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Jacobi matrices, orthogonal polynomials, Jost function, Jost solution, exponentially decaying perturbations, finite rank perturbations
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\begin{document}
\title[Decay and Analyticity]{Jost Functions and Jost Solutions for Jacobi Matrices,
II.~Decay and Analyticity}
\author[D. Damanik and B. Simon]{David Damanik$^{1,2}$ and Barry Simon$^{1,3}$}
\thanks{$^1$ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.
E-mail: damanik@caltech.edu; bsimon@caltech.edu}
\thanks{$^2$ Supported in part by NSF grant DMS-0227089}
\thanks{$^3$ Supported in part by NSF grant DMS-0140592}
\date{January 5, 2005}
\begin{abstract} We present necessary and sufficient conditions on the
Jost function for the corresponding Jacobi parameters $a_n -1$ and $b_n$
to have a given degree of exponential decay.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Among the most interesting results in spectral theory are those that give equivalent
sets of conditions --- one set involving recursion coefficients and the other
involving spectral data. Examples are Verblunsky's version \cite{V36} of the Szeg\H{o}
theorem (see \cite{OPUC1}), the strong Szeg\H{o} theorem written as a sum rule
(see \cite{OPUC1}), the Killip-Simon theorem \cite{KS} characterizing $L^2$
perturbations of the free Jacobi matrix, and Baxter's theorem \cite{Bax,OPUC1}.
Our goal in this paper is to present such an equivalence for Jacobi matrices
concerning exponential decay. That is, we consider orthogonal polynomials on
the real line (OPRL) whose recursion relation is
\begin{equation} \lb{1.1}
xp_n(x) = a_{n+1} p_{n+1}(x) + b_{n+1} p_n(x) + a_n p_{n-1}(x)
\end{equation}
for Jacobi parameters $\{a_n\}_{n=1}^\infty$, $\{b_n\}_{n=1}^\infty$. Here $p_n(x)$
are the orthonormal polynomials and $p_{-1}(x)\equiv 0$ (i.e., $a_0$ is not needed in
\eqref{1.1} for $n=0$).
\eqref{1.1} is often summarized by the Jacobi matrix
\begin{equation} \lb{1.1a}
J=\begin{pmatrix} b_1 & a_1 & 0 & \dots \\
a_1 & b_2 & a_2 & \dots \\
0 & a_2 & b_3 & \dots \\
\dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots
\end{pmatrix}
\end{equation}
By $J_0$ we mean the $J$ with $a_n \equiv 1$, $b_n\equiv 0$.
The model of what we will find here is the following result of Nevai-Totik \cite{NT89}
in the theory of orthogonal polynomials on the unit circle (OPUC):
\begin{theorem}[Nevai-Totik \cite{NT89}; see Section~7.1 of \cite{OPUC1}]\lb{T1.1}
Let $d\mu$ be a probability measure on $\partial\bbD$ obeying
\begin{equation} \lb{1.2}
d\mu = w(\theta)\, \f{d\theta}{2\pi} + d\mu_\s
\end{equation}
Fix $R>1$. Then the following are equivalent:
\begin{SL}
\item[{\rm{(1)}}] The Szeg\H{o} condition holds, $d\mu_\s=0$, and the Szeg\H{o}
function, $D(z)$, has $D(z)^{-1}$ analytic in $\{z\mid \abs{z} -\infty
\end{equation}
in which case $D$ is defined initially on $\bbD$ by
\begin{equation} \lb{1.5}
D(z) =\exp\biggl( \int \f{e^{i\theta}+z}{e^{i\theta}-z}\, \log(w(\theta))\,
\f{d\theta}{4\pi}\biggr)
\end{equation}
In \eqref{1.3}, $\alpha_n$ are the Verblunsky coefficients, that is, the recursion
coefficients for the monic OPUC, $\Phi_n$,
\begin{equation} \lb{1.6}
\Phi_{n+1}(z) = z\Phi_n(z) -\bar\alpha_n \Phi_n^*(z)
\end{equation}
with
\begin{equation} \lb{1.7}
\Phi_n^*(z) = z^n\, \ol{\Phi_n (1/\bar z)}
\end{equation}
See \cite{OPUC1,OPUC2,Szb,GBk,GBk1} for background on OPUC.
Also relevant to our motivation is the following simple result:
\begin{theorem}\lb{T1.2} Let $d\mu$ be a probability measure on $\partial\bbD$
obeying \eqref{1.2}. Then the following are equivalent:
\begin{SL}
\item[{\rm{(1)}}] The Szeg\H{o} condition holds, $d\mu_\s =0$, and the Szeg\H{o}
function, $D(z)$, has $D(z)^{-1}$ a polynomial of exact degree $n$.
\item[{\rm{(2)}}] $\alpha_j =0$ for $j\geq n$ and $\alpha_{n-1}\neq 0$.
\end{SL}
\end{theorem}
\begin{proof} {\ul{(2) $\Rightarrow$ (1).}} \ In this case (see
\cite[Theorem~1.7.8]{OPUC1}), $d\mu = \f{d\theta}{2\pi}
\abs{\varphi_n^* (e^{i\theta})}^{-2}$, so $D^{-1} =\varphi_n^*$ is
a polynomial.
