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\copyrightinfo{1996}{American Mathematical Society}
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\commby{Palle E. T. Jorgensen}
\date{December 8, 1994}

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\begin{document}

\title{On the Double Commutation Method}

\author{F. Gesztesy}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211, USA}
\email{gesztesyf@missouri.edu}

\author{G. Teschl}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211, USA and Department of Theoretical Physics, Technical
University of Graz, Graz, 8010, Austria}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\keywords{Commutation methods, Sturm-Liouville operators, eigenvalues}
\subjclass{Primary 34B24, 34L05; Secondary 34B20, 47A10}

\begin{abstract}
We provide a complete spectral characterization of the double commutation
method for general Sturm-Liouville operators which inserts any finite number
of prescribed eigenvalues into spectral gaps of a given background operator.
Moreover, we explicitly determine the transformation operator which links the
background operator to its doubly  commuted version (resulting in extensions and
considerably simplified proofs of spectral results even for the special case of
Schr\"odinger-type operators).
\end{abstract}

\maketitle




\section{Introduction}



Methods of inserting (and removing) eigenvalues in spectral gaps of a given
one-dimensional Schr\"odinger operator $H$ associated with differential
expressions of the type $-\frac{d^2}{dx^2} + q$ on $(a,\infty)$, $a \ge -\infty$
have recently attracted an enormous amount of attention. This is due to their
prominent role in diverse fields such as the inverse scattering approach
introduced by Deift and Trubowitz \cite{dt}, supersymmetric quantum mechanics
(see, e.g$.$, \cite{gss} and the references therein), level comparison theorems
(cf$.$ \cite{ba} and the literature cited therein), as a tool to construct
soliton solutions of the Korteweg-de Vries (KdV) hierarchy relative to (general)
KdV  background solutions (see, e.g$.$, \cite{bs}, \cite{de},
\cite{dt}, \cite{eak}, Ch. 4, \cite{fi}, \cite{gg}, \cite{gs}, \cite{gss},
\cite{km}-\cite{kum}, \cite{lv}, Sect. 6.6, \cite{rs}--\cite{sh2}), and in
connection with B\"acklund transformations for the KdV hierarchy (cf$.$, e.g$.$,
\cite{ek}, \cite{ef}, \cite{fm}, \cite{gs}, \cite{gu}, \cite{gw},
\cite{gss}, \cite{mk1}, \cite{mk2}, \cite{wa}).

Historically, these methods of inserting eigenvalues go back to Jacobi
\cite{ja} and Darboux \cite{da} with decisive later contributions by Crum
\cite{cr}, Schmincke \cite{sc}, and, especially, Deift \cite{de}. Two
particular methods, shortly to be discussed in an informal manner in
(\ref{tau})--(\ref{qdc}) below, turned out to be of special importance: The
single commutation method, also called the Crum-Darboux method \cite{cr},
\cite{da} (actually going back at least to Jacobi \cite{ja}) and the double
commutation method, to be found, e.g$.$, in the seminal work of Gel'fand and
Levitan \cite{gl}.

The single commutation method, although very simply implemented, has the
distinct disadvantage of relying on positivity properties of certain solutions 
of $H \phi = \lam \phi$ which confines its applicability to the insertion of
eigenvalues below the spectrum of $H$ (assuming $H$ to be bounded from below).
A complete spectral characterization of this method has been provided by Deift
\cite{de} (see also \cite{sc}) on the basis of unitary equivalence of $A^*
A|_{Ker(A)^\perp}$ and $AA^*|_{Ker(A^*)^\perp}$ for a densely defined closed
linear operator $A$ in a (complex, separable) Hilbert space.

The double commutation method on the other hand, allows one to insert
eigenvalues into {\em any} spectral gap of $H$. Although relatively simply
implemented also, a complete spectral characterization of the double commutation 
method for Schr\"odinger-type operators was only very recently achieved in
\cite{fg} on the basis of Weyl-Titch\-marsh $m$-function techniques.
In this note we follow up on the double commutation method and provide, in
particular,
\begin{itemize}
\item a complete spectral analysis of the double commutation method for general
Sturm-Liouville operators on arbitrary intervals $(a,b)$, $-\infty\le a < b \le
\infty$
\item a functional analytic approach which not only avoids the Weyl-Titchmarsh
theory employed in \cite{fg}, but in addition, considerably simplifies and
streamlines the corresponding proofs in \cite{fg}. 
\end{itemize}
We emphasize that our formulation of the double commutation method for general
Sturm-Liouville (SL) operators appears to be without precedent.

