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%%     journal="J. Reine und Angew. Math. 491, 1-15 (1997)",
%%     doi="10.1515/crll.1997.491.1",
%%     copyright="W. de Gruyter". 
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\begin{document}

\title{Spectral Deformations of Jacobi Operators}


\author{Gerald Teschl}
\address{Institut f\"ur Reine und Angewandte Mathematik\\
RWTH Aachen\\ 52056 Aachen\\ Germany}
\curraddr{Institut f\"ur Mathematik\\
Universit\"at Wien\\
Strudlhofgasse 4\\ 1090 Vienna\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\thanks{{\it J. Reine Angew. Math. {\bf 491}, 1--15 (1997)}}

\keywords{Jacobi operators, inverse spectral theory, commutation methods}
\subjclass{Primary 39A10, 39A70; Secondary 34B20, 47B39}


\begin{abstract}
We extend recent work concerning isospectral deformations for one-dimensional
Schr\"odinger operators to the case of Jacobi operators.  We provide a complete
spectral characterization of a new method that constructs isospectral deformations
of a given Jacobi operator $(H u)(n) = a(n) u(n+1) + a(n-1) u(n-1) - b(n) u(n)$.
Our technique is connected to Dirichlet data, that is, the spectrum of the operator
$H^{\infty}_{n_0}$ on $\ell^2 (-\infty,n_0) \oplus \ell^2 (n_0,\infty)$ with a
Dirichlet boundary condition at $n_0$. The transformation
moves a single eigenvalue of $H^{\infty}_{n_0}$ and perhaps flips which side of 
$n_0$ the eigenvalue lives. On the remainder of the spectrum the transformation
is realized by a unitary operator.
\end{abstract}


\maketitle




\section{Introduction}


Spectral deformations of Jacobi operators have proven useful in various
applications such as inverse spectral theory and construction of solutions for the
Toda and Kac van Moerbeke hierarchy \cite{bght}, \cite{gtjc}, \cite{TKvM},
\cite{ta}. In \cite{gstdef} a powerful new  spectral deformation method was
introduced for Schr\"odinger operators. The aim of the present paper is to develop
an analogous tool for Jacobi operators.

One approach to spectral deformations is to factor a given Jacobi operator
\begin{equation}
(H u)(n) = a(n) u(n+1) + a(n-1) u(n-1) - b(n) u(n),
\end{equation}
$u\in\db(H) \subseteq \lz$, into a product of first order difference operators
\begin{equation} \label{factorH}
H = A_\sig^* A_\sig + \lam, \quad \sig\in [-1,1].
\end{equation}
Interchanging the order of $A_{\sig}^*$ and $A_{\sig}$ produces a second
operator $H_\sig = A_\sig A_\sig^* + \lam$ whose spectral properties are closely
related to those of $H$. In fact, depending on the parameter $\sig$, one gets
operators which are either isospectral to $H$ or have the additional eigenvalue
$\lam$ \cite{deift}, \cite{gtjc} (see also \cite{TKvM}). 

Clearly, the special form of (\ref{factorH}) implies that  $H-\lam \ge 0$
and hence this single commutation method can only be applied to insert
eigenvalues below the spectrum of $H$. However, ignoring this fact and
performing two (suitable) commutation steps produces meaningful operators
$H_\gam$, $\gam>0$ (all intermediate operators are ill-defined unless $\lam$ is
below the spectrum of $H$). The operators $H_\gam$ are isospectral to $H$ except
for the additional eigenvalue $\lam$ (for details see \cite{gtjc}).

The idea of our new method is to perform two single commutation steps as before
(with possibly ill-defined intermediate operators), but now using different
choices for the parameter $\lam$ in the first respectively second step. The
investigation of the resulting transformed operator will be the task of this paper.

In order to further explain these ideas we need to introduce additional
notation. The Dirichlet operator $H^\infty_{n_0}$ is obtained  by
restricting $H$ to the subset of sequences $u\in\lz$ which satisfy  
$u(n_0)=0$ (see (\ref{defhinf}) below). It can be viewed as a rank one
resolvent perturbation (at infinite coupling) of $H$ implying that in each spectral
gap $(E_0,E_1)$ of $H$ there can be at most one eigenvalue $\mu_0$ of
$H^\infty_{n_0}$. However, note that special care has to be taken since the
resolvents of $H$ and $H^\infty_{n_0}$ live in different Hilbert spaces
(cf.\ \cite{gsroi}, \cite{gsxi} Appendix, or \cite{ttr} for details). Since
$H^\infty_{n_0}$ decomposes into a direct sum $H^\infty_{n_0}= H^\infty_{-,n_0}
\oplus H^\infty_{+,n_0}$ (with respect to the decomposition $\lz = \ell^2
(-\infty,n_0)
\oplus \ell^2 (n_0,\infty)$) there is a sign $\sig_0$ associated with each
$\mu_0$ such that
$\mu_0\in \sig(H^\infty_{\sig_0,n_0})$ ($\sig(.)$ denoting the spectrum of an
operator). 

Let $(E_0,E_1)$ be a spectral gap of $H$, $\mu_0\in
\sig(H^\infty_{\sig_0,n_0}) \cap (E_0,E_1)$ and pick $(\mu,\sig) \in (E_0,E_1)
\times \{\pm\}$. Then our transformation will send $H$ to an operator $H_\mus$ 
in such a way that $H, H_\mus$ are unitarily equivalent and the Dirichlet datum
$(\mu_0,\sig_0)$ will be shifted to $(\mu,\sig)$, whereas all other Dirichlet
eigenvalues remain unchanged. We will hence refer to this transformation as the
Dirichlet deformation method.