\smallskip
\noindent {\ul{(1) $\Rightarrow$ (2).}} \ $D(z)^{-1}$ is nonvanishing on
$\overline{\bbD}$, and so the measure has the form $\f{d\theta}{2\pi}\abs{D(z)}^2$ and so
has $\alpha_j =0$ for $j \geq n$ (\cite[Theorem~1.7.8]{OPUC1}). Thus $D(z)^{-1} =
\varphi_n^*(z)$, and since $\varphi_n$ has degree exactly $n$, $\Phi_n^* = \Phi_{n-1}^*
-\alpha_{n-1} z\Phi_{n-1}$ implies $\alpha_{n-1} \neq 0$.
\end{proof}
In our work, the spectral measure has the form
\begin{equation} \lb{1.8}
d\gamma(x) =f(x)\, dx + d\gamma_\s
\end{equation}
where $\supp f \subset [-2,2]$. We say $d\gamma_\s$ is regular if $\gamma_\s
([-2,2])=0$ and $d\gamma_\s$ has finite support (i.e., no embedded singular
spectrum and only finitely many bound states). The $m$-function associated to
$d\gamma$ is defined on $\bbC\backslash\supp(d\gamma)$ by
\begin{equation} \lb{1.9}
m(E) = \int \f{d\gamma(x)}{x-E}
\end{equation}
and $M$ is defined on $\bbD =\{z\mid\abs{z} <1\}$ by
\begin{equation} \lb{1.10}
M(z) = -m(z+z^{-1})
\end{equation}
Since $z\mapsto z+z^{-1}$ maps $\bbD$ to $\bbC\cup\{\infty\}\backslash [-2,2]$,
$M$ is analytic on $\bbD\backslash \{z\in\bbR\cap\bbD \mid z+z^{-1}$ is a
point mass of $d\gamma\}$ with simple poles at the missing points.
The Jost function, $u(z)$, is defined and analytic on $\bbD$ in many cases
and determined first by
\begin{equation} \lb{1.11}
\abs{u(e^{i\theta})}^2 \Ima M(e^{i\theta}) =\sin\theta
\end{equation}
where the functions at $e^{i\theta}$ are a.e.\ limits as $r\uparrow 1$ of
the functions at $re^{i\theta}$. The second condition on $u$ is that, for
$z\in\bbD$,
\begin{equation} \lb{1.12}
u(z) =0 \Leftrightarrow z+z^{-1}\text{ is a point mass of } d\gamma
\end{equation}
If one has the sufficient regularity of $\Ima M$ on $\partial\bbD$ and
$\gamma_\s$ is regular, \eqref{1.11}/\eqref{1.12} determine
$u$ via
\begin{equation} \lb{1.13x}
u(z) = \prod_{u(z_j)=0} \biggl( \f{z-z_j}{1-\bar z_j z}\biggr) \exp \biggl( \int
\f{e^{i\theta}+z}{e^{i\theta}-z}\, \log \biggl( \f{\sin\theta}{\Ima M(\theta)}\biggr)
\f{d\theta}{4\pi}\biggr)
\end{equation}
In addition, if the Jacobi parameters obey
\[
\sum_{n=1}^\infty \, \abs{a_n -1} + \abs{b_n} <\infty
\]
then the Jost function can be directly constructed using variation of parameters (see
Teschl \cite{Teschl}), perturbation determinants (see Killip-Simon \cite{KS}), or an
approach of Geronimo-Case \cite{GC80}. Since this latter approach is not well-known and
those authors do not provide the detailed estimates we will need, we have described this
approach in Appendix~A.
When there are zeros of $u$ in $\bbD$, then $u$ does not uniquely determine
$d\gamma$. $f$ is determined by \eqref{1.11} and
\begin{equation} \lb{1.12x}
f(2\cos\theta) =\Ima M(e^{i\theta})
\end{equation}
and the positions of the point masses are the zeros, but the weights, $w_j$, of the zeros
(i.e., the values of $\gamma (\{E_j\})=w_j$) are needed. The possible values of $w_j$ are
constrained by
\begin{equation} \lb{1.13}
\sum_j w_j + 2\int_0^\pi \f{\sin^2\theta}{\abs{u(e^{i\theta})}^2}\, d\theta =1
\end{equation}
by \eqref{1.12x}, \eqref{1.11}, and
\begin{equation} \lb{1.14}
\int_{-2}^2 f(E)\, dE = 2\int_0^\pi f(2\cos\theta) \sin\theta\, d\theta
\end{equation}
Thus, modulo some regularity issues, the knowledge of a $d\gamma$ with regular
$d\gamma_\s$ is equivalent to the knowledge of $u$ and the finite number of
weights $w_j$ constrained by \eqref{1.13}. Our main goal in this paper is to
describe what Jost functions and weights are associated to $a_n$'s and $b_n$'s
with a given rate of exponential decay or with finite support. We will view the
Jost function/weights as spectral data. This is justified by the following:
\begin{theorem}\lb{T1.3} Let $u$ be a function analytic in a neighborhood of
$\bar\bbD$ whose only zeros in this neighborhood lie in $\overline{\bbD} \cap \bbR$ with
those zeros all simple. For each zero in $\bbD\cap\bbR$, let a weight $w_j >0$ be given
so that \eqref{1.13} holds. Then there is a unique measure $d\gamma$ for which $u$ is the
Jost function and $w_j$ the weights.