Before starting our analysis in Section 2, we informally describe the single and
double commutation method for general Sturm-Liouville differential expressions.
Consider the differential expression ($p,p',k,k' \in AC_{loc}((a,b))$, $p,k>0$,
$q \in L^1_{loc}((a,b))$, $-\infty\le a < b \le
\infty$)
\begin{equation} \label{tau}
\tau = \frac{1}{k} \Big(- \frac{d}{dx} p \frac{d}{dx} + q \Big),
\end{equation}
and introduce
\begin{equation}
a = \frac{1}{k} \Big( \sqrt{kp}\frac{d}{dx} + \phi \Big),
\quad a^* = \frac{1}{k} \Big(- \frac{d}{dx}\sqrt{kp} + \phi \Big),
\end{equation}
where $\phi=(\sqrt{kp}\psi')/ \psi$ and $\psi>0$ satisfies $\tau \psi = \lam_1
\psi$. A straightforward calculation reveals
\begin{equation}
\tau = a^* a +\lam_1, \quad \hat{\tau} = a a^* +\lam_1 = \frac{1}{k} \Big(-
\frac{d}{dx} p \frac{d}{dx} + \hat{q} \Big),
\end{equation}
where
\begin{equation} \label{qsc}
\hat{q} = q - \frac{p''}{2} + \frac{(p')^2}{4p} + \frac{3(k')^2p}{4k^2} -
\frac{k''p}{2k} +\Big( \frac{(kp)'}{k} - 2 \frac{d}{dx} p\Big) \frac{d}{dx} \ln
\psi .
\end{equation}
Thus (taking proper domain considerations into account) we can define two
operators $H,\hat{H}$ on $L^2((a,b);k dx)$ associated with $\tau,\hat{\tau}$
which are unitarily equivalent when restricted to the orthogonal complement of
the eigenspaces corresponding to $\lam_1$, \cite{de} (cf.\ also \cite{sc}).
Moreover, $H-\lam_1, \hat{H}-\lam_1\ge 0$ which is equivalent to the existence of
the positive solution $\psi$ \cite{gz}. Formulas (\ref{tau})--(\ref{qsc})
constitute the single commutation method for general SL differential
expressions as discussed, e.g$.$, by Schmincke \cite{sc}. Next we assume that
$\psi$ is square integrable near $a$ and  consider two more expressions
$a_{\gam_1},a_{\gam_1}^*$ as above with $\psi_{\gam_1} = \psi/(1+\gam_1 \int_a^x
k(t) \psi(t)^2 dt)$. This implies
\begin{equation} \label{dc}
a a^* = a_{\gam_1}^* a_{\gam_1}, \quad \tau_{\gam_1} = a_{\gam_1} a_{\gam_1}^*
+ \lam_1 = \frac{1}{k} \Big(- \frac{d}{dx} p \frac{d}{dx} + q_{\gam_1} \Big),
\end{equation}
where
\begin{equation} \label{qdc}
q_{\gam_1} = q + \Big( \frac{(kp)'}{k} - 2 \frac{d}{dx} p\Big) \frac{d}{dx} \ln
\Big(1+\gam_1 \int_a^x k(t) \psi(t)^2 dt \Big).
\end{equation}
Now observe that $q_{\gam_1}$ is well defined even if $\psi$ has zeros, and
hence we expect $H,H_{\gam_1}$ (a SL operator associated with $\tau_{\gam_1}$)
to be closely related even in the case where all intermediate operators are
ill-defined. This was first shown in \cite{fg} for the special case of
Schr\"odinger-type operators where $k=p=1$. We shall prove this fact  for
general SL operators by explicitly computing a transformation operator for the
pair $\tau, \tau_{\gam_1}$ which, when restricted to the orthogonal complement
of the eigenspace of $\lam_1$, turns out to be unitary. Formulas
(\ref{tau})-(\ref{qdc}) sum up the double commutation method for general SL
differential expressions.

Our results are relevant in connection with  inverse scattering theory in
nonhomogeneous media (see, e.g$.$, \cite{ak} and the references cited therein)
and yield a direct construction of $N$-soliton solutions relative to arbitrary
background solutions of the (generalized) KdV hierarchy along the
methods of \cite{gs} (see also \cite{gu}).



\section{Construction of a transformation operator}
\setcounter{equation}{0}
\setcounter{thm}{0}


Let $k \in L^1_{loc}((a,b))$ with $k>0$. We pick $\hr = L^2((a,b);k dx)$ to be the underlying Hilbert space ($-\infty \le a < b \le \infty$) and define for $c \in
(a,b)$,
\begin{equation}
L^2_{loc}([a,b);k dx) = \{ f \in L^2_{loc}((a,b);k dx) | f \in L^2((a,c);k dx)
\}.
\end{equation}
Next, choose a positive number $\gam>0$, consider a fixed element $\psi \in
L^2_{loc}([a,b);k dx)$, and define the following (linear) transformation
\begin{equation} \label{defu}
U_\gam : \bay[t]{lcl} L^2_{loc}([a,b);k dx) &\to& L^2_{loc}([a,b);k dx)\\
f(x) &\mapsto& f_\gam(x) = f(x) - \gam \psi_\gam(x)
\int_a^x k(t) \ol{\psi(t)} f(t) dt, \eay
\end{equation}
where $\psi_{\gam}$ is defined by
\begin{equation}
\psi_\gam(x) = \frac{\psi(x)}{1 + \gam \int_a^x k(t) |\psi(t)|^2 dt}.
\end{equation}
By inspection, the inverse transformation is given by
\begin{equation}
U_\gam^{-1} : \bay[t]{lcl} L^2_{loc}([a,b);k dx) &\to& L^2_{loc}([a,b);k
dx)\\ g(x) &\mapsto& g(x) + \gam \psi(x) \int_a^x k(t)
\ol{\psi_\gam(t)} g(t) dt. \eay
\end{equation}
The restriction of $U_\gam$ to $\hr $ will be denoted by  $U_\gam$
as well. Note that we have
\begin{equation}
1 - \gam \int_a^x k(t) |\psi_\gam(t)|^2 dt = (1 + \gam \int_a^x
k(t) |\psi(t)|^2 dt)^{-1}.
\end{equation}

\begin{lem} \label{lemun}
The element $\psi_\gam$ fulfills
\begin{equation}
\psi_\gam \in \hr, \qquad \| \psi_\gam \|^2 = \frac{1}{\gam} 
\Big(1 -\lim_{x \to b} (1 + \gam \int_a^x k(t) |\psi(t)|^2 dt)^{-1} \Big).
\end{equation}
Denote by $P,P_\gam$ the orthogonal projections onto the one
dimensional subspaces of $\hr $ spanned by $\psi, \psi_\gam$ (set 
$P = 0$ if $\psi \not\in \hr $), then the operator $U_\gam$ is
unitary from $(1-P) \hr $ onto $(1-P_\gam) \hr $.
\end{lem}

\begin{proof}
The claims concerning $\psi_\gam$ are straightforward. Next we note that for
$x\in(a,b)$ and $f,f_\gam$ as in (\ref{defu}),
\begin{equation}
\int_a^x k(t) \ol{\psi_\gam(t)}  f_\gam(t) dt = \frac{\int_a^x k(t)
\ol{\psi(t)} f(t) dt}{1 + \gam\int_a^x k(t)|\psi(t)|^2 dt}.
\end{equation}
A direct calculation then shows
\begin{equation} \label{un1}
\int_a^x k(t)|f_\gam(t)|^2 dt = \int_a^x k(t)|f(t)|^2 dt - \gam \frac{
|\int_a^x k(t) \ol{\psi(t)} f(t) dt|^2}{1 + \gam\int_a^x
k(t)|\psi(t)|^2 dt},
\end{equation}
which proves the lemma if $\psi \in \hr$. Otherwise, consider $U_\gam$,
$U_\gam^{-1}$ on the dense subspace of square integrable elements with 
compact support in $(a,b)$ and take closures (cf$.$, e.g$.$, \cite{wd}, Theorem~6.13).
\end{proof}