As anticipated, this transformation is realized by two single commutations; using
$u_{\sig_0}(\mu_0,.)$, $u_{-\sig}(\mu,.)$ in the first, second factorization of
$H$, respectively. Here $u_\pm(z,.)$, $z\in\C\bs\sig(H)$ denote weak (i.e., formal)
solutions of $H u = z u$ being square summable near $\pm\infty$, respectively. By
\cite{gtjc} the operator $H_\mus$ is associated with the sequences
\bea
a_\mus(n) &=& a(n) \sqrt{\frac{W_\mus(n-1)
W_\mus(n+1)}{W_\mus(n)^2}},\\ b_\mus(n) &=& b(n) - \partial^*
\frac{a(n) u_{\sig_0}(\mu_0,n) u_{-\sig}(\mu,n+1)}{W_\mus(n)},
\eea
where
\begin{equation}
W_\mus(n) = \frac{W_n( u_{\sig_0}(\mu_0), u_{-\sig}(\mu))}{\mu-\mu_0},
\end{equation}
$W_n(.,..)$ denotes the (modified) Wronskian and $(\partial^*f)(n)=f(n)-f(n-1)$.
Clearly, $H_\mus$ is only well-defined if $W_\mus(n+1)W_\mus(n)>0$; but this
is ensured by \cite{gtosc}, Theorem~4.6.

In the special case, where $a,b$ are periodic (cf.\ \cite{vm}), these ideas have
been used in \cite{gtjc} to give the discrete analogue of the FIT-formula
derived in \cite{fit} for the isospectral torus of periodic Schr\"odinger operators.



\section{Preliminary definitions}
\setcounter{equation}{0}
\setcounter{thm}{0}


Throughout this paper we denote by  $\ell(I)=\ell(M,N)$,
$I=\{ n \in \Z | M < n< N \}$,   
$M,N \in \Z \cup \{\pm\infty \}$ the set of complex-valued sequences
$\{ u(n) \}_{n\in I}$ and by $\ell^p(I)$, $1 \le p \le \infty$
the sequences $u \in \ell(I)$ such that $|u|^p$ is summable over $I$.
The scalar product in the Hilbert space $\ell^2(I)$ will be denoted by
\begin{equation}
\spr{u}{v} = \sum_{n \in I} \ol{u(n)} v(n), \quad u,v \in \ell^2(I).
\end{equation}

We will be concerned with operators on $\lz$ associated with the
difference expression
\begin{equation}
(\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) +b(n) f(n),
\end{equation}
where $a,b \in \ell(\Z)$ satisfy
\bh \label{hab}
Suppose
\begin{equation}
a(n) \in \R \bs\{ 0\}, \quad b(n) \in \R, \quad n \in \Z.
\end{equation}
\eh
If $\tau$ is limit point ($l.p.$) at both $\pm\infty$ (cf., e.g., \cite{at},
\cite{be}), then $\tau$ gives rise to a unique self-adjoint operator $H$ when
defined maximally. Otherwise, we need to fix a boundary condition at each
endpoint where $\tau$ is limit circle ($l.c.$) (cf., e.g., \cite{at}, \cite{be}).
Throughout this paper we denote by $u_\pm(z,.)$, $z \in \C$, nontrivial solutions
of $\tau u = z u$ which satisfy the boundary condition at $\pm\infty$ (if any) with
$u_\pm(z,.) \in \ell^2_\pm(\Z)$, respectively. Here $\ell^2_\pm(\Z)$ denotes the
sequences in $\ell(\Z)$ being $\ell^2$ near $\pm\infty$. The solution $u_\pm(z,.)$
might not exist for $z \in \R$ (cf.\ \cite{gtosc}, Lemma~A.1), but if it exists it is
unique up to a constant multiple.

Picking a fixed $z_0 \in \C \bs \R$ we can characterize $H$ by
\begin{equation} \label{defh}
\ba{lccl} H :& \db(H) & \to & \lz \\ & f &\mapsto& \tau f \ea ,
\end{equation}
where the domain of $H$ is explicitly given by
\begin{equation}
\db(H) = \{ f \in \lz | \ba[t]{l} \tau f \in \lz, \: \lim_{n \to +\infty}
W_n(u_+(z_0),f) = 0, \\ \lim_{n \to -\infty}
W_n(u_-(z_0),f) = 0 \} \ea
\end{equation}
and
\begin{equation}
W_n(f,g) = a(n) \Big( f(n)g(n+1) - f(n+1)g(n) \Big)
\end{equation}
denotes the (modified) Wronskian. The boundary condition at $\pm\infty$ imposes
no additional restriction on $f$ if $\tau$ is $l.p.$ at $\pm\infty$ and can hence be
omitted in this case.

Next, denote by $P_{n_0}$ the orthogonal projection onto the one-dimensional
subspace spanned by $\delta_{n_0}$ in $\lz$, where $\delta_{n_0}(n)$ is $1$
for $n=n_0$ and $0$ else. The Dirichlet operator is now defined by
\begin{equation} \label{defhinf}
H^\infty_{n_0}  = (\id - P_{n_0}) H (\id - P_{n_0})
\end{equation}
in the Hilbert space $(\id - P_{n_0})\lz = \{ f\in \lz | \spr{\delta_{n_0}}{f}=0\}$.
Clearly, $H^\infty_{n_0}$ decomposes into a direct sum $H^\infty_{n_0}=
H^\infty_{-,n_0}\oplus H^\infty_{+,n_0}$ with respect to the decomposition   
$\lz = \ell^2 (-\infty,n_0) \oplus \ell^2 (n_0,\infty)$ and we have
$\sig_{ess}(H) = \sig_{ess}(H^\infty_{n_0}) = \sig_{ess}(H^\infty_{-,n_0}) \cup
\sig_{ess}(H^\infty_{+,n_0})$.

Without restriction we will only consider the case $n_0=0$ and abbreviate
$H^\infty_{\pm,0}=H_\pm$ to simplify notation. This enables us to formulate
our basic hypothesis.

\bh \label{hmusig}
(i). Let $(E_0,E_1)$ be a spectral gap of $H$, that is,
$(E_0,E_1)\cap \sig(H)= \{E_0,E_1\}$.\\
(ii). Suppose $\mu_0 \in \sig_d(H_{\sig_0})\cap [E_0,E_1]$.\\
(iii). Let $(\mu,\sig) \in [E_0,E_1]\times\{\pm\}$ and
$\mu \in (E_0,E_1)$ or $\mu\in\sig_d(H)$.
\eh

Here $\sig_d(H)$ denotes the discrete spectrum (i.e., $\sig_d(H) = \sig_d(H) \bs
\sig_{ess}(H)$).