\end{theorem}
Since this is peripheral to the main thrust of this paper, we do not give a
detailed proof, but note several remarks:
\begin{SL}
\item[1.] Related issues are discussed in Paper~I of this series \cite{Jost1}. \item[2.]
One first shows that the $M$ defined by $d\gamma$ has a meromorphic continuation to a
neighborhood of $\overline{\bbD}$; this is done in Theorem~13.7.1 of \cite{OPUC2}.
\item[3.] The methods we use in Sections~\ref{s2} and \ref{s3} then show that the $a_n
-1$ and $b_n$ decay exponentially. \item[4.] Thus, by the results of Appendix~A, a Jost
function, $\ti u$, exists. $u/\ti u$ has removable singularities, is nonzero on $\bbD$,
is analytic in a neighborhood of $\overline{\bbD}$, and on $\partial\bbD$, $\abs{u/\ti
u}=1$. Thus, $u=\bar u$.
\end{SL}
\smallskip
The perturbation determinant can be defined by
\begin{equation} \lb{1.15}
L(z) = \f{u(z)}{u(0)}
\end{equation}
This is obviously normalized by
\begin{equation} \lb{1.16}
L(0) =1
\end{equation}
which is simpler than \eqref{1.13}. Of course, $u(0)$ can be recovered from
$\{w_j\}_{j=1}^N$ and $L(z)$ by \eqref{1.15} and \eqref{1.13}. We note that
when $J-J_0$ is trace class, we have (see \cite{KS})
\begin{equation} \lb{1.17}
L(z) =\det (1+(J-J_0)[J_0 - (z+z^{-1})]^{-1})
\end{equation}
Our goal in this paper is to prove four theorems: two in the simple case
where there is no point spectrum and two in the general case. In each pair,
one describes finite support perturbations and one, exponential decay.
We begin with the case of no bound states:
\begin{theorem}\lb{T1.3A} If $a_n =1$ and $b_n =0$ for large $n$, then
$L(z)$ is a polynomial. Conversely, any polynomial $L(z)$ which obeys
\begin{SL}
\item[{\rm{(i)}}] $L(z)$ is nonvanishing on $\overline{\bbD}\backslash\{\pm 1\}$
\item[{\rm{(ii)}}] If $+1$ and/or $-1$ are zeros, they are simple \item[{\rm{(iii)}}]
$L(0)=1$
\end{SL}
is the perturbation determinant of a unique Jacobi matrix and it obeys
$a_n =1$ and $b_n =0$ for all large $n$.
\end{theorem}
{\it Remark.} By Theorem~\ref{TA.1.1}, there is a precise relation between the degree of
$L$ and the range of $(a_n - 1, b_n)$.
\begin{theorem} \lb{T1.4} Let $R>1$. If
\begin{equation} \lb{1.18}
\lim_{n\to\infty}\, (\abs{a_n -1} + \abs{b_n})^{1/2n} \leq R^{-1}
\end{equation}
then $L(z)$ has an analytic continuation to $\{z\mid\abs{z}1$ and $u(z_j)=0$ with $\abs{z_j}>R^{-1}$. We say the weight at $z_j$
is canonical if and only if
\begin{equation} \lb{1.20}
\ti w_j \equiv \lim_{z\to z_j}\, (z-z_j) M(z) = - (z_j -z_j^{-1}) [u'(z_j)\, \ol{u(1/\bar
z_j)}\,]^{-1}
\end{equation}
\smallskip
Here are our main theorems on the general case:
\begin{theorem}\lb{T1.5} If $a_n =1$ and $b_n =0$ for large $n$, then $L(z)$
is a polynomial and all the weights are canonical. Conversely, if $L$ is a
polynomial obeying
\[
(\text{\rm{i}}^\prime) \qquad\qquad L(z) \text{ is nonvanishing on }
\overline{\bbD}\backslash\bbR
\]
and {\rm{(ii)--(iii)}} of Theorem~\ref{T1.3A}, then there is at most one set of
Jacobi parameters with $a_n =1$ and $b_n =0$ for $n$ large that has that $L$
as perturbation determinant. Moreover, the weights associated to this set
are the canonical ones. If these canonical weights lead to $w_j >0$, then there
is a set of Jacobi parameters with $a_n =1$ and $b_n =0$ for large $n$.