Using, e.g$.$, the polarization identity, we obtain in addition
\bea \nn
\int_a^x k(t) \ol{g_\gam(t)} f_\gam(t) dt &=& \int_a^x k(t)
\ol{g(t)} f(t) dt \\ \label{sprgam} && {}- \gam \frac{\int_a^x k(t)
\ol{\psi(t)} f(t) dt \int_a^x k(t) \psi(t) \ol{g(t)} dt}{1 + \gam \int_a^x k(t)
|\psi(t)|^2 dt}.
\eea


\section{Inserting a single eigenvalue}
\setcounter{equation}{0}
\setcounter{thm}{0}


Now we turn to the SL differential expression
\begin{equation} \label{stli}
\tau = \frac{1}{k} \Big(- \frac{d}{dx} p \frac{d}{dx} + q \Big),
\end{equation}
where the coefficients $p,q,k$ are real-valued
satisfying
\begin{equation}
p^{-1},q,k \in L^1_{loc}((a,b)), \quad k>0, \quad kp \in  AC_{loc}((a,b)).
\end{equation}
We are interested in self-adjoint operators $H$ associated with $\tau$ and
separated boundary conditions.

\bigskip
\noindent{\bf Hypothesis (H.3.1).}\quad\addtocounter{thm}{1}
Let $\lam \in \R$ and suppose $\psi$ satisfies the following conditions.
\begin{list}{(\roman{me}).}{\usecounter{me}\leftmargin9mm}
\item $\psi, p\psi' \in AC_{loc}((a,b))$ and $\psi$
is a real-valued solution of $\tau \phi = \lam \phi$.
\item $\psi$ is square integrable near $a$ and fulfills the
boundary condition (of $H$) at $a$ and $b$ if any (i.e$.$, if $\tau$ is limit
circle ($l.c.$) at $a$ respectively $b$).
\end{list}

Sufficient conditions for the existence of a function $\psi$ satisfying
(H.3.1) are
\begin{list}{(\alph{me}).}{\usecounter{me}\leftmargin9mm}
\item $\lam_1 \in \sigma_{p}(H)$ ($\sigma_{p}(.)$ the point
spectrum, i.e$.$, the set of eigenvalues), or
\item $\tau$ is $l.c.$ at $a$ but not at $b$, or
\item $\sigma(H) \ne \R$ (and $\lam \in \R \bs \sigma(H)$), or
\item $\sigma(H^c_-) \ne \R$ (and $\lam \in \R \bs \sigma(H^c_-)$)
where $H^c_-$ is a restriction of $H$ to $L^2((a,c))$ with $c \in (a,b)$
(finite) and, e.g$.$, a Dirichlet boundary condition at $c$.
\end{list}

If $\lam_1\in \R$ and $\psi(\lam_1,.)$ obeys (H.3.1), it follows that $H$ is
explicitly given by
\begin{equation} \label{bgop}
H f = \tau f,  \qquad \db(H) = \bay[t]{l} \{ f \in \hr |\, f,pf' \in
AC_{loc}((a,b)) ; \tau f \in \hr  ;\\ W_a(\psi(\lam_1),f)=0
\mbox{ if $\tau$ is $l.c.$  at $a$}\\ W_b(\psi(\lam_1),f)=0
\mbox{ if $\tau$ is $l.c.$  at $b$}\} \eay
\end{equation}
with $W_x(u,v) = p(x) (u(x)v(x)' - u(x)' v(x) )$ the (modified)
Wronskian of $u,v \in AC_{loc}((a,b))$.

We now use Lemma \ref{lemun} with $\psi=\psi(\lam_1)$, $\gam=\gam_1$ to prove

\begin{thm} \label{thm}
Let $\lam_1 \in \R$ and $\psi(\lam_1,.)$ be a solution satisfying (H.3.1).
Define $U_{\gam_1}$, $P(\lam_1)$, $P_{\gam_1}(\lam_1)$ as in (\ref{defu}) and
Lemma \ref{lemun} in terms of $\psi(\lam_1,.)$ and set $\psi_{\gam_1}(\lam_1,.)
= U_{\gam_1} \psi(\lam_1,.)$. Then the operator $H_{\gam_1}$ defined by
\begin{equation}
H_{\gam_1} f = \tau_{\gam_1} f, \quad \db(H_{\gam_1}) = \bay[t]{l} \{ f \in 
\hr |\, f,pf' \in AC_{loc}((a,b)) ; \tau_{\gam_1} f \in \hr  ;\\
W_a(\psi_{\gam_1}(\lam_1),f)=W_b(\psi_{\gam_1}(\lam_1),f)=0 \}, \eay
\end{equation}
with
\begin{equation}
q_{\gam_1} = q + \Big( \frac{(kp)'}{k} - 2 \frac{d}{dx} p \Big)
\frac{d}{dx} \ln\Big(1 + \gam_1 \int_a^x \! k(t) \psi(\lam_1,t)^2 dt\Big)
\end{equation}
is self-adjoint. Moreover, $H_{\gam_1}$ has the eigenfunction
$\psi_{\gam_1}(\lam_1) = U_{\gam_1} \psi(\lam_1)$ associated with the
eigenvalue $\lam_1$. If $\psi(\lam_1) \not\in \hr $ (and hence
$\tau$ is limit point ($l.p.$) at $b$) we have
\begin{equation}
 H_{\gam_1} (1-P_{\gam_1}(\lam_1)) = U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1))
\end{equation}
and thus
\begin{equation}
\bay{rcl@{\qquad}rcl}
\sigma(H_{\gam_1}) &=& \sigma(H) \cup \{ \lam_1\}, & \sigma_{ac}(H_{\gam_1})
&=& \sigma_{ ac}(H), \\  \sigma_{p}(H_{\gam_1}) &=& \sigma_{p}(H) \cup \{
\lam_1\}, & \sigma_{sc}(H_{\gam_1}) &=& \sigma_{sc}(H).
\eay 
\end{equation}
(Here $\sigma_{ac}(.),\sigma_{sc}(.)$ denotes the absolutely and singularly
continuous spectrum, respectively.) If $\psi(\lam_1) \in \hr $ there
is a unitary operator $\ti{U}_{\gam_1} = U_{\gam_1} \oplus
\sqrt{1+\gam_1 \| \psi(\lam_1) \|^2} U_{\gam_1}$ on $(1-P(\lam_1)) \hr 
\oplus P(\lam_1) \hr $ such that
\begin{equation}
H_{\gam_1} = \ti{U}_{\gam_1} H \ti{U}_{\gam_1}^{-1}.
\end{equation}
\end{thm}