\br
Clearly, if $\mu_0$ is an eigenvalue of two of the operators
$H, H_-, H_+$, then it is also one of the third. Hence if
$\mu_0 \in \sig_d(H_{\sig_0})\bs \sig_d(H_{-\sig_0})$ then
$\mu_0\in(E_0,E_1)$ and if $\mu_0 \in \sig_d(H_{\sig_0})\cap
\sig_d(H_{-\sig_0})$ then $\mu_0\in \{E_0,E_1\}$. (The choice of
$\sig_0$ in the latter case is irrelevant). Condition (ii) thus says that
$\mu_0=E_{0,1}$ is only allowed if $E_{0,1}$ is a discrete eigenvalue of $H$.
Similar in (iii) for $\mu$.
\er

Our next objective is to define the operator $H_\mus$ of the Introduction. Since
$H_{(\mu_0,\sig_0)}=H$, we will assume $\mus \ne (\mu_0,\sig_0)$
without restriction.

Due to our assumption (H.\ref{hmusig}) we can find solutions 
$u_{\sig_0}(\mu_0,.)$, $u_{-\sig}(\mu,.)$ (cf. \cite{gtosc}, Lemma~A.1)
and define
\begin{equation}
W_\mus(n) = \left\{ \ba{l@{,\quad}l} \frac{W_n( u_{\sig_0}(\mu_0),
u_{-\sig}(\mu))}{\mu-\mu_0} & \mu\ne\mu_0\\
\sum\limits_{m=\sig_0\infty}^n u_{\sig_0}(\mu_0,m)^2 & \mus =
(\mu_0,-\sig_0)\ea \right. ,
\end{equation}
where $\sum_{m=+\infty}^n = -\sum_{m=n+1}^{\infty}$.
The motivation for the case $\mus = (\mu_0,-\sig_0)$ follows from
(assuming $u_{-\sig}(\mu,m)$ holomorphic w.r.t. $\mu$)
\bea \nn
\lim_{\mu\to\mu_0} \frac{W_n(u_{\sig_0}(\mu_0),
u_{\sig_0}(\mu))}{\mu-\mu_0} &=& \lim_{\mu\to\mu_0}
\sum_{m=\sig_0\infty}^n u_{\sig_0}(\mu_0,m) u_{\sig_0}(\mu,m)\\
&=& \sum_{m=\sig_0\infty}^n u_{\sig_0}(\mu_0,m)^2.
\eea
{}From the proof of \cite{gtosc}, Theorem~4.6 we infer

\bl
Suppose  (H.\ref{hmusig}), then
\begin{equation}
W_\mus(n+1) W_\mus(n) >0, \quad n \in \Z.
\end{equation}
\el
Thus the sequences 
\bea
a_\mus(n) &=& a(n) \sqrt{\frac{W_\mus(n-1) W_\mus(n+1)}{W_\mus(n)^2}},\\
b_\mus(n) &=& b(n) - \partial^* \frac{a(n) u_{\sig_0}(\mu_0,n)
u_{-\sig}(\mu,n+1)}{W_\mus(n)}
\eea
are both well-defined and we can consider the associated difference expression
 \begin{equation}
(\tau_\mus u)(n) = a_\mus(n) u(n+1) + a_\mus(n-1) u(n-1) + b_\mus(n) u(n).
\end{equation}
The next lemma collects some basic properties which follow either from
\cite{gtjc}, Section~3 (choosing $N=2$) or can be verified directly.

\bl
Let
\begin{equation}
(A_\mus u)(z,n) = \frac{W_\mus(n) u(z,n) - \frac{1}{z-\mu_0} u_{-\sig}(\mu,n)
W_n(u_{\sig_0}(\mu_0),u(z))}{\sqrt{W_\mus(n-1)W_\mus(n)}},
\end{equation}
where $u(z)$ solves $\tau u = z u$ for $z \in \C \bs \{ \mu_0 \}$. Then we have
\begin{equation}
\tau_\mus (A_\mus u)(z,n) = z (A_\mus u)(z,n)
\end{equation}
and
\begin{equation}
|(A_\mus u)(z,n)|^2 = |u(z,n)|^2 + \frac{1}{|z-\mu|^2}
\frac{u_{-\sig}(\mu,n)}{u_{\sig_0}(\mu_0,n)} \partial^*
\frac{|W_n(u_{\sig_0}(\mu_0),u(z))|^2}{W_\mus(n)}.
\end{equation}
Moreover, the sequences
\bea
u_{\mu_0}(n) &=&
\frac{u_{-\sig}(\mu,n)}{\sqrt{W_\mus(n-1)W_\mus(n)}},\\
u_\mu(n) &=&
\frac{u_{\sig_0}(\mu_0,n)}{\sqrt{W_\mus(n-1)W_\mus(n)}}
\eea
satisfy $\tau_\mus u = \mu_0 u$, $\tau_\mus u = \mu u$ respectively. Note also
\begin{equation}
u_{\sig_0}(\mu_0,0) = u_\mu(0) = 0
\end{equation}
and
\begin{equation} \label{produmuumun}
u_{\mu_0}(n) u_\mu(n) = \partial^* \frac{1}{W_\mus(n)}.
\end{equation}