\end{theorem}
{\it Remark.} It is easy to construct polynomial $L$'s which are not the perturbation
determinant of any finite support Jacobi parameters, although they are perturbation
determinants. For example, if $L(z_0) =L(z_0^{-1}) =0$ for some $z_0 \in (0,1)$,
\eqref{1.20} cannot hold. Thus
\[
L(z) = (1-2z)(1-\tfrac12\,z)
\]
is a perturbation determinant but not for a Jacobi matrix of finite support. There
are also examples where the canonical weights are negative.
\begin{theorem}\lb{T1.6} Let $R>1$. If \eqref{1.18} holds, then $L(z)$ has an
analytic continuation to $\{z\mid\abs{z}R^{-1}$ are canonical. Conversely, if $L(z)$ is a function
analytic in $\{z\mid\abs{z}R^{-1}$ are canonical.
\end{theorem}
These four theorems have a direct part (i.e., going from $\{a_n, b_n\}_{n=1}^\infty$
to $L(z)$) and an inverse part. The direct parts (except for the importance of
canonical weights) are well-known. We provide a proof of all but the canonical
weights in Appendix~A. The canonical weight result is proven in Section~\ref{s3}.
The inverse parts are more subtle --- and the main content of this paper. The
no bound state results appear in Section~\ref{s2} and the bound state results
in Section~\ref{s3}. Our approach is based on the use of coefficient stripping,
that is, relating $u,M$ for $\{a_n,b_n\}_{n=1}^\infty$ to $u,M$ for
$\{\ti a_n, \ti b_n\}_{n=1}^\infty$ where $\ti a_n=a_{n+1}$, $\ti b_n =
b_{n+1}$. Section~\ref{s2} will rely on a remarkably simple contraction argument,
Section~\ref{s3} on the fact that coefficient stripping only preserves
analyticity if weights are canonical.
One can wonder if one can't at least prove the no bound state results by appealing
to the Nevai-Totik theory and the Szeg\H{o} mapping (see Section~13.1 of \cite{OPUC2})
relating OPUC and OPRL. Indeed, we will show in Section~\ref{s2} that our method can
be used to prove the inverse part of their result. There is a difficulty with blind
use of the Szeg\H{o} map, already seen by the fact that $J_0$ does not map into
Verblunsky coefficients with exponential decay (see Example~13.1.3 of \cite{OPUC2}).
This can be understood by noting that the Jost function, $u$, for $d\gamma$ and
the Szeg\H{o} function, $D$, for $\mu= \Sz^{-1}(d\gamma)$ are related by
\begin{equation} \lb{1.21}
D(z)^{-1} = \f{2^{-1/2} u(z)}{1-z^2}
\end{equation}
Thus, $D(z)^{-1}$ is not analytic where $u$ is, unless $u(+1)=u(-1)=0$. In
that case, one can use Nevai-Totik to obtain Theorems~\ref{T1.3A} and \ref{T1.4}.
There are two strategies for dealing with the general case. First (and our
original proof), one can add extra $a$'s and $b$'s at the start to produce
$u(+1) = u(-1)=0$. Second, $\mu\mapsto\Sz(\mu)$ is one of four maps (see
Section~13.2 of \cite{OPUC2}). $\Sz_2$ maps onto all Jacobi matrices
with spectrum on $[-2,2]$ and with $u(1)\neq 0\neq u(-1)$ and has no division
factor. $\Sz_3$ and $\Sz_4$ divide by $1-z$ and $1+z$ and are onto all matrices
with $u(-1)\neq 0$ and $u(1)\neq 0$. In this way, one can always find a $\mu$
with $\gamma =\Sz_j(\mu)$ so the $D$-function for $\mu$ is analytic.
It should also be possible to prove the inverse results we need using the
Marchenko equation. That said, we prefer the approach in Section~\ref{s2}.
Surprisingly, the four main results of this paper appear to be new, although for
Schr\"odinger operators with Yukawa potentials, there are related results in Newton
\cite{New} and Chadan-Sabatier \cite{CS}. Geronimo \cite{Ge94} has a paper closely
related to our theme here, but he makes an a priori hypothesis about $M$ that means his
results are not strictly Jacobi-parameter hypotheses on one side. So he does not have our
results, although it is possible that one can modify his methods to prove them.
An analog of our results on what are Jost functions for Jacobi matrices of finitely
supported Jacobi parameters is the study of the sets of allowed resonance positions for
half-line Schr\"odinger operators with compactly supported potentials. There is a large
literature on this question \cite{Fad,Fro,Koro1,Koro2,Sim,Zwo1,Zwo2}. In particular in
\cite{Koro1,Koro2}, Korotyaev makes some progress in classifying all Jost functions in
this case.
We announced the results in \cite{DKS} and some of them have been presented
in \cite{OPUC2}, but we note an error in \cite{OPUC2}: Theorem~13.7.4 is
wrong because, when stating existence of a finite-range solution, it fails
to require $u(z_j^{-1})\neq 0$ and that the canonical weights be positive.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Case of No Bound States} \lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our goal in this section is to prove Theorems~\ref{T1.3A} and \ref{T1.4}.
We suppose we have a set of Jacobi parameters $\{a_n,b_n\}_{n=1}^\infty$ with
Jost function, $u(z)\equiv u^{(0)}(z)$, and $M$-function, $M(z)\equiv M^{(0)}(z)$.