\begin{proof}
It suffices to prove $ H_{\gam_1} (1-P_{\gam_1}(\lam_1)) = U_{\gam_1} H U_{\gam_1}^{-1}
(1-P_{\gam_1}(\lam_1))$. Let $f$ be square integrable near $a$ (with $f,pf' \in
AC_{loc}((a,b))$) such that $\tau f$ is also square integrable near
$a$ and $f$ fulfills the boundary condition at $a$ (if any). Then a
straightforward calculation shows
\begin{equation}
\tau_{\gam_1} (U_{\gam_1} f) = U_{\gam_1} (\tau f)
\end{equation}
and we only have to check the boundary conditions at $a$ and $b$. Equation
(\ref{un1}) shows that $\tau_{\gam_1}$ is $l.c.$ at $a$, respectively $b$, if
$\tau$ is. The formula
\begin{equation} \label{wrongamp}
W_x(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = \frac{W_x(\psi(\lam_1),f)}{1 + \gam_1
\int_a^x k(t)\psi(\lam_1,t)^2 dt}
\end{equation}
reveals $W_a(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) =0$ for $f \in \db(H)$.
Furthermore, we claim that
\begin{equation}
W_b(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = 0, \qquad f \in \db(H).
\end{equation}
This is clear if $\psi(\lam_1) \in \hr$, otherwise, i.e., if  
$\psi(\lam_1) \not\in \hr $ we use
\begin{equation} \label{wrbc}
|W_x(\psi_{\gam_1}(\lam_1),U_{\gam_1} f)|^2 = \frac{|\int_a^x k(t)\psi(\lam_1,t) (\lam_1 - \tau)
f(t) dt|^2}{(1 + \gam_1 \int_a^x k(t)\psi(\lam_1,t)^2 dt)^2}.
\end{equation}
The right hand side of (\ref{wrbc}) tends to zero for $ f \in \db(H)$ as can be
seen from (\ref{un1}) and the fact that $U_{\gam_1}$ is unitary. Combining the
last results yields
\begin{equation}
(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H) \subseteq (1-P_{\gam_1}(\lam_1))
\db(H_{\gam_1}).
\end{equation}
But $(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H)$  cannot be properly contained in
$(1-P_{\gam_1}(\lam_1)) \db(H_{\gam_1})$ by the property of self-adjoint
operators being maximal.
\end{proof}

\begin{rem} \label{remrm}
(i) If $H$ has an eigenfunction $\psi(\lam_1)$ we can remove this
eigenfunction from the spectrum upon choosing $\gam_1 = - \| \psi(\lam_1) \|^{-2}$.
Then the corresponding element $\psi_{\gam_1}(\lam_1)$ is not in $\hr$, implying
that $\tau_{\gam_1}$ is $l.p.$ at $b$.\\
(ii) The double commutation method has the pleasant feature of leaving $p$
invariant. This is in sharp contrast to the double commutation method for Jacobi
operators \cite{gt} (the finite difference analog for (\ref{stli})).\\
(iii) Let $u$ (with $u,pu' \in AC_{loc}((a,b))$)
fulfill $\tau u = z u$ (with $z \in \C \bs \{ \lam_1 \}$) and let
\begin{equation}
v(z,x) = u(z,x) + \frac{\gam_1}{z-\lam_1} \psi_{\gam_1}(\lam_1,x)
W_x(\psi(\lam_1),u(z)).
\end{equation}
Then $v,pv' \in AC_{loc}((a,b))$ and $v$ fulfills $\tau_{\gam_1} v = z v$.
If $u$ is square integrable near $a$ and fulfills the boundary condition at
$a$ (if any) we have $v=U_{\gam_1}u$. We also note
\begin{equation} \label{normv}
|v(z,x)|^2 = |u(z,x)|^2 - \frac{\gam_1 k(x)^{-1}}{|z-\lam_1|^2} \frac{d}{dx}
\left( \frac{|W_x(\psi(\lam_1),u(z))|^2}{1 + \gam_1 \int_a^x k(t)
\psi(\lam_1,t)^2 dt} \right),
\end{equation}
and if $\hat{u}$, $\hat{v}$ are constructed analogously then
\bea \nn
W_x(v(z),\hat{v}(\hat{z})) &=& W_x(u(z),\hat{u}(\hat{z}))
- \frac{\gam_1}{1 + \gam_1 \int_a^x k(t) \psi(\lam_1,t)^2 dt} \times \\ &&
\frac{z-\hat{z}}{(z-\lam_1)(\hat{z}-\lam_1)} 
W_x(\psi(\lam_1),u(z))W_x(\psi(\lam_1),\hat{u}(\hat{z})).
\eea
(iv) Writing $U_{\gam_1} f$ as
\begin{equation}
(U_{\gam_1}f)(x) =  f(x) - \int_a^x \frac{\gam_1 k(t) \psi(\lam_1,x) \psi(\lam_1,t)}{1 + \gam_1 \int_a^x k(s) \psi(\lam_1,s)^2 ds} f(t) dt,
\end{equation}
we see that $U_{\gam_1}$ is the transformation operator for $H,H_{\gam_1}$ in
the terminology of \cite{lv} and \cite{mc}.\\
(v) The limiting case $\gam=\infty$ can be handled analogously producing an
unitarily equivalent operator.
\end{rem}

Finally we discuss conditions for $\tau_{\gam_1}$ to be $l.p.$ at
$a,b$. Let $c  \in (a,b)$ and let $H^c_+$ denote a self-adjoint operator 
associated with $\tau$ on $(c,b)$ and the boundary condition induced by
$\psi(\lam_1)$ at $c$ (i.e$.$, $W_c(f,\psi(\lam_1))=0$, $f \in \db(H^c_+)$).