In addition, let $u(z),\hat{u}(z)$ satisfy $\tau u = z u$, then
\bea\label{wronmusigon}
&& W_{\mus,n}(u_{\mu_0},A_\mus u(z)) =
\frac{W_n(u_{-\sig}(\mu),u(z))}{W_\mus(n)},\\
&& W_{\mus,n}(u_\mu,A_\mus u(z)) = \frac{z-\mu}{z-\mu_0}
\frac{W_n(u_{\sig_0}(\mu_0),u(z))}{W_\mus(n)},\\ \nn
&& W_{\mus,n}(A_\mus u(z),A_\mus \hat{u}(\hat{z})) = \frac{z-\mu}{z-\mu_0}
W_n(u(z),\hat{u}(\hat{z}))\\ && \quad{}+
\frac{z-\hat{z}}{(z-\mu_0)(\hat{z}-\mu_0)} \frac{W_n(u_{\sig}(\mu),u(z))
W_n(u_{-\sig_0}(\mu_0),\hat{u}(\hat{z}))}{W_\mus(n)},
\eea
where
\begin{equation}
W_{\mus,n}(u,v) = a_\mus(n) \Big( u(n) v(n+1) - u(n+1) v(n)\Big).
\end{equation}
\el

Having these preliminaries out of the way, we will now define operators
associated with $\tau_\mus$ by introducing suitable boundary conditions
(since $\tau_\mus$ is not necessarily $l.p.$ at $\pm\infty$). We single
out the following three situations, which are the only ones where the spectra of
$H$ and $H_\mus$ are closely related.

Let $\om\in\{\pm\}$ and
\begin{equation} \label{defbchmus}
BC_\om(f) = \left\{ \ba{l@{\mbox{ if }}l} 
\lim\limits_{n\to\om\infty} W_n(u_\om,f) = 0 & \tau_\mus \mbox{
is $l.c.$ at }\om\infty\\
0 & \tau_\mus \mbox{ is $l.p.$ at }\om\infty \ea \right. ,
\end{equation}
where $u_\om$ is chosen according to one of the following cases:\\[2mm]
{\bf Case I:} $\tau$ is $l.p.$ at $\om\infty$. Choose $u_\om= u_\mu$
or $u_\om= u_{\mu_0}$.\\
{\bf Case II:} $\tau$ is $l.c.$ at $\om\infty$.\\
\hspace*{1cm}(i). If  $\om=\sig =\sig_0$ choose $u_\om= u_\mu$.\\
\hspace*{1cm}(ii). If  $-\om=\sig =\sig_0$ choose $u_\om= u_{\mu_0}$.\\
\hspace*{1cm}(iii). If  $\sig =-\sig_0$ and $\mu\in\sig_d(H)$ choose $u_\om=
u_{\mu_0}$.\\
\hspace*{1cm}(iv). If  $\sig =-\sig_0$ and $\mu_0\in\sig_d(H)$ choose
$u_\om= u_\mu$.\\[2mm]
(Note that in Case~II (iii) and (iv) are the same if $\mus = (\mu_0,-\sig_0)$.)

Using this boundary conditions we define
\begin{equation} \label{defhmus}
\ba{lccl} H_\mus :& \db(H_\mus) & \to & \lz \\ & f &\mapsto& \tau_\mus f \ea ,
\end{equation}
where the domain of $H_\mus$ is explicitly given by
\begin{equation} \label{dbhmus}
\db(H_\mus) = \{ f \in \lz | \tau_\mus f \in \lz,\, BC_-(f)=BC_+(f)=0 \}.
\end{equation}
As always, there is no boundary condition at $\om\infty$ in (\ref{dbhmus}) if
$\tau_\mus$ is $l.p.$ at $\om\infty$, $\om\in\{\pm\}$. Furthermore,
$H_{\mus,\pm}$ denote the corresponding Dirichlet half-line operators with
respect to the base point $n_0=0$.




\section{ Half-line operators}
\setcounter{equation}{0}
\setcounter{thm}{0}


In this section we will give a complete spectral characterization of the
half-line operators $H_{\mus,\pm}$. In addition, this will provide all
necessary results for the investigation of $H_\mus$.

To begin with we compute the Weyl $\ti{m}$-functions
\begin{equation}
\ti{m}_\pm(z) = \mp\frac{u_\pm(z,1)}{a(0) u_\pm(z,0)}.
\end{equation}
We recall that $\ti{m}_\pm(z)$ are equivalently given by
\begin{equation}
\ti{m}_\pm(z) =  \frac{\pm1}{a(0)} \lim_{n\to\pm\infty}
\frac{W_n(c(z),f)}{W_n(s(z),f)},
\end{equation}
where $c(z,n)$, $s(z,n)$ form a fundamental system for $\tau$  (i.e., $(\tau - z)
c(z) = (\tau - z) s(z) =0$, $s(z,0)=c(z,1)=0$, and $s(z,1)=c(z,0)=1$) 
and $f$ is arbitrary if $\tau$ is $l.p.$ at $\pm\infty$ respectively
solves $(\tau-\lam)f=0$, $\lam\in\R$, and satisfies the boundary condition at
$\pm\infty$ if $\tau$ is $l.c.$ at $\pm\infty$ (cf.\ \cite{gtjc}, Appendix B).