Associated to Jacobi parameters $\{a_{k+n}, b_{k+n}\}_{k=1}^\infty$, we have
corresponding Jost function, $u^{(n)}(z)$, and $M$-function, $M^{(n)}(z)$. $u^{(n)}(z)$
is the solution of a difference equation at $0$ where the solution is asymptotic to $z^n$
as $n\to\infty$. It follows that
\begin{equation} \lb{2.1}
u_n(z) = a_n^{-1} z^n u^{(n)}(z)
\end{equation}
obeys (see \eqref{A.1.37})
\begin{equation} \lb{2.2}
a_n u_{n+1} + (b_n -(z+z^{-1}))u_n +a_{n-1} u_{n-1} =0
\end{equation}
Moreover (see \eqref{A.1.36} and \eqref{A.1.38}),
\begin{equation} \lb{2.3}
M^{(n)}(z) = \f{u_{n+1}(z)}{a_n u_n(z)}
\end{equation}
This leads to the following set of update formulae:
\begin{align}
u^{(n+1)}(z) &= a_{n+1} z^{-1} u^{(n)}(z) M^{(n)}(z) \lb{2.4} \\
M^{(n)}(z)^{-1} &= z+z^{-1} -b_{n+1} -a_{n+1}^2 M^{(n+1)}(z) \lb{2.5}
\end{align}
Since $M(z)=\langle\delta_0, (z+z^{-1} - J)^{-1} \delta_0\rangle$, we see
\begin{equation} \lb{2.6}
\f{M^{(n)}(z)}{z} = 1+ O(z)
\end{equation}
so that \eqref{2.5} implies
\begin{equation} \lb{2.7}
\biggl( \f{M^{(n)}(z)}{z}\biggr)^{-1} = 1-b_{n+1} z - (a_{n+1}^2 -1)
z^2 + O(z^3)
\end{equation}
which means
\begin{equation} \lb{2.8}
\log \biggl( \f{M^{(n)}(z)}{z}\biggr) = b_{n+1} z +
((a_{n+1}^2 -1) + \tfrac12\, b_{n+1}^2)z^2 + O(z^3)
\end{equation}
There is an additional feature we will need. Suppose $u(z)$ is analytic in
$\{z\mid\abs{z}1$. Define
\begin{equation} \lb{2.9}
f^\sharp (z) = \ol{f(1/\bar z)}
\end{equation}
for $z\in\bbA_R =\{z\mid R^{-1} <\abs{z} 0$ on $\bbD\cap\bbC_+$, so $\Real M>0$ for $z\in (0,1)$ and
$\Real M <0$ for $z\in (-1,0)$. It follows $u^{(1)}$ is nonvanishing on
$\overline{\bbD}\backslash \{-1,1\}$. And (ii) above shows that even if $u(\pm 1)$ is
zero, $M$ has a compensating pole.
\end{proof}
We will also need
\begin{theorem} \lb{T2.2} If the Jost function of Jacobi data $\{a_n,
b_n\}_{n=1}^\infty$ has finitely many zeros in $\bbD$ and the only zeros
on $\partial\bbD$ are at $\pm 1$ and those are simple, then
\begin{equation} \lb{2.8a}
\abs{a_n-1} + \abs{b_n} \to 0
\end{equation}
and
\begin{equation} \lb{2.8b}
M^{(n)}(z)\to z
\end{equation}
uniformly on compacts of $\bbD$. In particular, for each $\rho <1$,
\begin{equation} \lb{2.8c}
\sup_{\abs{z}\leq \rho}\, \biggl| \f{M^{(n)}(z)}{z}\biggr| \to 1
\end{equation}
\end{theorem}
\begin{proof} Since the weight of the spectral measure is given by \eqref{1.12x}
and \eqref{1.11}, the Szeg\H{o} condition holds and so does the quasi-Szeg\H{o}
condition of \cite{KS}. This plus finite spectrum show $\sum_{n=1}^\infty
\abs{a_n -1}^2 + \abs{b_n}^2 <\infty$ by the work of Killip-Simon \cite{KS}.
Thus \eqref{2.8a} holds.
That implies the corresponding Jacobi matrix $J^{(n)}$ converges in norm
to $J_0$ so the resolvents converge, which implies \eqref{2.8b}. \eqref{2.8c}
is a consequence of $M^{(n)}(z)/z\to 1$ uniformly.
\end{proof}
We now combine \eqref{2.4} and \eqref{2.10} to write the critical update
equation:
\begin{equation} \lb{2.9a}
u^{(n+1)} (z) = a_{n+1} (1-z^{-2}) (u^{(n)\sharp}(z))^{-1}
+ a_{n+1} z^{-2} u^{(n)}(z) N_n^\sharp(z)
\end{equation}
where
\begin{equation} \lb{2.10a}
N_n(z) = \f{M^{(n)}(z)}{z}
\end{equation}
so
\begin{equation} \lb{2.11a}
N_n^\sharp (z) = zM^{(n)\sharp}(z)
\end{equation}
\eqref{2.9a} looks complicated because of the $u^{(n)\sharp}$ term. But consider
expanding all functions in a Laurent series near $\{z\mid\abs{z}=R_1\}$
for $1\ell/2$,
then $|||u^{(n)}|||_R =0$, that is, $u^{(n)}$ is a constant. But then the weight in
$M$ is the free one, that is, $a_{j+n}\equiv 1$, $b_{j+n}\equiv 0$ for $j\geq 0$.