\bigskip
\noindent{\bf Hypothesis (H.3.4).}\quad\addtocounter{thm}{1}
Suppose one of the following spectral conditions (i)--(iii) holds.
\begin{list}{(\roman{me}).}{\usecounter{me}\leftmargin9mm}
\item $\sig_{ess} (H^c_+) \ne \emptyset$.
\item $\sig(H^c_+)=\sig_d(H^c_+)=\{\lam_{n}\}_{n\in \Z}$ with $\sum_{n\in
\Z} (1+\lam^2_{n})^{-1}=\infty$.
\item $H^c_+$ is bounded from below and $\int^b |k(x)/p(x)|^{1/2} dx = \infty$.
\end{list}

All conditions (i)--(iii) imply that $\tau$ is $l.p.$ at $b$. This is clear
for (i),(ii) since if $H^c_+$ were $l.c.$ at $b$ its resolvent would be a
Hilbert-Schmidt operator contradicting (i),(ii). For (iii) this follows from
\cite{fg}, Lemma C.1.

\begin{thm} \label{thmlp}
\begin{list}{(\roman{me}).}{\usecounter{me}\leftmargin2mm}
\item $\tau_{\gam_1}$ is $l.p.$ at $a$ if and only if $\tau$ is and $l.c.$ at $b$
if $\tau$ is.
\item Assume (H.3.4), then $\tau_{\gam_1}$ is $l.p.$ at $b$.
\end{list}
\end{thm}

\begin{proof}
(i) follows from (\ref{normv}). For (ii) consider the doubly  commuted operator
$H^c_{+,\gam_1^c}$ of $H^c_+$, where $\gam_1^c = \gam_1/(1+\gam_1 \int_a^c
\psi(\lam_1,t)^2 dt)$. Then $\tau_{\gam_1}|_{(c,b)} = \tau_{\gam_1^c}^c$ and
$H^c_{+,\gam_1^c}$ also satisfies (H.3.4). Hence $\tau_{\gam_1}$ is $l.p.$ at
$b$ as claimed.
\end{proof}

\begin{rem}
(i) Removing an eigenvalue (cf.\ Remark \ref{remrm} (i)) from an operator which
is $l.c.$ at $b$ yields an operator which is $l.p.$. Thus $\tau_{\gam_1}$ is not
necessarily $l.p.$ if $\tau$ is. Moreover, this shows that one cannot insert
additional eigenvalues into an operator which is $l.c.$ at $b$ (remove this
eigenvalue again to obtain a contradiction).\\
(ii) Clearly we can interchange the role of $a$ and
$b$. One only has to substitute $a \to b$ in the text and $\int_a \to \int^b$ in
the formulas.\\
(iii) As long as $\tau$ is $l.c.$ at $a$ our method can be used to insert
additional eigenvalues into the spectrum of $H$. In the special case of
Schr\"odinger operators where $k=p=1$ this has first been observed by Gel'fand
and Levitan \cite{gl} in connection with Wigner-von Neumann examples. More
recent studies can be found in \cite{em}, \cite{rc}, \cite{th} (see also
\cite{eak}, Section 4.4).\\
(iv) Theorems \ref{thm} and \ref{thmlp} have first been derived in \cite{fg}
for the special case $p=k=1$ and under somewhat more restrictive spectral
conditions (such as $\lam_1 \in \R \bs \sigma(H)$).
\end{rem}


\begin{thm}
Assume (H.3.1) and let $m_\pm(z,\alpha)$,  $m_{\pm,\gam_1}(z,\alpha)$ denote the
Weyl $m$-functions of $H$, $H_{\gam_1}$, respectively associated with the boundary
condition $\sin(\alpha) f(c) + \cos(\alpha) (pf')(c) =0$, $c\in (a,b)$. Then we have
\begin{equation}
m_{\pm,\gam_1}(z,\ti{\alpha}) =  \frac{1+\ti{\beta}^2}{1+\beta^2}
\Big( m_\pm(z,\alpha) - \frac{\ti{\gam}_1}{z - \lam_1}  + \beta \Big) -\ti{\beta},
\end{equation}
where $\beta = \cot(\alpha)$,
\begin{equation}
\ti{\gam}_1 = \frac{\gam_1}{1+\gam \int_a^c k(t) \psi(\lam_1,t)^2 dt}, 
\quad \ti{\beta} = \cot(\ti{\alpha}) = \cot(\alpha) + \ti{\gam} \sin^2(\alpha).
\end{equation}
\end{thm}

