanics}{J.~Math.\ Phys.}{28}{1987}{1512--1525}
\gap
\item{\bufi.}\ref{M.~Buys and A.~Finkel}{The inverse periodic
problem for
Hill's
equation with a finite-gap potential}{J.~Diff.~Eqs.}{55}{1984}
{257--275}
\gap
\item{\crum.}\ref{M.M.~Crum}{Associated Sturm-Liouville systems}
{Quart.~J.~Math.~Oxford (2)}{6}{1955}{121--127}
\gap
\item{\dar.}\ref{G.~Darboux}{Sur une proposition relative aux
\'equations
lin\'eaires}
{C.R.~Acad.~Sci.~(Paris)}{94}{1882}{1456--1459}
\gap
\item{\deift.}\ref{P.A.~Deift}{Applications of a commutation
formula}{Duke
Math.~J.}
{45}{1978}{267--310}
\gap
\item{\detr.}\ref{P.~Deift and E.~Trubowitz}{Inverse scattering on
the line}
{Commun.~Pure Appl.~Math.}{32}{1979}{121--251}
\gap
\item{\eastkalf.} M.S.P.~Eastham and H.~Kalf, {\it{Schr\"odinger-Type
Operators
with Continuous Spectra}}, Pitman, Boston, 1982.
\gap
\item{\ehkn.}\ref{F.~Ehlers and H.~Kn\"orrer}{An algebro-geometric
interpretation
of the B\"acklund transformation of the Korteweg-de Vries equation}
{Comment.~Math.~Helv.}{57}{1982}{1--10}
\gap
\item{\erfl.}\ref{N.M.~Ercolani and H.~Flaschka}{The geometry of
the Hill
equation
and of the Neumann system}{Philos.~Trans.~Roy.~Soc.~London Ser.~A}
{315}{1985}
{405--422}
\gap
\item{\fit.}\ref{A.~Finkel, E.~Isaacson, and E.~Trubowitz}{An explicit
solution of
the inverse problem for Hill's equation}{SIAM J.~Math.~Anal.}{18}
{1987}{46--53}
\gap
\item{\fir.}\ref{N.E.~Firsova}{On solution of the Cauchy problem for the
Korteweg-de
Vries equation with initial data the sum of a periodic and a rapidly
decreasing function}
{Math.~USSR Sbornik}{63}{1989}{257--265}
\gap
\item{\flmcl.} H.~Flaschka and D.W.~McLaughlin, {\it{Some comments on
B\"acklund
transformations, canonical transformations, and the inverse scattering
method}},
Lecture Notes in Math., Vol.~515, (R.M.~Miura, ed.), pp.~252--295,
Springer, Berlin,
1976.
\gap
\item{\ggkm.}\ref{C.S.~Gardner, J.M.~Greene, M.D.~Kruskal, and
R.M.~Miura}
{Korteweg-de Vries equation and generalizations, VI. Methods for
exact solution}
{Commun.~Pure Appl.~Math.}{27}{1974}{97--133}
\gap
\item{\gellev.}\ref{I.M.~Gel'fand and B.M.~Levitan}{On the
determination of a
differential equation from its spectral function}{Amer.~Math.~Transl.
~Ser.~2}{1}
{1955}{253--304}
\gap
\item{\gdekker} F.~Gesztesy, {\it{On the modified Korteweg-de Vries
equation}} in
``Differential Equations with Applications in Biology, Physics, and
Engineering,"
(J.A.~Goldstein, F.~Kappel, and W.~Schappacher, eds.), pp.~139--183,
Marcel
Dekker, New York, 1991.
\gap
\item{\gcam.} F.~Gesztesy, {\it{Quasi-periodic, finite-gap solutions
of the
modified
Korteweg-de Vries equation}} in ``Ideas and Methods in Mathematical
Analysis,
Stochastics, and Applications," (S.~Albeverio, J.E.~Fenstad,
H.~Holden, and
T.~Lindstr\o m, eds.), Vol.~1, pp.~428--471, Cambridge Univ.~Press,
Cambridge, 1992.