Of course, $L$ is a polynomial if and only if $u$ is.
\end{proof}
{\it Remark.} This proof and the direction in the appendix allow us to relate
the degree of the polynomial $u$ to the support of $J-J_0$.
\smallskip
\eqref{2.14} also implies
\begin{proposition} \lb{P2.3} For $10$,
\begin{equation} \lb{2.17}
\sup_{\abs{z}\leq 1}\, \abs{u^{(n)}(z) -u^{(n)}(0)}\, \abs{R-\delta}^{-2n}
\to 0
\end{equation}
which in turn, using $u^{(n)}(0)\to 1$ (by \eqref{2.8b}), implies
\[
1=2\int_0^\pi \f{\sin^2\theta}{\abs{u^{(n)}(e^{i\theta})}^2}\, d\theta
= \f{1}{\abs{u^{(n)}(0)}^2} + O(\abs{R-\delta}^{-2n})
\]
which implies
\begin{equation} \lb{2.18}
u^{(n)}(0) =1+O(\abs{R-\delta}^{-2n})
\end{equation}
Thus the difference between the free weight and the weight for $f^{(n)}$ is
$O(\abs{R-\delta}^{-2n})$, so
\begin{equation} \lb{2.19}
\limsup \biggl(\, \sup_{\abs{z}\leq\f12}\, \biggl|\f{M^{(n)}(z)}{z} -1\biggr|
\biggr)^{1/n} \leq R^{-2}
\end{equation}
By \eqref{2.8}
\begin{align*}
\limsup \abs{b_n}^{1/n} &\leq R^{-2} \\
\limsup \abs{(a_{n+1}^2-1) + \tfrac12\, b_{n-1}^2}^{1/n} & \leq R^{-2}
\end{align*}
which implies \eqref{1.18}.
\end{proof}
That completes what we want to say about OPRL with no bound states. As an aside,
we show how the ideas of this section provide an alternate to the hard (i.e.,
inverse spectral) side of the Nevai-Totik theorem, Theorem~\ref{T1.1}. Their
proof is shorter but relies on a magic formula (see (2.4.36) of \cite{OPUC1})
\[
d\mu_\s = 0\Rightarrow \alpha_n = -\kappa_\infty \int \ol{\Phi_{n+1}(e^{i\theta})}\,
D(e^{i\theta})^{-1}\, d\mu(\theta)
\]
Our proof will exploit or develop the relative Szeg\H{o} function, $\delta_0 D$,
of Section~2.9 of \cite{OPUC1}. Our goal is to prove
\begin{theorem}\lb{T2.4} Let $d\mu$ be a measure on $\partial\bbD$ with
$d\mu_\s =0$ and so that the Szeg\H{o} condition holds. Suppose $D(z)^{-1}$
has an analytic continuation to $\{z\mid\abs{z}1$. Then
\begin{equation} \lb{2.20}
\limsup_{n\to\infty}\, \abs{\alpha_n}^{1/n} \leq R^{-1}
\end{equation}
\end{theorem}
So we suppose the Szeg\H{o} condition holds, which is equivalent to
\begin{equation} \lb{2.21}
\sum_{n=0}^\infty \, \abs{\alpha_n}^2 <\infty
\end{equation}
Let $d\mu_n$ be the measure with Verblunsky coefficients $\{\alpha_{k+n}\}_{k=0}^\infty$
and $D^{(n)}$ its Szeg\H{o} function. $F^{(n)}$ and $f^{(n)}$ are defined by
\begin{align}
F^{(n)}(z) &= \int \f{e^{i\theta}+z}{e^{i\theta}-z}\, d\mu^{(n)}(\theta) \lb{2.22} \\
F^{(n)}(z) &= \f{1+z f^{(n)}(z)}{1-zf^{(n)}(z)} \lb{2.23}
\end{align}
Geronimus' theorem (see \cite{OPUC1}) says that the relation between the $f$'s
is given by the Szeg\H{o} algorithm,
\begin{equation} \lb{2.24x}
f^{(n)}(z) \equiv \f{\alpha_n + zf^{(n+1)}(z)}{1+\bar\alpha_n zf^{(n+1)}(z)}
\end{equation}
and the equivalent
\begin{equation} \lb{2.25x}
zf^{(n+1)}(z) = \f{f^{(n)}(z)-\alpha_n}{1-\bar\alpha_n f^{(n)}(z)}
\end{equation}
In Section~2.9 of \cite{OPUC1}, the relative Szeg\H{o} function is defined by
($\rho_n = (1-\abs{\alpha_n}^2)^{1/2}$)
\begin{equation} \lb{2.24}
(\delta_nD) (z) = \f{1-\bar\alpha_n f^{(n)}(z)}{\rho_n} \, \,
\f{1-zf^{(n+1)}(z)}{1-zf^{(n)}(z)}
\end{equation}
and it is proven that
\begin{equation} \lb{2.25}
(\delta_n D)(z) = \f{D^{(n)}(z)}{D^{(n+1)}(z)}