\begin{proof}
Consider the sequences
\begin{equation}
\phi_{\alpha,\gam_1}(z,x), \quad \theta_{\alpha,\gam_1}(z,x) + \ti{\gam}_1 \Big(
\frac{1}{z-\lam_1} + \sin^2(\alpha) \frac{1-\beta \ti{\beta}}{1+\ti{\beta}^2} \Big)
\phi_{\alpha,\gam_1}(z,x)
\end{equation}
constructed from the fundamental system $\theta_\alpha(z,x)$,
$\phi_\alpha(z,x)$ for $\tau$, that is,
\begin{equation}
\theta_\alpha(z,c) = p(c)\phi_\alpha'(z,c) = \cos(\alpha), \quad
p(c)\theta_\alpha'(z,c) = -\phi_\alpha(z,c) = \sin(\alpha).
\end{equation}
They form a fundamental system for $\tau_{\gam_1}$
corresponding to the initial conditions associated with
$\ti{\alpha}$ up to constant multiples. Now use (\ref{wrongamp}) to
evaluate 
\bea \nn
m_{\pm,\gam_1}(z,\ti{\alpha}) &=& -\lim_{x \to \genfrac{}{}{0pt}{}{b}{a}}
\frac{W_x(\psi_{\gam_1}(\lam_1),\theta_{\gam_1,\ti\alpha}(z))}{W_x(\psi_{\gam_1}(\lam_1),
\phi_{\gam_1,\ti\alpha}(z))}\\
&=& - \frac{1+\ti{\beta}^2}{1+\beta^2}
\frac{W_x(\psi_{\gam_1}(\lam_1),\theta_{\alpha,\gam_1}(z) + \ti{\gam}_1 (\dots) 
\phi_{\alpha,\gam_1}(z))}{W_x(\psi_{\gam_1}(\lam_1),\phi_{\alpha,\gam_1}(z))},
\eea
where $\theta_{\gam_1,\ti\alpha}(z,x)$, $\phi_{\gam_1,\ti\alpha}(z,x)$ is the
fundamental system  for $\tau_{\gam_1}$ corresponding to $\ti\alpha$ at $c$.
\end{proof}










\section{Inserting finitely many eigenvalues}
\setcounter{equation}{0}
\setcounter{thm}{0}


Finally we demonstrate how to iterate this method. We choose a given
background operator $H$ (with coefficients $k, p, q$) and pick
$\gam_1>0$, $\lam_1 \in \R$. Now choose $\psi(\lam_1)$, as in Section 3 to
define the transformation $U_1$ and the operator $H_{\gam_1}$. Next we choose
$\gam_2>0$, $\lam_2 \in \R$ and another function $\psi(\lam_2)$ to define
$\psi_{\gam_1}(\lam_2) = U_{\gam_1} \psi(\lam_2)$ and corresponding
operators $U_{\gam_1,\gam_2}$ and $H_{\gam_1,\gam_2}$. Applying this
procedure $N$ times leads to

\begin{thm} \label{dcommN}
Let $H$ be the background operator (\ref{bgop}) and let $ \gam_j>0, \:  
\lam_j \in \R$, $1 \le j \le N$, be such that there exist corresponding solutions
$\psi(\lam_j,x)$ of $\tau \phi = \lam_j \phi$ satisfying (H.3.1). We set
$\psi_{\gam_1,\dots,\gam_\ell}(\lam_j) = U_{\gam_1,\dots,\gam_\ell}
\cdots U_{\gam_1} \psi(\lam_j)$ and define the following matrices ($1 \le \ell
\le N$)
\begin{equation}
C_\ell(x) =  \left\{ \delta_{r,s} + \sqrt{\gam_r
\gam_s} \int\limits_a^x k(t) \psi(\lam_r,t) \psi(\lam_s,t) dt \right\}_{1 \le
r,s \le \ell},
\end{equation}
\begin{equation}
\label{cij} C_{\ell;i,j}(x) = \left\{ \bay{c@{\quad}l}
C_{\ell-1}(x)_{r,s}, & {\scriptstyle r,s \le \ell-1}\\
\sqrt{\gam_s} \int\limits_a^x k(t) \psi(\lam_i,t) \psi(\lam_s,t) dt, & 
{\scriptstyle s\le \ell-1, r=\ell}\\ \sqrt{\gam_r} \int\limits_a^x k(t)
\psi(\lam_r,t) \psi(\lam_j,t) dt, & {\scriptstyle r \le \ell-1, s=\ell}\\
\int\limits_a^x k(t) \psi(\lam_i,t) \psi(\lam_j,t) dt, & {\scriptstyle
r=s=\ell} \eay \right\}_{1 \le r,s \le \ell} ,
\end{equation}
\begin{equation}
\Psi_\ell(\lam_j,x) = \left\{ \bay{c@{\quad}l}
C_\ell(x)_{r,s}, & {\scriptstyle r,s \le \ell}\\ \sqrt{\gam_s} \int\limits_a^x
k(t) \psi(\lam_j,t) \psi(\lam_s,t) dt, &  {\scriptstyle s \le \ell, r=\ell+1}\\
\sqrt{\gam_r} \psi(\lam_r,x), & {\scriptstyle r \le \ell, s=\ell+1}\\
\psi(\lam_j,x), & {\scriptstyle r=s=\ell+1}
\eay \right\}_{1 \le r,s \le \ell+1} .
\end{equation}
Then we have (set $C_0(x)=1$)
\begin{equation} \label{cln}
1 + \gam_\ell \int_a^x k(t) \psi_{\gam_1,\dots,\gam_{\ell-1}}(\lam_\ell,t)^2 dt =
\frac{\det C_\ell(x)}{\det C_{\ell-1}(x)}
\end{equation}
and
\begin{equation}
q_{\gam_1,\dots,\gam_N} =  q + \Big( \frac{(kp)'}{k} - 2 \frac{d}{dx} p\Big)
\frac{d}{dx} \ln \det C_N.
\end{equation}
Furthermore, we obtain
\begin{equation} \label{sprl}
\int_a^x k(t) \psi_{\gam_1,\dots,\gam_{\ell-1}}(\lam_i,t) 
\psi_{\gam_1,\dots,\gam_{\ell-1}}(\lam_j,t) dt =
\frac{\det C_{\ell;i,j}(x)}{\det C_{\ell-1}(x)}
\end{equation}
and
\begin{equation} \label{phil}
\psi_{\gam_1,\dots,\gam_\ell}(\lam_j,x) = \frac{\det
\Psi_\ell(\lam_j,x)}{\det C_\ell(x)}.
\end{equation}
The spectrum of $H_{\gam_1,\dots,\gam_N}$ is given by
\begin{equation}
\bay{rcl@{\quad}rcl}
\sigma(H_{\gam_1,\dots,\gam_N}) &=& \sigma(H) \cup \{ \lam_j\}_{j=1}^N, &
\sigma_{ac}(H_{\gam_1,\dots,\gam_N}) &=& \sigma_{ ac}(H),\\ \sigma_{p}(H_{\gam_1,\dots,\gam_N}) &=& \sigma_{p}(H) \cup \{ \lam_j \}_{j=1}^N , &  \sigma_{sc}(H_{\gam_1,\dots,\gam_N}) &=& \sigma_{sc}(H).
\eay 
\end{equation}
Moreover,
\bea \nn
&& H_{\gam_1,\dots,\gam_N} (1-\sum\limits_{j=1}^N
P_{\gam_1,\dots,\gam_N}(\lam_j))
\\ && \quad = (U_{\gam_1,\dots,\gam_N} \cdots U_{\gam_1}) H  (U_{\gam_1}^{-1}
\cdots U_{\gam_1,\dots,\gam_N}^{-1}) (1- \sum\limits_{j=1}^N
P_{\gam_1,\dots,\gam_N}(\lam_j)),
\eea
where $P_{\gam_1,\dots,\gam_N}(\lam_j)$ denotes the projection onto the 
one-dimensional subspace span\-ned by $\psi_{\gam_1,\dots,\gam_N}(\lam_j)$.
\end{thm}

