\input amstex
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\documentstyle{amsppt}
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\baselineskip=15 pt
%\NoBlackBoxes
\TagsOnRight
\def\gap{\vskip 0.1in\noindent}
\def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.}
%References
\def\am {1} % Agranovich-Marchenko
\def\akh {2} % Akhiezer
\def\aki {3} % Antony-Krishna I
\def\akii {4} % Antony-Krishna II
\def\bak {5} % Baker
\def\bggk {6} % Battig-Grebert-Guillot-Kappeler
\def\ber {7} % Berezanskii
\def\bfr {8} % Bloch-Flaschka-Ratiu
\def\bg {9} % Boley-Golub
\def\boorg {10} % Boor-Golub
\def\borg {11} % Borg
\def\borgun {12} % Borg unique
\def\bght {13} % Bulla-Gesztesy-Holden-Teschl
\def\casi {14} % Case, II
\def\casev {15} % Case, V
\def\casii {16} % Case, one dimension
\def\casiii {17} % Case, transport
\def\casiv {18} % Case, scattering
\def\casevi {19} % Case, VI
\def \cc {20} % Case-Chui
\def\ck {21} % Case-Kac
\def\data {22} % Date-Tanaka
\def\dn {23} % Deift-Nanda
\def\dt {24} % Deift-Trubowitz
\def\fe {25} % Ferguson
\def\fl {26} % Flaschka
\def\fh {27} % Fu-Hochstadt
\def\gg {28} % Gasymov-Guseinov
\def\gl {29} % Gelfand-Levitan
\def\gc {30} % Geronimo-Case
\def\geho {31} % Gesztesy-Holden
\def\gr {32} % Gesztesy-Renger
\def\gsxi {33} % Gesztesy-Simon, xi function
\def\gsun {34} % Gesztesy-Simon, uniqueness paper
\def\gsac {35} % Gesztesy-Simon, ac paper
\def\gsdis {36} % Gesztesy-Simon, discrete spectrum
\def\gt {37} % Gesztesy-Teschl
\def\gkt {38} % GesztesyKrishna-Teschl
\def\gla {39} % Gladwell
\def\gh {40} % Gragg-Harrod
\def\gw {41} % Gray-Wilson
\def\gui {42} % Guseinov I
\def\guii {43} % Guseinov II
\def\guiii {44} % Guseinov III
\def\hald {45} % Hald
\def\harch {46} % Hochstadt - Arch. Math.
\def\hspec {47} % Hochstadt - spectral data
\def\hmixed {48} % Hochstadt - mixed given data
\def\hl {49} % Hochstadt-Liberman
\def\kmi {50} % Kac-van Moerbeke I
\def\kmii {51} % Kac-van Moerbeke II
\def\kmiii {52} % Kac-van Moerbeke III
\def\la {53} % Landau
\def\levs {54} % Levinson
\def\lev {55} % Levitan - AMS Transl.
\def\levbook {56} % Levitan book
\def\lg {57} % Levitan-Gasymov
\def\ls {58} % Levitan-Sargsjan
\def\maun {59} % Marchenko - uniqueness
\def\mar {60} % Marchenko
\def\mr {61} % Masson-Repka
\def\van {62} % van Moerbeke
\def\mm {63} % Moerbeke-Mumford
\def\pt {64} % Poschl-Trubowitz
\def\simon {65} % Simon - Vancouver
\def\syi {66} % Sodin-Yuditskii I
\def\syii {67} % Sodin-Yuditskii II
\def\sze {68} % Szego
\def\te {69} % Teschl
\def\tom {70} % Tomei
\def\zmrs {71} % Zakhariev et al.
\topmatter
\title M-functions and Inverse Spectral
Analysis for Finite and Semi-Infinite
Jacobi Matrices
\endtitle
\rightheadtext{M-functions and Inverse
Spectral Analsysis}
\author Fritz Gesztesy$^1$ and Barry Simon$^2$
\endauthor
\leftheadtext{F.~Gesztesy and B.~Simon}
\thanks$^1$ Department of Mathematics,
University of Missouri, Columbia,
MO~65211, USA.