\bth \label{thmddmwmt}
Let $\ti{m}_\pm(z)$, $\ti{m}_{\mus,\pm}(z)$ denote the
Weyl $\ti{m}$-functions of $H$, $H_\mus$ respectively. Then we have
\begin{equation} \label{ddmwmt}
\ti{m}_{\mus,\pm}(z) = \frac{1}{1 +\gam_\mus} \Big( \frac{z-\mu_0}{z-\mu}
\ti{m}_\pm(z) \mp \frac{\gam_\mus}{z - \mu} \Big),
\end{equation}
where
\begin{equation} \label{defgammus}
\gam_\mus =\left\{ \ba{cl}  -\sig (\mu-\mu_0) \ti{m}_{-\sig}(\mu), &
\mu\ne\mu_0\\ -
\frac{\sig_0 u_{\sig_0}(\mu_0,\sig_01)^2}{\sum\limits_{n\in\sig_0\N}
u_{\sig_0}(\mu_0,n)^2}, & \mus=(\mu_0,-\sig_0)\ea \right. .
\end{equation}
\eth
\bpf
We first note that
\bea
c_\mus(z,n)  &=& \frac{z-\mu_0}{z-\mu} (A_\mus c)(z,n) -
\frac{\gam_\mus}{z-\mu} a(0) (A_\mus s)(z,n), \\
s_\mus(z,n)  &=& \sqrt{1+\gam_\mus}  (A_\mus s)(z,n)
\eea
constructed from the fundamental system $c(z,n)$,
$s(z,n)$ for $\tau$  form a fundamental system for $\tau_\mus$
corresponding to the same initial conditions. Furthermore, note
\begin{equation}
\frac{W_\mus(1)}{W_\mus(0)} =1 +\gam_\mus,
\quad \frac{W_\mus(0)}{W_\mus(-1)} =1.
\end{equation}
Now the result follows upon evaluating
\begin{equation}
\ti{m}_{\mus,\pm}(z) =  \frac{\pm1}{a_\mus(0)} \lim_{n\to \pm\infty}
\frac{W_{\mus,n}(c_\mus(z),u_\om)}{W_{\mus,n}(s_\mus(z),u_\om)}.
\end{equation}
Using (\ref{wronmusigon}) one obtains for
$u_\om= u_\mu(n), u_{\mu_0}(n)$,
$v_\om= u_{\sig_0}(\mu_0,n), u_{\sig_0}(\mu_0,n)$ respectively
(according to Case I or  II above)
\begin{equation}
\ti{m}_{\mus,\pm}(z) = \frac{1}{1+\gam_\mus} \Big( \frac{z-\mu_0}{z-\mu}
\frac{\pm1}{a(0)} \lim_{n\to\pm\infty}
\frac{W_n(c(z),v_\om)}{W_n(s(z),v_\om)}
\mp \frac{\gam_\mus}{z - \mu} \Big)
\end{equation}
the claim follows.
\epf

Observe that even if there seems to be some freedom in the choice of the boundary
condition $BC_\om$ at first sight, Theorem ~\ref{thmddmwmt} shows that
different choices give rise to the same operator $H_\mus$ (since
$\ti{m}_{\mus,\pm}(z)$ determine $H_\mus$ uniquely). This fact will be used in the
proof of Lemma~\ref{lemddmlp}. As a second consequence we note

\bk \label{corumuspm}
The sequences
\begin{equation}
u_{\mus,\pm}(z,n) = (A_\mus u_\pm)(z,n), \quad z \in\C\bs\{ \mu, \mu_0\},
\end{equation}
are square summable near $\pm\infty$ and satisfy the boundary condition of
$H_{\mus,\pm}$ at $\pm\infty$ (if any). Moreover, the same is true for
\bea
u_{\mus,-\sig}(\mu,n) &=& W_\mus(n) u_{\mu_0}(n) -
u_\mu(n) \sum\limits_{m=-\sig\infty}^n u_{-\sig}(\mu,m)^2,\\
u_{\mus,\sig_0}(\mu_0,n) &=& W_\mus(n) u_\mu(n) -
u_{\mu_0}(n) \sum\limits_{m=\sig_0\infty}^n u_{\sig_0}(\mu_0,m)^2,
\eea
and
\bea
u_{\mus,\sig}(\mu,n) &=&  u_\mu(n), \quad \mu\not\in\sig_d(H),\\
u_{\mus,-\sig_0}(\mu_0,n) &=& u_{\mu_0}(n), \quad \mu_0 \not\in\sig_d(H).
\eea
If $\mu$ or $\mu_0\in\sig_d(H)$ one has to replace the last formulas by
\bea
u_{\mus,\sig}(\mu,n) &=& W_\mus(n) u_{\mu_0}(n) - u_\mu(n)
\sum\limits_{m=\sig\infty}^n u_{-\sig}(\mu,m)^2,\\
u_{\mus,-\sig_0}(\mu_0,n) &=& W_\mus(n) u_\mu(n) - u_{\mu_0}(n)
\sum\limits_{m=-\sig_0\infty}^n u_{\sig_0}(\mu_0,m)^2,
\eea
respectively.
\ek

\bpf
Follows immediately form
\begin{equation}
u_{\mus,\pm}(z,n) = \frac{c_\pm(z)}{1+\gam_\mus} \frac{z-\mu_0}{z-\mu} \Big(
\frac{c_\mus(z,n)}{a_\mus(0)} \mp \ti{m}_{\mus,\pm} (z) s_\mus(z,n) \Big)
\end{equation}
if
\begin{equation}
u_\pm(z,n) = c_\pm(z)\Big(\frac{c(z,n)}{a(0)} \mp \ti{m}_\pm(z) s(z,n) \Big).
\end{equation}
If $z=\mu,\mu_0$ one can assume $u_\pm(z)$ holomorphic with respect to $z$
near $\mu,\mu_0$ (\cite{gtosc}, Theorem~A.1) and  consider limits (compare
\cite{gtosc}, Theorem~A.3).
\epf

Next we are interested in the pole structure  of $\ti{m}_{\mus,\pm}(z)$ near
$z=\mu,\mu_0$. A straightforward investigation of (\ref{ddmwmt}) (invoking the
Herglotz property of $\ti{m}_{\mus,\pm}(z)$)  shows 

\bk
We have
\begin{equation}
\ti{m}_{\mus,\om}(z) = \left\{ \ba{cl} -\frac{\gam_\mu}{z-\mu} + O(z-\mu)^0, &
\om=\sig \\ O(z-\mu)^0, & \om=-\sig \ea \right. , \:\: \om\in\{\pm\},
\end{equation}
where
\begin{equation} \label{defgamwm}
\gam_\mu = \left\{ \ba{cl} 
(\mu-\mu_0)(\ti{m}_+(\mu) + \ti{m}_-(\mu)), & \mu\ne\mu_0\\
\frac{u_-(\mu,-1)^2}{\sum_{n=-\infty}^{-1} u_-(\mu,n)^2} +
\frac{u_+(\mu,1)^2}{\sum_{n=1}^{+\infty} u_+(\mu,n)^2}, &
\mus=(\mu_0,-\sig_0)\ea\right. \ge 0. 
\end{equation}
Moreover, $\gam_\mu=0$ if $\mu\in\sig_d(H)\bs\{ \mu_0\}$ and $\gam_\mu>0$
else. If $\mu\ne\mu_0$, then
\begin{equation}
\ti{m}_{\mus,\pm}(z) = O(z-\mu_0)^0
\end{equation}
and note $\gam_\mu = a(0)^{-2} (\mu-\mu_0) G(\mu,0,0)^{-1}$.
\ek