\begin{proof}
It suffices to prove (\ref{sprl}), (\ref{phil}) which requires a
straightforward induction argument using Sylvester's determinant identity
(\cite{gant}, Sect.\ II.3). The resulting identity
\bea \nn
\det C_\ell(x) \det C_{\ell;i,j}(x) - \gam_\ell \det C_{\ell;\ell,j}(x) \det
C_{\ell;i,\ell}(x) \\ = \det C_{\ell-1}(x) \det C_{\ell+1;i,j}(x),
\eea
together with (\ref{sprgam}) then proves (\ref{sprl}). Similarly,
\bea \nn
\det C_\ell(x) \det \Psi_{\ell-1}(\lam_j,x) - \gam_\ell \det
\Psi_{\ell-1}(\lam_\ell,x) \det C_{\ell;\ell,j}(x)\\ = \det C_{\ell-1}(x) \det
\Psi_\ell(\lam_j,x),
\eea
and (\ref{defu}) prove (\ref{phil}). The rest then
follows from these two equations and Theorem \ref{thm}.
\end{proof}

\begin{rem}
(i) For any element $f$ which is square integrable near $-\infty$,
$f_{\gam_1,\dots,\gam_\ell} = U_{\gam_1,\dots,\gam_\ell} \cdots U_{\gam_1} f$ is
given by substituting $\psi(\lam_j) \to f$ in (\ref{phil}). Likewise we get the
scalar product of $f_{\gam_1,\dots,\gam_\ell}$ and $g_{\gam_1,\dots,\gam_\ell}$
from (\ref{sprl}) by  substituting $\psi(\lam_i) \to f$ and $\psi(\lam_j) \to g$
in (\ref{cij}).\\
(ii) Equation (\ref{phil}) can be rephrased as
\bea \nn
&&(\sqrt{\gam_1} \psi_{\gam_1,\dots,\gam_N}(\lam_1,x), \dots,
\sqrt{\gam_N} \psi_{\gam_1,\dots,\gam_N} (\lam_N,x)) \\ && \quad =
(C_N(x))^{-1} ( \sqrt{\gam_1} \psi(\lam_1,x), \dots,
\sqrt{\gam_N} \psi(\lam_N,x) ),
\eea
where $(C_N(x))^{-1}$ denotes the inverse of $C_N(x)$.
\end{rem}

Clearly Theorem \ref{thmlp} extends (by induction) to this more general
situation.

\begin{thm}
(i). $\tau_{\gam_1,\dots,\gam_N}$ is $l.p.$ at $a$ if and only if $\tau$ is and
$l.c.$ at $b$ if $\tau$ is.\\
(ii). Assume (H.3.4), then $\tau_{\gam_1,\dots,\gam_N}$ is $l.p.$ at $b$.
\end{thm}