E-mail: mathfg\@mizzou1.missouri.edu
\endthanks
\thanks$^2$ Division of Physics, Mathematics,
and Astronomy, California
Institute of
Technology, Pasade- \linebreak na, CA~91125,
USA. E-mail: bsimon\@caltech.edu
\endthanks
\thanks This material is based upon work
supported by the National Science
Foundation under Grant Nos.~DMS-9623121
and DMS-9401491.
\endthanks
%\date August 30, 1996
%\enddate
\abstract We study inverse spectral analysis
for finite and semi-infinite Jacobi
matrices $H$.
Our results include a new proof of the central
result of the inverse theory
(that the spectral
measure determines $H$). We prove an
extension of Hochstadt's theorem (who
proved
the result in the case $n=N$) that $n$
eigenvalues of an $N\times N$ Jacobi
matrix, $H$,
can replace the first $n$ matrix elements in
determining $H$ uniquely. We
completely
solve the inverse problem for $(\delta_n,
(H-z)^{-1}\delta_n)$ in the $N<\infty$
case.
\endabstract
\endtopmatter
\document
\vskip 0.2in
\flushpar{\bf {\S 1. Introduction}}
\vskip 0.1in
There is an enormous literature on
inverse spectral problems for
$-\frac{d^2}{dx^2}+V(x)$
(see [\am, \gl, \levbook--\mar, \pt] and
references therein), but considerably
less for their discrete
analog, the infinite and semi-infinite Jacobi
matrices (see, e.g., [\aki, \akii,
\bggk--\bfr,
\bght--\data, \dt, \fl--\gg, \gc, \gr,
\gt, \gkt, \gui--\guiii, \kmi--\kmiii,
\mr--\mm, \syi, \syii,
\te--\zmrs]) and even less for finite
Jacobi matrices (where references include,
e.g., [\bg, \boorg,
\dn, \fe, \gla, \gh, \gw, \hald--\hmixed]).
Our goal in this paper is to study
the last two problems
using one of the most powerful tools from
the spectral theory of
$-\frac{d^2}{dx^2} +V(x)$,
the $m$-functions of Weyl.
Explicitly, we will study finite
$N\times N$ matrices of the form:
$$
H=\pmatrix
b_1 & a_1 & 0 & 0 & \cdot &
\cdot & \cdot \\
a_1 & b_2 & a_2 & 0 & \cdot &
\cdot & \cdot \\
0 & a_2 & b_3 & a_3 & \cdot &
\cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot &
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot &
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot &
0 & a_{N-1} & b_N
\endpmatrix \tag 1.1
$$
and the semi-infinite analog $H$ defined on
$$
\ell^2 (\Bbb N) = \biggl\{ u = (u(1),
u(2), \dots) \biggm| \sum^\infty_{n=1}
|u(n)|^2 <
\infty \biggr\}
$$
given by
$$\alignedat2
(Hu)(n) &= a_n u(n+1) + b_n u(n) + a_{n-1}
u(n-1), && \qquad n \geq 2 \\
&= a_1 u(2) + b_1 u(1), &&\qquad n=1.
\endalignedat \tag 1.2
$$
In both cases, the $a$'s and $b$'s are real
numbers with $a_n >0$.
To avoid inessential technical complications,
we will only consider the case
where $\sup_n [|a_n| + |b_n|] <\infty$ in which
case $H$ is a map from $\ell^2$ to
$\ell^2$ and defines a bounded self-adjoint
operator.
In the semi-infinite case, we will set
$N=\infty$. At times to have unified
notation, we will use
something like $1\leq j < N+1$ to indicate
$1\leq j \leq N$ in the finite case
and $1\leq j <
\infty$ in the semi-infinite case.
It will sometimes be useful to consider
the $b$'s and $a$'s as a single sequence
$b_1, a_1,
b_2, a_2, \dots := c_1, c_2,
\dots$, that is,
$$
c_{2n-1} = b_n, \quad c_{2n} = a_n,
\qquad n=1,2, \dots \, . \tag 1.3
$$
What makes Jacobi matrices special among
all matrices is that the eigenvalue
condition $Hu =
\lambda u$ is a second-order difference
equation. The $n=1$ case of (1.2) can be
thought of as
forcing the Dirichlet boundary condition
$u(0)=0$. Thus, any possible non-zero
solution of
$Hu =\lambda u$ must have $u(1)\neq 0$,
which implies
\roster
\item"\rom{(i)}" Eigenvalues of $H$ must be
simple (or else a linear combination
would
vanish at $n=1$).