Using the previous corollary plus weak convergence of $\pi^{-1}
\im(\ti{m}_\pm(\lam+\I\eps)) d\lam$ to the corresponding spectral
measure $d\ti{\rho}_\pm(\lam)$ as $\eps\downarrow 0$ implies

\bl \label{lemddmsm}
Let $d\ti{\rho}_\pm(\lam)$ and $d\ti{\rho}_{\mus,\pm}(\lam)$ be the respective
spectral measures of $\ti{m}_\pm(z)$ and $\ti{m}_{\mus,\pm}(z)$. Then we have
\begin{equation}
d\ti{\rho}_{\mus,\pm}(\lam) = \frac{1}{1 +\gam_\mus}
\Big( \frac{\lam-\mu_0}{\lam-\mu} d\ti{\rho}_\pm(\lam) + \left\{ \ba{lc}
\gam_\mu \, d\Theta(\lam-\mu), & \sig=\pm \\ 0, & \sig = \mp \ea\right\} \Big),
\end{equation}
where $\gam_\mus, \gam_\mu$ are defined in (\ref{defgammus}),
(\ref{defgamwm}) and $d\Theta$ is the unit point measure concentrated at 0.
\el


Let $P_\pm(\mu_0)$, $P_{\mus,\pm}(\mu)$ denote the orthogonal projections
onto the subspaces spanned by $u_{\sig_0}(\mu_0,.)$,
$u_\mu(.)$ in $\ell^2(\pm\N)$ respectively. Then the above results clearly
imply

\bth \label{thmddmhl}
The operators $(\id -P_{\pm}(\mu_0))H_\pm$ and
$(\id- P_{\mus,\pm}(\mu))H_{\mus,\pm}$ are unitarily equivalent.
Moreover, we have $\mu\not\in\sig(H_{\mus,-\sig})$ and
$\mu_0\not\in\sig(H_{\mus,\pm})\bs\{ \mu \}$.

If $\mu\not\in\sig_d(H)$ or $\mus=(\mu_0,-\sig_0)$, then
$\mu\in\sig(H_{\mus,\sig})$ and thus
\begin{equation}
\sig(H_{\mus,\pm}) = \Big(\sig(H_\pm) \bs \{\mu_0\} \Big)  \cup
\left\{\ba{cl}\{\mu\}, & \sig=\pm \\ \emptyset, & \sig=\mp\ea \right. .
\end{equation}
Otherwise, that is, if $\mu\in\sig_d(H)\bs\{\mu_0\}$, then 
$\mu\not\in\sig(H_{\mus,\sig})$ and thus
\begin{equation}
\sig(H_{\mus,\pm}) = \sig(H_\pm) \bs \{\mu_0\}.
\end{equation}
\eth

In essence, Theorem~\ref{thmddmhl} says that, as long as
$\mu\not\in\sig_d(H)\bs\{\mu_0\}$, the Dirichlet datum $(\mu_0,\sig_0)$ is
rendered into $\mus$, whereas everything else remains unchanged. If 
$\mu\in\sig_d(H)\bs\{\mu_0\}$, that is, if we are trying to move $\mu_0$ to an
eigenvalue, then $\mu_0$ is removed. This latter case reflects the fact that we
cannot move $\mu_0$ to an eigenvalue $E$ without moving the Dirichlet
eigenvalue on the other side of $E$ to $E$ at the same time.

We end this section with a few additions.

\br
(i). For $f \in \ell(\N)$ set
\bea \nn
(A_{\mus,+} f)(n) &=& \sqrt{\frac{W_\mus(n)}{W_\mus(n-1)}} f(n)\\
&&{}-u_{-\sig_0,\mus}(\mu_0,n) \sum_{j=1}^n u_{\sig_0}(\mu_0,j) f(j),\\ \nn
(A_{\mus,+}^{-1} f)(n) &=& \sqrt{\frac{W_\mus(n-1)}{W_\mus(n)}} f(n)\\
&&{}- u_{-\sig}(\mu,n) \sum_{j=1}^n u_{\sig,\mus}(\mu,j) f(j).
\eea
Then we have $A_{\mus,+} A_{\mus,+}^{-1} = A_{\mus,+}^{-1} A_{\mus,+}
=\id_{\ell(\N)}$ and
\begin{equation}
\tau_{\mus,+} = A_{\mus,+} \tau_+ A_{\mus,+}^{-1}.
\end{equation}
Similarly, for $f \in \ell(-\N)$ set
\bea \nn
(A_{\mus,-} f)(n) &=& \sqrt{\frac{W_\mus(n)}{W_\mus(n-1)}} f(n)\\
&&{}- u_{-\sig_0,\mus}(\mu_0,n) \sum_{j=n+1}^1 u_{\sig_0}(\mu_0,j) f(j),\\ \nn
(A_{\mus,-}^{-1} f)(n) &=& \sqrt{\frac{W_\mus(n-1)}{W_\mus(n)}} f(n)\\
&&{}- u_{-\sig}(\mu,n) \sum_{j=n+1}^1 u_{\sig,\mus}(\mu,j) f(j).
\eea
Then we have $A_{\mus,-} A_{\mus,-}^{-1} = A_{\mus,-}^{-1} A_{\mus,-}
=\id_{\ell(-\N)}$ and
\begin{equation}
\tau_{\mus,-} = A_{\mus,-} \tau_- A_{\mus,-}^{-1}.
\end{equation}
(ii). Note that the case $\mus=(\mu_0,-\sig_0)$ corresponds to the
double commutation method with $\gam=\infty$ (cf. \cite{gtjc}, Section ~4).
Furthermore, the operators $A_{\mus,\pm}$ are unitary when restricted to
proper  subspaces of $\ell^2(\pm\N)$ in this case.\\
(iii). Due to the factor $\frac{z-\mu_0}{z-\mu}$ in front of
$\ti{m}_{\mus,\pm}(z)$, all norming constants (i.e., the negative residues at each
pole of $\ti{m}_{\mus,\pm}(z)$) are altered.\\
(iv).  Clearly our transformation preserves reflectionless properties.
\er