\end{equation}
which we write as
\begin{equation} \lb{2.26}
D^{(n+1)}(z)^{-1} = D^{(n)}(z)^{-1} (\delta_n D)(z)
\end{equation}
It will be useful to rewrite \eqref{2.24} using \eqref{2.25x} to get
\begin{equation} \lb{2.27}
(\delta_n D)(z) = \f{1-\bar\alpha_n f^{(n)}(z) - f^{(n)}(z) + \alpha_n}
{\rho_n (1-zf^{(n)}(z))}
\end{equation}
Using
\begin{equation} \lb{2.27a}
f^{(n)}(z) = \f{1}{z}\, \f{F^{(n)}(z)-1}{F^{(n)}(z)+1}
\end{equation}
one finds
\begin{equation} \lb{2.28b}
(\delta_n D)(z) = \tfrac12\, z^{-1} M^{(n)}(z)
\end{equation}
where
\begin{equation} \lb{2.28c}
M^{(n)}(z) = z(1+\alpha_n) (F^{(n)}(z) +1) - (1+\bar\alpha_n)
(F^{(n)}(z) -1)
\end{equation}
Interestingly enough, $M^{(n)}(z)$ for $n=0$ appears in the theory of minimal
Carath\'eodory functions on the hyperelliptic Riemann surfaces that occur in the analysis
of OPUC with periodic Verblunsky coefficients (see (11.7.76) in \cite{OPUC2}); a related
function appears in Geronimo-Johnson \cite{GJo1}. While \cite{OPUC1,OPUC2} introduced
both $\delta_0D$ and $M^{(0)}(z)$, its author appears not to have realized the relation
\eqref{2.28b}. $\delta_nD$ is nonsingular at $z=0$ since $M^{(n)}(z) = 0$ at $z=0$ (since
$F^{(n)}(0)=1$). We note that where Section~11.7 of \cite{OPUC2} uses $M(z)$, it could
use $(\delta_0 D)(z)$. The difference is the $0$ at $0_+$ is moved to $\infty_-$ and the
pole at $\infty_+$ to $0_-$. We note that the relation between $M$ and $\delta_0D$ is
hinted at in \eqref{2.24}. In gaps in $\supp (d\mu)$ in $\partial\bbD$, $\delta_0D$ has
poles at zeros of $1-zf$ and zeros at zeros of $1-zf_1$. By \eqref{2.23}, $\delta_0 D$
has poles at poles of $F$ and zeros at poles of $F^{(1)}$, which is the critical property
that $M$ needs in the analysis of Section~11.7 of \cite{OPUC2}.
We will also need the analytic continuation of
\begin{equation} \lb{2.28}
\Real F^{(n)} (e^{i\theta}) = \abs{D^{(n)}(e^{i\theta})}^2
\end{equation}
namely,
\begin{equation} \lb{2.29}
F^{(n)} + (F^{(n)})^\sharp = 2D^{(n)} (D^{(n)})^\sharp
\end{equation}
where $\sharp$ is given by \eqref{2.9}.
\begin{theorem}\lb{T2.5} Let $R>1$. If $D^{-1}$ is analytic in $\{z\mid
\abs{z}R^{-1}$, $M(z)$ may have a
pole at $1/z_j$ due to the $(u^\sharp)^{-1}$ term. $uM$ will then have a pole
(unless $u$ has a zero, which we will see does not help). The way to avoid this
is to arrange for $M^\sharp$ to have a compensating pole, and this will happen
precisely if the weight at $z_j$ is canonical! Here is the detailed result:
\begin{theorem}\lb{T3.2} Let $u$ be a Jost function and be analytic in $\{z\mid
\abs{z} R^{-1}$. Moreover,
\begin{SL}
\item[{\rm{(i)}}] If $u$ has a zero of order $k\geq 1$ at $z_j^{-1}$,
then $M$ has a pole of order $k+1$ there.
\item[{\rm{(ii)}}] If $u(z_j^{-1})\neq 0$, then $M$ has a pole at $z_j^{-1}$
if and only if the weight at $z_j$ is {\em not} canonical.
\end{SL}
In particular, $u^{(1)} =uM$ is analytic in $\{z\mid\abs{z}R^{-1}$ are canonical.
\end{theorem}
\begin{proof} The zeros at $z_j$ are simple, so if $u$ has a zero of order
$k\geq 1$ at $z_j^{-1}$, then $(u(z) u^\sharp (z))^{-1}$ has a pole of
order $k+1$. Since $M^\sharp(z)$ has only simple poles in $\{z\mid\abs{z}
>1\}$, $M(z)$ has a pole of order $k+1$. This proves (i).