\item"\rom{(ii)}" Eigenfunctions must be
non-vanishing at $n=1$.
\endroster
Thus for $N<\infty$, $H$ has eigenvalues
$\lambda_1 < \cdots < \lambda_N$ and
associated
orthonormal eigenvectors $\varphi_1,
\dots, \varphi_N$ with $\varphi_j (1) \neq
0$. For
$N=\infty$, the proper way of encompassing
(i), (ii) is that $\delta_1$ is a
cyclic vector for
$H$ ($\delta_j$ is the vector in $\ell^2$
with $\delta_j (n) =1$ (resp.~$0$) if
$n=j$ (resp.~$n
\neq j$)).
The spectral measure $d\rho$ for the pair
$(H, \delta_1)$ is defined by
$(\delta_1, H^\ell
\delta_1 ) = \int \lambda^\ell \, d\rho
(\lambda)$. Since our $H$'s are bounded,
$d\rho$ is
a measure of bounded support. In case
$N<\infty$,
$$
d\rho(\lambda) = \sum^N_{j=1} |\varphi_j
(1)|^2 \delta(\lambda - \lambda_j) \,
d\lambda
\qquad (\varphi_j, \varphi_k) = \delta_{j,k}.
\tag 1.4
$$
The central fact of the inverse theory is
that $d\rho$ determines the $a$'s and
$b$'s and any
$d\rho$ can occur for a unique $H$.
(If $N<\infty$, $d\rho$ has support at
exactly $N$ points.
If $N= \infty$, $d\rho$ must have infinite
support.) The usual proof of this
central fact is via
orthogonal polynomials, and has been
rediscovered by many people (see references
in the
appendix). For the reader's convenience, we
have a brief appendix presenting
this approach.
One purpose of this paper is to present in
Section~3 a new approach to the
central result based
on $m$-functions and trace formulas. Given
$\rho$, one forms $m(z) = \int d\rho
(\lambda) \,
(\lambda -z)^{-1}$. $m(z)$ has an asymptotic
expansion at infinity given by
$$
m(z) \sim -\frac{1}{z} - \frac{b_1}{z^2} -
\frac{a^2_1 + b^2_1}{z^3} +
O(z^{-4}).
\tag 1.5
$$
Thus, one easily recovers $b_1$ and $a_1$
(recall $a_1 > 0$) from $m(z)$. Now
define
$m_1 (z)$ by
$$
(-m(z))^{-1} = z-b_1 + a^{2}_1 m_1 (z).
\tag 1.6
$$
It turns out that $m_1 (z)$ is the spectral
measure for the Jacobi matrix
obtained by removing
the top row and left-most column of $H$.
An obvious inductive procedure obtains
$b_2, a_2,
\dots\,$.
The $m$-functions defined by this method,
which we call $m_+ (z,n)$ (so $m(z) :=
m_+ (z,0)$,
$m_1 (z) := m_+ (z,1)$, etc.), is one class
of $m$-functions defined by
$$
m_+ (z,n) = (\delta_{n+1}, (H_{[n+1, N]}
-z)^{-1} \delta_{n+1}), \tag 1.7
$$
where $H_{[n+1, N]}$ is the matrix with
the top $n$ rows and left $n$ columns
removed and
thought of as acting on $\ell^2 (\{n+1, n+2,
\dots, N\})$. There is a second
$m$-function that
plays a role,
$$
m_- (z,n) = (\delta_{n-1},
(H_{[1, n-1]}-z)^{-1} \delta_{n-1}), \tag 1.8
$$
where $H_{[1, n]}$ is the $n\times n$ upper
left corner of $H$.
Section~2 relates these $m$-functions to
solutions of the second-order
difference equation and
obtains relations between $m_\pm (z,n)$
and $m_\pm (z,n+1)$ (of which (1.6) is a
special
case). Section~2 also contains some
critical formulas expressing the diagonal
Green's functions
$G(z, n,n) := (\delta_n, (H-z)^{-1}
\delta_n)$ in terms of $m_+ (z)$ and $m_-
(z)$.