\section{ Full-line operators}
\setcounter{equation}{0}
\setcounter{thm}{0}


Having the results of the previous section at our disposal we can now easily
deduce all spectral properties of the operator $H_\mus$. We recall the
Weyl $M$-matrix
\bea \nn
M(z) &=& \Big(\spr{\delta_j}{(H-z)^{-1}\delta_k} \Big)_{0\le j,k \le 1}
- \frac{1}{2a(n)} \left( \ba{cc} 0 &1 \\ 1 & 0\ea \right)\\
&=& \frac{1}{\ti{m}_+(z) + \ti{m}_-(z)} \left( \ba{cc} -\frac{1}{a(0)^2} &
\frac{\ti{m}_+(z) - \ti{m}_-(z)}{2a(0)}\\ \frac{\ti{m}_+(z) - \ti{m}_-(z)}{2a(0)} &
\ti{m}_+(z) \ti{m}_-(z) \ea \right),
\eea
associated with $H$. Then Theorem~\ref{thmddmwmt} yields

\bth
Given $H, H_\mus$ the respective Weyl $M$-matrices $M(z), M_\mus(z)$ are
related by
\bea
M_{\mus,0,0}(z) &=& \frac{1}{(1+\gam_\mus)^2} \frac{z-\mu}{z-\mu_0}
M_{0,0}(z),\\ M_{\mus,0,1}(z) &=& \frac{1}{1+\gam_\mus} \Big( M_{0,1}(z) +
\frac{\gam_\mus}{z-\mu_0} a(0) M_{0,0}(z) \Big),\\
\nn M_{\mus,1,1}(z) &=& \frac{z-\mu_0}{z-\mu} M_{1,1}(z) - 2
\frac{\gam_\mus}{z-\mu_0} a(0) M_{0,1}(z)\\ && {} +
\frac{\gam_\mus^2}{(z-\mu_0)(z-\mu)} a(0)^2 M_{0,0}(z).
\eea
Moreover, $M_{\mus,j,k}(z,m,n)$, $j,k\in\{0,1\}$ are holomorphic near
$z=\mu,\mu_0$. 
\eth

Given the connection between $M(z)$ and $M_\mus(z)$ we can compute the
corresponding Herglotz matrix measure of $M_\mus(z)$ as in
Lemma~\ref{lemddmsm}.

\bl
The matrix measures $d\rho, d\rho_\mus$ corresponding to $M(z), M_\mus(z)$
are related by
\bea
d\rho_{\mus,0,0}(\lam) &=& \frac{1}{(1+\gam_\mus)^2}
\frac{\lam-\mu}{\lam-\mu_0} d\rho_{0,0}(\lam),\\
d\rho_{\mus,0,1}(\lam) &=& \frac{1}{1+\gam_\mus} \Big( d\rho_{0,1}(\lam) +
\frac{\gam_\mus}{\lam-\mu_0} a(0) d\rho_{0,0}(\lam) \Big),\\
\nn d\rho_{\mus,1,1}(\lam) &=& \frac{\lam-\mu_0}{\lam-\mu} d\rho_{1,1}(\lam)
- 2 \frac{\gam_\mus}{\lam-\mu_0} a(0)  d\rho_{0,1}(\lam)
\\ && {} + \frac{\gam_\mus^2}{(\lam-\mu_0)(\lam-\mu)}
a(0)^2 d\rho_{0,0}(\lam). 
\eea
Equivalently
\bea \nn
d\rho_\mus(\lam) &=& \frac{1}{(z-\mu)(z-\mu_0)} \times \\
&& \left(\ba{cc} \frac{z-\mu}{1+\gam_\mus} & 0 \\ a(0)\gam_\mus &
z-\mu_0 \ea\right) d\rho(\lam) \left(\ba{cc} \frac{z-\mu}{1+\gam_\mus} &
a(0)\gam_\mus \\ 0 & z-\mu_0 \ea\right).
\eea
\el

This finally leads to our main theorem

\bth
Let $H, H_\mus$ be defined as in (\ref{defh}), (\ref{defhmus}) respectively.
Denote by $P(\mu_0)$, $P(\mu)$ the orthogonal projections corresponding to
the spaces spanned by $u_{\sig_0}(\mu_0,.)$, $u_{-\sig}(\mu,.)$ in $\lz$
respectively. Then $(\id-P(\mu_0)-P(\mu)) H$ and $H_\mus$ are unitarily
equivalent. In particular,  $H$ and $H_\mus$ are unitarily equivalent
if $\mu,\mu_0 \not\in\sig_d(H)$. 
\eth

\br \label{remddmcd}
By inspection, Dirichlet deformations produce the commuting diagram\\
\begin{picture}(10,2.5)
\put(4.95,2){$(\mu_1, \sigma_1)$}
\put(3,0.5){$(\mu_0, \sigma_0)$}
\put(7,0.5){$(\mu_2, \sigma_2)$}
\put(3.6,0.8){\vector(1,1){1.2}}
\put(6.3,2){\vector(1,-1){1.2}}
\put(4.5,0.6){\vector(1,0){2.1}}
\end{picture}\\
for $(\mu_j, \sigma_j)$, $0\le j\le 2$ according to (H.\ref{hmusig}).\\
\er

\br
We have seen in Theorem~\ref{thmddmhl} that the Dirichlet deformation method
cannot create a situation where a discrete eigenvalue $E_0$ of $H$ is rendered into
a Dirichlet eigenvalue (i.e., moving $\mu_0$ to the eigenvalue $E_0$).
However, 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 $H_-\oplus H_+$).