\gap
\item{\gjfa.} \ref{F.~Gesztesy}{A complete spectral characterization
of the
double
commutation method}{J.~Funct.~Anal.}{117}{1993}{401--446}
\gap
\item{\gkt.} \ref{F.~Gesztesy, M.~Krishna, and G.~Teschl}{On
isospectral setsof Jacobi operators}{Commun.~Math.~Phys.}{181}{2009}{631-645}
\gap
\item{\gnow.} \ref{F.~Gesztesy, R.~Nowell, and W.~P\"otz}{One-dimensional
scattering theory for quantum systems with nontrivial spatial
asymptotics}{Differential Integral Equations}{10}{1997}{521--546}
\gap
\item{\gss.}\ref{F.~Gesztesy, W.~Schweiger, and B.~Simon}{Commutation
methods
applied to the mKdV-equation}{Trans.~Amer.~Math.~Soc.}{324}{1991}
{465--525}
\gap
\item{\gstrans.} \ref{F.~Gesztesy and B.~Simon}{Uniqueness theorems
in inverse
spectral theory for one-dimensional Schr\"odinger operators}
{Trans.~Amer.~Math.~Soc.}{348}{1996}{349--373}
\gap
\item{\gsacta.}\ref{ F.~Gesztesy and B.~Simon}{The xi function}{Acta
Math.}{176}
{1996}{49--71}
\gap
\item{\gst.} \ref{F.~Gesztesy, B.~Simon, and G.~Teschl}{Zeros of the
Wronskian and
renormalized oscillation theory}{Am.~J.~Math.}{118}{1996}{571--594}
\gap
\item{\gsvi.}\ref{F.~Gesztesy and R.~Svirsky}{(m)KdV-solitons on the
background of
quasi-periodic finite-gap solutions}{Memoirs Amer.~Math.~Soc.}{118}{1995}
{no.~563}
\gap
\item{\gtproc.} \ref{F.~Gesztesy and G.~Teschl}{On the double commutation
method}{Proc.~Amer. Math.~Soc.}{124}{1996}{1831--1840}
\gap
\item{\gtdiff.} \ref{F.~Gesztesy and G.~Teschl}{Commutation methods for
Jacobi operators}{J.~Diff.~Eqs.}{128}{1996}{252--299}
\gap
\item{\gwei.} F.~Gesztesy and R.~Weikard, {\it{Spectral deformations
and soliton
equations}} in ``Differential Equations with Applications in Mathematical
Physics,"
(W.F.~Ames, E.M.~Harrell, and J.V.~Herod, eds.), pp.~101--139,
Academic Press,
Boston, 1993.
\gap
\item{\gzh.}\ref{F.~Gesztesy and Z.~Zhao}{On critical and subcritical
Sturm-Liouville
operators}{J.~Funct. Anal.}{98}{1991}{311--345}
\gap
\item{\grma.} H.~Grosse and A.~Martin, {\it{Particle Physics and the
Schr\"odinger
Equation}}, Cambridge University Press, Cambridge, 1997.
\gap
\item{\iwa.}\ref{K.~Iwasaki}{Inverse problem for Sturm-Liouville
and Hill's
equations}
{Ann.~Mat.~Pura Appl. (4)}{149}{1987}{185--206}
\gap
\item{\jac.}\ref{C.G.J.~Jacobi}{Zur Theorie der Variationsrechnung
und der
Differentialgleichungen}{J.\ Reine Angew.~Math.}{17}{1837}{68--82}
\gap
\item{\kaymo.}\ref{J.~Kay and H.E.~Moses}{Reflectionless transmission
through
dielectrics and scattering potentials}{J.~Appl.~Phys.}{27}{1956}
{1503--1508}
\gap
\item{\kumi.}\ref{E.A.~Kuznetsov and A.V.~Mikhailov}{Stability of
stationary waves
in nonlinear weakly dispersive media}{Soviet Phys.~JETP}{40}{1975}
{855--859}
\gap
\item{\lei.}\ref{W.~Leighton}{On self-adjoint differential equations of
second order}
{J.~London Math.~Soc.}{27}{1952}{37--47}
\gap
\item{\levbook.} B.M.~Levitan, {\it{Inverse Sturm-Liouville Problems}},
VNU
Science
Press, Utrecht, 1987.