Section~4 contains one of the most
intriguing results of this paper. In
[\hmixed], Hochstadt
proved the remarkable result that for a
finite Jacobi matrix, a knowledge of all
but the first
$N$ $c$'s and the $N$-eigenvalues, that is,
of $c_{N+1}, c_{N+2}, \dots,
c_{2N-1}$ and
$\lambda_1, \dots, \lambda_N$, determines
$H$ uniquely. In Section~4, we extend
this by
showing that $c_{n+1}, \dots, c_{2N-1}$
and {\it any} $n$ eigenvalues of $H$
determine
$H$ uniquely for any $n=1,2,\dots, N$.
After a brief interlude in Section~5
obtaining the straightforward analog of
Borg's two-spectra
theorem [\borg] (see also [\borgun, \levs,
\lev, \lg, \maun]) first considered
in the Jacobi context
by Hochstadt [\harch, \hspec] (see also
[\boorg, \fh, \gh, \gw, \hald, \hmixed,
\te]), we turn in
Section~6 to the question of determining
$H$ from a diagonal Green's function
element
$(\delta_n, (H-z)^{-1} \delta_n)$ when
$N< \infty$. If $n=1$ or $N$, the central
inverse
spectral theory result says $G(z,n,n)$
uniquely determines $H$. For other $n$,
there are always
at least $\binom{N-1}{n-1}$ different $H$'s
compatible with a given $G(z, n,n)$.
Generically,
there are precisely that many $H$'s.
Section~6 has a complete analysis.
In a final Section~7 we present some
results and conjectures about the inverse
problem when
$a_n \equiv 1$.
\vskip 0.3in
\flushpar{\bf {\S 2. $\boldkey m$-Function
Formulas}}
\vskip 0.1in
Let $H$ be a finite or semi-infinite
Jacobi matrix of the type described in
Section~1. We
begin by defining some special functions
of a complex variable $z$ which we will
call
$\{P(z,n)\}^{N+1}_{n=1}$ and
$\{\psi_+(z,n)\}^N_{n=0}$. The $P(z,n)$'s are
polynomials
of degree $n-1$ defined by the pair of
conditions
$$\gather
a_n P(z,n+1) + b_n P(z,n) + a_{n-1} P(z,n-1)
= z P(z,n), \qquad 1\leq n < N+1,
\tag 2.1 \\
P(z,0) = 0, \qquad P(z,1) = 1.
\endgather
$$
(For convenience, we define $a_N := 1$ in
order to define $P(z,N+1)$ in case $N
< \infty$.)
Clearly, (2.1) defines $P(z,n)$ inductively
as a polynomial of the claimed
degrees. Again,
inductively it is clear that
$$
P(z,j+1) = \frac{1}{a_1 \dots a_j}\, z^j +
\text{ lower degree in $z$}. \tag 2.2
$$
As explained in the appendix, the $P$'s are
intimately related to the spectral
measure for $H$
(as defined in Section~1).
\proclaim{Proposition 2.1}
{\rom{ ([\ber], p.~542)}}
$$
P(z,j+1) = (a_1 \dots a_j)^{-1}
\det(z- H_{[1,j]}), \qquad j \geq 1, \tag 2.3
$$
where $H_{[1,j]}$ is the $j\times j$ matrix
in the upper left corner of $H$.
\endproclaim
\demo{Proof} By (2.2), $a_1 \dots a_j
P(z,j+1)$ and $\det (z-H_{[1,j]})$ are
monic
polynomials of degree $j$. Thus, it suffices
to show they have the same zeros
and multiplicities.
But $P(z,j+1)=0$ if and only if there is a
vector $v = (v_1, \dots, v_j)$ with
$v_1 = 1$ so that
$(H_{[1,j]}-z)v=0$. As explained in
Section~1, every eigenvector of $H_{[1,j]}$
has $v_1
\neq 0$. Thus, the zeros of $P(z,j+1)$ are
precisely the eigenvalues of
$H_{[1,j]}$. Since the
eigenvalues are simple, the multiplicities
are all one. \qed
\enddemo
In case $N<\infty$, $\psi_+(z,n)$ is
defined via
$$\gather
a_n \psi_+(z,n+1) + b_n \psi_+(z,n) +
a_{n-1} \psi_+(z,n-1) = z \psi_+(z,n),
\, \, \,
n=0, \dots, N-1, \tag 2.4 \\
\psi_+(z,N) = 1, \qquad \psi_+(z,N+1) = 0,
\endgather
$$
where again for convenience we define
$a_0 =1$ to enable us to define
$\psi_+(z,0)$. $\psi_+$
is just like $P$ but run from the other
end. By the same reasoning,
$$
\psi_+(z,N-j) = \frac{1}{a_{N-1} \dots
a_{N-j}} \, \det (z-H_{[N-j+1, N]}) \tag
2.5
$$
is a polynomial of degree $j$.