If $u(z_j^{-1})\neq 0$, both $(z-z^{-1}) [u(z)u^\sharp(z)]^{-1}$ and $M^\sharp (z)$ have
simple poles at $z_j^{-1}$. Their residues cancel if and only if \eqref{1.20} holds.
\end{proof}
This gets us one step if the weights are canonical. We get beyond that because
automatically weights after that are canonical!
\begin{theorem}\lb{T3.3} Let $u$ be a Jost function and be analytic in $\{z\mid
\abs{z}R^{-1}$ implies that $u^{(1)}
(\ti z_j^{-1})\neq 0$ and the weight of $M^{(1)}$ is canonical.
\end{theorem}
\begin{proof} Zeros of $u$ in $\bbD$ are cancelled by poles of $M$\!, so
$u^{(1)}$ has zeros precisely at points $\ti z_j$ where $M(\ti z_j)=0$. Since $[M(\ti
z_j) -M^\sharp(\ti z_j)] u^\sharp (\ti z_j) u (\ti z_j) = \ti z_j -\ti z_j^{-1} \neq 0$,
$M^\sharp (\ti z_j) \neq 0$. By \eqref{2.5}, $M(\ti z_j^{-1}) \neq 0$ implies
$M^{(1)}(z)$ is regular at $\ti z_j^{-1}$. By a small calculation, $u^{(1)}, M^{(1)}$
obey \eqref{2.10}, so the weight must be canonical.
\end{proof}
{\it Remark.} There is a potentially puzzling feature of Theorem~\ref{T3.3}.
If stripping a Jacobi parameter pair cannot produce noncanonical weights,
how can they occur? After all, we can add a parameter pair before $J$ and
then remove it. The resolution is that adding a parameter pair also shrinks
the region of analyticity if $J$ has a noncanonical weight. Essentially,
noncanonical weights produce poles in a meromorphic $u^{(n)}(z)$ with
residues which are a ``resonance eigenfunction." Lack of a canonical
weight in $u^{(n)}$ is a sign that this resonance eigenfunction function
vanishes at $n$, but then it will not at $n+1$ or $n-1$.
\begin{proof}[Proof of Theorem~\ref{T1.5}] If some weight is not canonical,
$u^{(1)}$ is not entire, and so the Jacobi parameters cannot have finite support. If all
weights are positive, one can normalize $u$ so \eqref{1.13} holds with the canonical
weights and obtain a positive weight since the necessary canonical weights are all
positive. With canonical weights, $uM$ is entire and by \eqref{3.1} and $M(z) =O(z)$ at
$z=0$, we see that $M(z) =O(1/z)$ at $z=\infty$ so long as $u(z) =O(z^\ell)$ with $\ell
>2$. Thus $u^{(1)}$ is a polynomial of degree at most $1$ less than $u$. Iterating and
using Theorem~\ref{T3.1}, we eventually get a polynomial of $u^{(k)}(z)$ with no zeros in
$\bbD$ and so, by Theorem~\ref{T1.3A}, a finite-range set of $\{a_n, b_n\}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T1.6}] If weights are canonical, we can
iterate to a $u^{(k)}$ nonvanishing in $\bbD$, and use Theorem~\ref{T1.4}.
If some weight is not canonical, $u^{(1)}$ is not analytic in $\{z\mid\abs{z}
1$. $a_n^2 -1$ will enter in estimates, but since $\abs{a_n-1}
\leq \abs{a_n^2-1} \leq (1+\sup_n \abs{a_n}) (\abs{a_n-1})$ in all these
estimates, $\abs{a_n-1}$ can replace $\abs{a_n^2 -1}$ with no change.
In all cases, $\sum \abs{a_n-1}<\infty$, so $\prod_{j=1}^n a_j$ is uniformly bounded
above and below, and hence $c$ (resp., $g$) and $C$ (resp., $G$) are comparable.
We will first prove bounds and use them to control convergence:
\begin{theorem} \lb{TA.1.2}
\begin{SL}
\item[{\rm{(i)}}] Let \eqref{A.1.21} hold. Then for each $z\in\overline{\bbD} \backslash
\{\pm 1\}$,
\begin{equation} \lb{A.1.24}
\sup_n \, [\abs{G_n(z)} + \abs{C_n(z)}] \equiv A_0(z) < \infty
\end{equation}
where $A_0(z)$ is bounded uniformly on compact subsets of $\overline{\bbD}
\backslash\{\pm 1\}$. \item[{\rm{(ii)}}] Let \eqref{A.1.22} hold. Then for some constant
$A_1$,
\begin{align}
\sup_{n,z\in\overline{\bbD}}\, \abs{G_n(z)} &\leq A_1 \lb{A.1.25} \\
\sup_{n,z\in\overline{\bbD}}\, \f{\abs{C_n (z)}}{1+n} &\leq A_1 \lb{A.1.26}
\end{align}
\item[{\rm{(iii)}}] Let \eqref{A.1.23} hold and let $R>1$. Then there is some constant
$A_2$ such that for all $z$ with $\abs{z}