(ii) As a consequence of step (i), there is now another eigenvalue
$\tilde\mu$ of $H_-\oplus H_+$ 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 double commutation method at the beginning of this remark to
change $\sigma_0$ into any allowed value.

Theorem~4.4 of \cite{gtjc} then shows that the resulting operator is unitarily
equivalent to the original operator $H$, and Theorem~5.4 of \cite{gtjc} then proves
that the remaining Dirichlet eigenvalues remain invariant.
\er

Next, we provide two limit point results. The first, although trivial from a
technical point of view, nevertheless will apply in a great variety of situations.

\bl
Given $H, H_\mus$, let $\om\in\{\pm\}$ and suppose that
one of the following conditions (1)--(2) holds.
\begin{itemize}
\item[(1).] $\sigma_{ess}(H_\om)\ne\emptyset$.
\item[(2).] $\sigma(H_\om)= \sigma_d(H_\om) =
\{E_{\om, n}\}_{n\in\N}$ with $\sum_{n\in\N}
(1+E^2_{\om, n})^{-1} =\infty$.
\end{itemize}
Then, both $\tau$ and $\tau_\mus$ are $l.p.$ at $\om\infty$.
\el

\bpf
A simple consequence of the fact that $\tau$ $l.c.$ at $\om\infty$
implies that the resolvent of $H_\om$ is Hilbert--Schmidt.
\epf

Our second limit point result is more tailored toward the Dirichlet
deformation method .

\bl \label{lemddmlp}
Assume that $\mu_0,\mu\in (E_0, E_1)$ and $\sig=\sig_0$.
Then $\tau_\mus$ is in the limit point (resp., $l.c.$) case at $\om\infty$ if
and only if $\tau$ is $l.p$ (resp., $l.c.$) at $\om\infty$, $\om\in \{\pm\}$.
\el

\bpf
It suffices to consider $\mu\neq\tilde\mu$ .
Assume that $\tau$ is $l.p$ at $\om\infty$ and suppose the
contrary for $\tau_\mus$, that is, suppose $\tau_\mus$ is $l.c.$ at
$\om\infty$. The fact that both choices $u_\om=u_\mu$ and
$u_\om=u_{\mu_0}$ in (\ref{defbchmus}) yield equivalent boundary condition
(since they yield the same operator) implies
\begin{equation}
\lim\limits_{n\to\om\infty} W_n(u_{\mu_0},u_\mu)=0.
\end{equation}
In other words, $u_{\mu_0},u_\mu$ both satisfy the boundary condition
$BC_\om(u_{\mu_0})=BC_\om(u_\mu)=0$ at $\om\infty$.
If $-\om=\sig_0$ we infer that $\mu\in\sig(H_\mus)=\sig(H)$ since
$BC_{-\om}(u_\mu)=0$ by Corollary~\ref{corumuspm}. But this contradicts
$\mu\in(E_0,E_1)$.  Similarly, if $\om=\sig_0$ we infer that
$\mu_0\in\sig(H_\mus)=\sig(H)$ since $BC_{-\om}(u_{\mu_0})=0$ by
Corollary~\ref{corumuspm}. But this contradicts $\mu_0\in(E_0,E_1)$. By
symmetry in $\tau$ and $\tau_\mus$, the proof is complete.
\epf

\br
If $\sig=-\sig_0$ then the Dirichlet deformation method does not necessarily
preserve the $l.p.$ property. By Remark~\ref{remddmcd} it suffices to consider the
case $\mu=\mu_0$. Take an operator $H$ being $l.c.$ at $\sig_0\infty$ and
such that $\sig_{ess}(H_{-\sig_0})\ne\emptyset$. Then $\tau\mus$ is $l.p.$ at
$\sig\infty$ since $u_\mu \not\in\ell^2(\sig\N)$ (by (\ref{produmuumun})) and
$l.p.$ at
$-\sig\infty$ since $\sig_{ess}(H_{\mus,-\sig_0})\ne\emptyset$. 
\er

Finally, we briefly comment on how to iterate Dirichlet deformation method (see
\cite{gtjc}, Section~3). Suppose
\begin{equation}
(E_{0,j}, E_{1,j}), \quad (\mu_{0,j}, \sig_{0,j}), (\mu_j, \sig_j) \in [E_{0,j},
E_{1,j}]\times\{\pm\}
\end{equation}
satisfy (H.\ref{hmusig}) for each $j=1,\dots, N$, $N\in\N$. Then the
Dirichlet deformation result after $N$ steps, denoted by
$H_{(\mu_1,\sig_1),\dots,(\mu_N,\sig_N)}$, is associated with the sequences
\bea \nn
&& a_{(\mu_1,\sig_1),\dots,(\mu_N,\sig_N)}(n) = 
\sqrt{a(n-N) a(n+N)}\\
&& \qquad\times \frac{ \sqrt{ C_{n-N}(U_{1,\dots,N})
C_{n-N+2}(U_{1,\dots,N})}}{C_{n-N+1}(U_{1,\dots,N})},\\
\nn && b_{(\mu_1,\sig_1),\dots,(\mu_N,\sig_N)}(n) = 
b(n) - \partial^*\,  a(n) \frac{D_{n-N+1}(U_{1,\dots,N})}{C_{n-N+1}(U_{1,\dots,N})},
\eea
where $(U_{1,\dots,N}) = (u_{\sig_{0,1}}(\mu_{0,1}),u_{\sig_1}(\mu_1),\dots,
u_{\sig_{0,N}}(\mu_{0,N}),u_{\sig_N}(\mu_N))$ and $C_n, D_n$ are given by
\bea
C_n(u_1,\dots,u_N) &=& \det\{ u_i(n+j-1)\}_{1\le i,j\le N},\\
D_n(u_1,\dots,u_N) &=& \det\left\{\ba{cl} u_i(n+j-1), & j<N \\
u_i(n+N), & j=N \ea\right\}_{1\le i,j\le N}.
\eea

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\end{document}