In case $N=\infty$, $\psi_+(z,n)$ initially
is only defined in the region
$\text{Im}\, (z)\neq 0$
by requiring (2.4) and
$$
\psi_+(z,0) = 1, \qquad \sum^\infty_{n=0}
|\psi_+(z,n) |^2 < \infty. \tag 2.6
$$
It is a standard argument that when $H$ is
bounded and self-adjoint, there is a
solution that
is $\ell^2$ at infinity unique up to
constant multiples (and everywhere
non-vanishing so one
can normalize it by $\psi_+(z,0) =1$).
Given any two sequences $u(n), v(n)$,
define the (modified) Wronskian $W(u,v)$
by
$$
W(u,v) (n) = a_n [u(n) v(n+1) - u(n+1) v(n)].
$$
For any two solutions of (2.1), $W$ is
constant. The Green's function is defined
by
$(1 \leq m, n < N+1)$
$$
G(z,m,n) = (\delta_m, (H-z)^{-1} \delta_n)
\tag 2.7
$$
for $\text{Im}\, (z) \neq 0$. We will also
sometimes use $(j \leq m,n\leq k)$
$$
G_{[j,k]} (z,m,n) = (\delta_m,
(H_{[j,k]} -z)^{-1} \delta_n).
$$
We have the following standard formula:
\proclaim{Proposition 2.2}
$$
G(z,m,n) = [W (P(z,\cdot\,),
\psi_+(z,\cdot\,))]^{-1} P(z,\min(m,n))
\psi_+(z,\max(m,n)).
\tag 2.8
$$
\endproclaim
\demo{Proof} One easily checks that
if $G(z,m,n)$ is defined by (2.8),
then
$$
\sum_k (H_{m,k} - z\delta_{m,k})
G(z,k,n) = \delta_{m,n}.
$$
In the finite case, the choice of
$P$, $\psi_+$ ensures that the equation
holds
at the points
where $n$ or $m$ equals $1$ or $N$.
In the infinite case, the choice of $P$
ensures the
equation holds at $n$ or $m$ equals $1$,
and the choice of $\psi_+$ ensures that
$\sum_n
G(z,k,n) f_n$ is $\ell^2$ in $k$ for any
finite support sequence $\{f_n\}$. In
either case, it
follows that $G(z,m,n)$ is indeed the
matrix of the resolvent. \qed
\enddemo
We can now define the most basic
$m$-function (there will be more later),
$$
m(z)= (\delta_1, (H-z)^{-1} \delta_1).
\tag 2.9
$$
By (2.8), we claim
\proclaim{Proposition 2.3}
$$
m(z) = -\frac{\psi_+(z,1)}
{a_0 \psi_+(z,0)}\, . \tag 2.10
$$
\endproclaim
\remark{Remark} By our convention,
$a_0 =1$, but we carry it along for the
general definition
of $m(z,n)$ later (cf.~(2.14)).
\endremark
\demo{Proof} $P(z,0)=0$, $P(z,1) =1$
so (2.8) becomes
$$
G(z,1,1) = \frac{\psi_+(z,1)}
{-a_0 \psi_+(z,0)}\, . \qed
$$
\enddemo
In terms of the spectral measure
$d\rho$,
$$
m(z) = \int \frac{d\rho (\lambda)}
{\lambda -z}\, . \tag 2.11
$$
\proclaim{Theorem 2.4} If $N$ is finite,
then
$$
m(z) = -\frac{\prod^{N-1}_{\ell=1}
(z-\nu_{\ell})}{\prod^N_{j=1}
(z-\lambda_j)}\,
,
\tag 2.12
$$
where $\lambda_1 < \cdots < \lambda_N$
are the eigenvalues of $H$ and $\nu_1 <
\cdots <
\nu_{N-1}$ are the eigenvalues of
$H_{[2,N]}$.