\gap
\item{\levsbor.}\ref{B.M.~Levitan}{Sturm-Liouville operators on the
whole line,
with the same discrete spectrum}{Math.~USSR Sbornik}{60}{1988}{77--106}
\gap
\item{\mar.} V.A.~Marchenko, {\it{Sturm-Liouville Operators and
Applications}},
Birkh\"auser, Basel, 1986.
\gap
\item{\mckcpam.}\ref{H.P.~McKean}{Variation on a theme of Jacobi}
{Commun.~Pure
Appl.~Math.}{38}{1985}{669--678}
\gap
\item{\mckrev.}\ref{H.P.~McKean}{Geometry of KdV {\rom{(1):}} Addition
and
the unimodular spectral classes}{Rev.~Mat.~Iberoamericana}{2}{1986}
{235--261}
\gap
\item{\mckjsp.}\ref{H.P.~McKean}{Geometry of KdV {\rom{(2):}} Three
examples}
{J.~Statist.~Phys.}{46}{1987}{1115--1143}
\gap
\item{\mckspm.} H.P.~McKean, {\it{Is there an infinite-dimensional
algebraic geometry?
Hints from KdV}} in ``Theta Functions," (L.~Ehrenpreis and
R.C.~Gunning, eds.),
Proc.~Symp. Pure Math., Vol. 49, pp.~27--37, Amer.~Math.~Soc.,
Providence,
RI, 1989.
\gap
\item{\mcknew}\ref{H.P.~McKean}{Geometry of KdV {\rom{(3):}} Determinants
and unimodular isospectral flows}{Commun.~Pure Appl.~Math.}{45}{1992}
{389-415}
\gap
\item{\mcktr.}\ref{H.P.~McKean and E.~Trubowitz}{The spectral class
of the
quantum-mechanical harmonic oscillator}{Commun.~Math.~Phys.}{82}{1982}
{471--495}
\gap
\item{\mckmoe.}\ref{H.P.~McKean and P.~van Moerbeke}{The spectrum
of Hill's
equation}{Invent.~Math.}{30}{1975}{217--274}
\gap
\item{\nmpz.} S.~Novikov, S.V.~Manakov, L.P.~Pitaevskii, and
V.E.~Zakharov,
{\it{Theory of Solitons}}, Consultants Bureau, New York, 1984.
\gap
\item{\potr.} J.~P\"oschel and E.~Trubowitz, {\it{Inverse Spectral
Theory}},
Academic Press, Boston, 1987.
\gap
\item{\ratr.}\ref{J.~Ralston and E.~Trubowitz}{Isospectral sets for
boundary value
problems on the unit interval}{Ergod.~Th.~Dynam.~Sys.}{8}{1988}
{301--358}
\gap
\item{\schm.}\ref{U.-W.~Schmincke}{On Schr\"odinger's factorization
method for
Sturm-Liouville operators}{Proc.~Roy.~Soc.~Edinburgh Sect.~A}{80}
{1978}
{67--84}
\gap
\item{\simon.} B.~Simon, {\it{Spectral analysis of rank one
perturbations
and applications}}
in CRM Proc. Lecture Notes, Vol. 8 (J.~Feldman, R.~Froese, and
L.~Rosen, eds.),
pp.~109--149, Amer.~Math. Soc., Providence, RI, 1995.
\gap
\item{\te.} G.~Teschl, {\it{Oscillation and renormalized
oscillation theory
for Jacobi
operators}}, J. Diff. Eq. {\bf 129}, 532--558 (1996).
\gap
\item{\tpre.} G.~Teschl, {\it{Spectral deformations of
Jacobi operators}},
J. Reine und Angew. Math. {\bf 491}, 1--15 (1997).

\enddocument