\begin{thebibliography}{XXXX}
\bi{ak} T. Aktosun, M. Klaus, and C. van der Mee, {\em Scattering and inverse
scattering in one-dimensional nonhomogeneous media}, J. Math. Phys. {\bf 32},
1717-1744 (1992).
\bi{ba} B. Baumgartner, {\em Level comparison theorems}, Ann. Phys. (N.Y.)
{\bf 168}, 484--526 (1986).
\bi{bs} R. F. Bikbaev and R. A. Sharipov, {\em Asymptotics at $t\to\infty$ of
the solution to the Cauchy problem for the Korteweg--de Vries equation in
the class of potentials with finite--gap behavior as $x\to\pm\infty$},
Theoret. Math. Phys. {\bf 78}, 244--252 (1989).
\bi{cr} M. M. Crum, {\em Associated Sturm--Liouville systems}, Quart. J.
Math. Oxford (2) {\bf 6}, 121--127 (1955).
\bi{da} G. Darboux, {\em Sur une proposition relative aux \'{e}quations
lin\'{e}aires}, C. R. Acad. Sci. (Paris) {\bf 94}, 1456--1459 (1882).
\bi{de} P. A. Deift, {\em Applications of a commutation formula}, Duke
Math. J. {\bf 45}, 267--310 (1978).
\bi{dt} P. Deift and E. Trubowitz, {\em Inverse scattering on the line},
Commun. Pure Appl. Math. {\bf 32}, 121--251 (1979).
\bi{eak} M. S. P. Eastham and H. Kalf, {\em Schr\"{o}dinger--Type Operators
with Continuous Spectra}, Pitman, Boston, 1982.
\bi{em} M. S. P. Eastham and J. B. McLeod, {\em The existence of eigenvalues
embedded in the continuous spectrum of ordinary differential operators},
Proc. Roy. Soc. Edinburgh {\bf 79A}, 25--34 (1977).
\bi{ek} F. Ehlers and H. Kn\"{o}rrer, {\em An algebro--geometric
interpretation of the B\"acklund transformation of the Korteweg--de
Vries equation}, Comment. Math. Helv. {\bf 57}, 1--10 (1982).
\bi{ef} N. M. Ercolani and H. Flaschka, {\em The geometry of the Hill
equation and of the Neumann system}, Phil. Trans. Roy. Soc. London {\bf
A315}, 405--422 (1985).
\bi{fi} N. E. Firsova, {\em 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 {\bf 63}, 257--265
(1989).
\bi{fm} H. Flaschka and D. W. McLaughlin, {\em Some comments on B\"acklund
transformations, canonical transformations, and the inverse scattering
method}, in ``B\"acklund Transformations, the Inverse Scattering
Method, Solitons, and their Applications'', R. M. Miura (ed.), Lecture
Notes in Math. {\bf 515}, Springer, Berlin, 1976, p. 252--295.
\bi{gant} F. R. Gantmacher, {\em The Theory of Matrices}, Vol. 1,
Chelsa, New York, 1990.
\bi{gg} C. S. Gardner, J. M. Greene, M. D. Kruskal, and  R. M. Miura, {\em
Korteweg--de Vries equation and generalizations. VI. Methods for exact
solution}, Commun. Pure Appl. Math. {\bf 27}, 97--133 (1974).
\bi{gl} I. M. Gel'fand and B. M. Levitan, {\em On the determination of a
differential equation from its spectral function}, Amer. Math. Soc.
Transl. Ser 2, {\bf 1}, 253--304 (1955).
\bi{fg} F. Gesztesy, {\em A complete spectral characterization of the
double commutation method}, J. Funct. Anal. {\bf 117}, 401--446 (1993).
\bi{gs} F. Gesztesy and R. Svirsky, {\em (m)KdV-Solitons on the background of
quasi-periodic finite-gap solutions}, Memoirs Amer. Math. Soc. {\bf 118},
No. 563 (1995).
\bi{gt} F. Gesztesy and G. Teschl, {\em Commutation methods for Jacobi
operators}, J. Diff. Eqs. {\bf 128}, 252--299 (1996).
\bi{gu} F. Gesztesy and K. Unterkofler, {\em Isospectral deformations for
Sturm-Liouville and Dirac-type operators and associated nonlinear evolution
equations}, Rep. Math. Phys. {\bf 31}, 113-137 (1992).
\bi{gw} F. Gesztesy and R. Weikard, {\em Spectral deformations and soliton
equations}, in ``Differential Equations with Applications in Mathematical
Physics'', W. F. Ames, E. M. Harrell, J. V. Herod (eds.), Academic Press,
Boston, 1993, p. 101--139.
\bi{gz} F. Gesztesy and Z. Zhao, {\em On critical and subcritical
Sturm-Liouville operators}, J. Funct. Anal.
{\bf 98}, 311-345 (1991).
\bi{gss} F. Gesztesy, W. Schweiger, and B. Simon, {\em Commutation methods
applied to the mKdV-equation}, Trans. Amer. Math. Soc. {\bf 324}, 465--525
(1991).
\bi{ja} C. G. J. Jacobi, {\em Zur Theorie der Variationsrechnung und der
Differentialgleichungen}, J. Reine Angew. Math. {\bf 17}, 68--82 (1837).
\bi{km} I. Kay and H. E. Moses, {\em Reflectionless transmission through
dielectrics and scattering potentials}, J. Appl. Phys. {\bf 27},
1503--1508 (1956).
\bi{kr} I. M. Krichever, {\em Potentials with zero coefficient of
reflection on a background of finite--zone potentials}, Funct. Anal.
Appl. {\bf 9}, 161--163 (1975).
\bi{kum} E. A. Kuznetsov and A. V. Mikhailov, {\em Stability of solitary
waves in nonlinear weakly dispersive media}, Sov. Phys. JETP {\bf 40},
855--859 (1975).
\bi{mk1} H. P. McKean, {\em Geometry of KdV (1): Addition and the
unimodular spectral classes}, Rev. Mat. Iberoamericana {\bf 2}, 235--261 (1986).
\bi{mk2} H. P. McKean, {\em Geometry of KdV (2): Three examples}, J. Stat.
Phys. {\bf 46}, 1115--1143 (1987).
\bi{lv} B. M. Levitan, {\em Inverse Sturm--Liouville Problems}, VNU Science
Press, Utrecht, 1987.
\bi{mc} V. A. Marchenko, {\em Sturm-Liouville Operators and Applications},
Birkh\"auser, Basel, 1986.
\bi{rc} R. R. del Rio Castillo, {\em Embedded eigenvalues of Sturm
Liouville operators}, Commun. Math. Phys. {\bf 142}, 421--431 (1991).
\bi{rs} A. V. Rybin and M. A. Sall', {\em Solitons of the Korteweg--de Vries
equation on the background of a known solution}, Theoret. Math. Phys.
{\bf 63}, 545--550 (1985).
\bi{sc} U. -W. Schmincke, {\em On Schr\"{o}dinger's factorization method
for Sturm--Liouville operators}, Proc. Roy. Soc. Edinburgh {\bf 80A},
67--84 (1978).
\bi{sh1} R. A. Sharipov, {\em Finite--zone analogues of $N$--multiplet
solutions of the Korteweg--de Vries equation}, Russ. Math. Surv. {\bf
41:5}, 165--166 (1986).
\bi{sh2} R. A. Sharipov, {\em Soliton multiplets of the Korteweg--de Vries
equation}, Sov. Phys. Dokl. {\bf 32}, 121--123 (1987).
\bi{th} C. Thurlow, {\em The point--continuous spectrum of second order,
ordinary differential operators}, Proc. Roy. Soc. Edinburgh {\bf 84A},
197--211 (1979).
\bi{wa} H. D. Wahlquist, {\em B\"acklund transformations of potentials
of the Korteweg--de Vries equation and the interaction of solitons with
cnoidal waves}, in ``B\"acklund Transformations, the Inverse Scattering
Method, Solitons, and their Applications'', R. M. Miura (ed.), Lecture
Notes in Math. {\bf 515}, Springer, Berlin, 1976, p. 162--183.
\bi{wd} J. Weidmann, {\em Linear Operators in Hilbert Spaces}, Springer, New
York, 1980.
\end{thebibliography}

\end{document}