\endproclaim
\demo{Proof} By (2.5) and (2.10),
$$
m(z) = -\frac{\det (z-H_{[2,N]})}
{\det(z-H)}\, .
$$
This can be viewed as a cofactor formula
for the matrix elements of
$(H-z)^{-1}$. \qed
\enddemo
\proclaim{Corollary 2.5} If $N$ is finite,
$\{\lambda_j\}^N_{j=1} \cup
\{\nu_{\ell}\}^{N-1}_{\ell=1}$ uniquely
determine $H$. Any set of real
$\lambda$'s
and $\nu$'s are allowed as long as
$$
\lambda_1 < \nu_1 < \lambda_2 < \nu_2 <
\cdots < \lambda_N. \tag 2.13
$$
\endproclaim
\demo{Proof} By (2.12), the $\lambda$'s
and $\nu$'s determine $m$, and then by
(2.11),
they determine $d\rho$. By Theorem~A.6
(in the appendix), $d\rho$ determines the
$a$'s and
$b$'s. That any $\nu$'s, $\lambda$'s are
allowed follows from the fact that if
$$
m(z) = \sum^N_{j=1} \frac{\alpha_j}
{\lambda_j -z} \, ,
$$
then
$\alpha_j >0$ for all $j$ is equivalent
to (2.13). \qed
\enddemo
\remark{Remark} The result (proven, e.g.,
in [\boorg, \gh, \gw, \hald--\hmixed])
is an analog
of Borg's result for Sturm-Liouville
operators that two spectra determine the
potential. Its
analog for $N=\infty$ (see [\fh]) is that
Krein's spectral shift function for
the pair
$(H, H_{[2,\infty)})$ determines $m(z)$
(cf.~[\gsun, \te]).
\endremark
\definition{Definition 2.6} $m_+(z,n) =
(\delta_{n+1}, (H_{[n+1, N]} -z)^{-1}
\delta_{n+1})$,
$n=0,1, \dots, N-1$, where $H_{[n+1, N]}$
is interpreted as $H_{[n+1, \infty)}$
if $N=\infty$.
\enddefinition
Thus, $m(z) := m_+(z,0)$, and by the same
calculation that led to (2.10),
$$
m_+(z,n) = -\psi_+(z,n+1) \big/ [a_n
\psi_+(z,n)]. \tag 2.14
$$
>From (2.4), we deduce the following Ricatti
equation (more precisely, an analog
of what is a
Ricatti equation in the continuum case),
$$
a^2_n m_+(z,n) + \frac{1}{m_+(z,n-1)}
=b_n-z. \tag 2.15
$$
It is also useful to have an analog of
the $m$-function but starting at $1$
instead of at $N$ or
$\infty$.
\definition{Definition 2.7} $m_-(z,n)=
(\delta_{n-1}, (H_{[1, n-1]} -z)^{-1}
\delta_{n-1})$,
$n=2,3,\dots, N+1$.
\enddefinition
We immediately have analogs of (2.14)
and (2.15), viz.,
$$\gather
m_-(z,n) = -P(z,n-1) \big/ [a_{n-1}
P(z,n)], \tag 2.16 \\
a^2_{n-1} m_-(z,n) + \frac{1}{m_-(z,n+1)}
= b_n -z. \tag 2.17
\endgather
$$
The usefulness of having both $m_+ (z)$
and $m_- (z)$ is that we can use them to
express
$G(z,n,n)$. We claim
\proclaim{Theorem 2.8}
$$\align
G(z,n,n) &= \frac{-1}{a^2_n m_+(z,n) +
a^2_{n-1} m_-(z,n) + z-b_n} \tag 2.18 \\
&= \frac{-1}{a^2_{n-1} m_-(z,n) - \frac{1}
{m_+(z,n-1)}} \tag 2.19 \\
&= \frac{-1}{a^2_n m_+(z,n) - \frac{1}
{m_-(z,n+1)}}, \qquad n=1,2, \dots \, .
\tag 2.20
\endalign
$$
\endproclaim
\demo{Proof} It suffices to prove (2.19),
for then (2.18) follows from (2.15)
and then (2.20)
follows from (2.17).
To prove (2.19), use (2.8) evaluating the
Wronskian at $n-1$ to see that
$$\align
G(z,n,n) &= \frac{1}{a_{n-1} \left(
\frac{P(z,n-1)}{P(z,n)} -
\frac{\psi_+(z,n-1)}{\psi_+ (z,n)}
\right)} \\
&= \frac{1}{-a^2_{n-1} m_-(z,n) +
(m_+(z,n-1))^{-1}}
\endalign
$$
by (2.14) and (2.16). \qed
\enddemo
\proclaim{Theorem 2.9} Let $N \in
\Bbb N$. At any eigenvalue $\lambda_j$ of $H$
we infer
that
$$
m_-(\lambda_j,n+1) = [a^2_n
m_+(\lambda_j,n)]^{-1}, \qquad 1
\leq n \leq N-1,
\tag 2.21
$$
where equality in {\rom{(2.21)}} includes
the case that both sides equal
infinity.
\endproclaim
\demo{Proof} At first sight, this would
seem to be a triviality. For $G(z,n,n)$
has poles at
$\lambda_j$ and thus the denominator in
(2.20) must vanish. But there is a
subtlety. It can
happen that at an eigenvalue $\lambda_j$
of $H$, $P(\lambda_j,n)
=\psi_+(\lambda_j,n) = 0$
and $G(z,n,n)$ then also vanishes at
$\lambda_j$.
Thus we consider two cases: First
$\varphi_j (n) \neq 0$ ($\varphi_j$ the
eigenvector of $H$
associated with $\lambda_j$). In that case
$G(z,n,n)$ has a pole as $z\to
\lambda_j$ and so
by (2.20), (2.21) must hold (although both
sides will be infinite if $\varphi_j
(n+1) =0$).
In the second case, $\varphi_j (n) =0$.
Then both sides of (2.21) are zero, and
so (2.21) holds.
(However, the denominator of (2.20) is
$\infty - \infty$ and happens to be
$\infty$ so that
$G(z,n,n)$ vanishes, but (2.21) still
holds.) \qed
\enddemo
\vskip 0.3in
\flushpar{\bf {\S 3. Trace Formulas and
a New Approach to the Inverse Problem}}
\vskip 0.1in
In this section, we will use $m$-functions
to show how to recover a Jacobi
matrix from the
spectral function $d\rho$. The more usual
approach via orthogonal polynomials is
sketched
in the appendix. Our approach is new,
although iterated $m$-functions are
equivalent to a
continued fraction expansion of $m(z)$, and
so the work of Masson and Repka
[\mr] is not
unrelated to our approach.
We begin with
\proclaim{Theorem 3.1} Near $z=\infty$,
$$
m(z) = -\frac{1}{z} - \frac{b_1}{z^2} -
\frac{a^2_1 + b^2_1}{z^3} + O(z^{-4}).
\tag 3.1
$$
\endproclaim
\demo{First Proof} By the basic definition
of $m(z)$ (see (2.9)) and the norm
convergent
expansion (since $H$ is bounded):
$$\align
(H-z)^{-1} &= -z^{-1} (1-z^{-1}H)^{-1} \\
&= -z^{-1} -z^{-2} H-z^{-3} H^2 + O(z^{-4}).
\endalign
$$
We have
$$
m(z) = -z^{-1} - z^{-2} (\delta_1,
H\delta_1) -z^{-3} \| H\delta_1 \|^2
+O(z^{-4}).
$$
Clearly, $(\delta_1, H\delta_1) = b_1$ and
$\|H\delta_1\|^2 = \| a_1 \delta_2 +
b_1 \delta_1\|^2
= a^2_1 + b^2_1$. \qed
\enddemo
\demo{Second Proof} By (2.15),
$$
m(z) = \frac{1}{b_1 - z - a^2_1 m_+(z,1)}\, .
$$
But $m_+(z,1) = -\frac{1}{z} +
O(z^{-2})$. Thus,
$$\align
m(z) &= -\frac{1}{z} \biggl( 1 -\frac{b_1}{z}
- \frac{a^2_1}{z^2} +
O(z^{-3})\biggr)^{-1} \\
&=- \frac{1}{z} \biggl( 1 + \frac{b_1}{z}
+ \frac{a^2_1}{z^2} + \biggl(
\frac{b_1}{z}
\biggr)^2 + O(z^{-3})\biggr). \qed
\endalign
$$
\enddemo
In terms of the spectral measure $d\rho$,
(3.1) becomes
$$\align
b_1 &= \int \lambda \, d\rho(\lambda),
\tag 3.2 \\
a^2_1 &= \int \lambda^2 \, d\rho(\lambda)
- \biggl( \int \lambda \,
d\rho(\lambda) \biggr)^2,
\tag 3.3
\endalign
$$
formulas implicit in the orthogonal
polynomial approach.
In case $N<\infty$, there is a direct way
to interpret (3.1) as generating trace
formulas:
\proclaim{Theorem 3.2} Assume $N \in \Bbb N$
and let $\lambda_1, \dots,
\lambda_N$ be
the eigenvalues of $H$ and $\nu_1, \dots,
\nu_{N-1}$, the eigenvalues
of
$H_{[2,N]}$. Then
$$\align
b_1 &= \sum^N_{j=1} \lambda_j -
\sum^{N-1}_{\ell =1} \nu_\ell, \tag 3.4 \\
2a^2_1 + b^2_1 &= \sum^N_{j=1} \lambda^2_j
- \sum^{N-1}_{\ell =1} \nu^2_\ell.
\tag 3.5
\endalign
$$
\endproclaim
\demo{Proof} Write (see (2.12))
$$\align
m(z) &= -\frac{\prod^{N-1}_{\ell=1}
(z-\nu_\ell)}{\prod^N_{j=1}
(z-\lambda_j)} =
-\frac{1}{z} \prod^{N-1}_{\ell=1}
\biggl(1-\frac{\nu_\ell}{z}\biggr)
\prod^N_{j=1}
\biggl( 1-\frac{\lambda_j}{z}
\biggr)^{-1} \\
&= -\frac{1}{z} - \frac{\alpha}{z^2}
- \frac{\beta}{z^3} + O(z^{-4}),
\endalign
$$
where
$$\align
\alpha &= \sum^N_{j=1} \lambda_j -
\sum^{N-1}_{\ell=1} \nu_\ell, \tag 3.6 \\
\beta &= \sum^N_{j=1} \lambda^2_j +
\sum^N_{j \beta$ and $\xi (\lambda)= 0$
if $\lambda <
\alpha$), so we can assume that $0\in
(\alpha, \beta)$. Then $Q(z)$ is analytic
in $\Bbb C
\backslash [\alpha,\beta]$ and on
$(\alpha, \beta)$:
$$\alignat2
\frac{1}{\pi}\, \text{Im}\,
(Q(\lambda +i0)) &= \xi (\lambda),
&& \qquad \lambda
< 0 \\
&= \xi (\lambda) -1, && \qquad \lambda >0.
\endalignat
$$
By (3.13), for $R$ sufficiently large,
$$\align
b_1 &= \frac{1}{2\pi i} \oint _{|z|=R}
Q(z) \, dz = -\int^\beta_\alpha
\frac{1}{\pi}\,
\text{Im}\, (Q(\lambda+i0)) \,
d\lambda \\
&= - \int^0_\alpha d\lambda +
\int^\beta_\alpha (1-\xi(\lambda))
\, d\lambda =
\text{(3.11)}
\endalign
$$
and
$$\align
2a^2_1 + b^2_1 &= \frac{1}{2\pi i}
\oint _{|z|=R} 2z Q(z)\, dz =
-\int^\beta_\alpha
\frac{1}{\pi} \, 2\lambda \,
\text{Im}\, (Q(\lambda +i0)) \, d\lambda \\
&= -\int^0_\alpha 2\lambda \,
d\lambda + \int^\beta_\alpha 2
\lambda(1-\xi(\lambda))\,
d\lambda = \text{(3.12).} \qed
\endalign
$$
\enddemo
(3.2)--(3.5), (3.8), (3.9), (3.11),
(3.12), etc.~clearly underscore that
one can
derive an infinite
sequence of such trace formulas which
are precisely the well-known invariants of
the hierarchy
of Toda lattices. A systematic approach
to these trace formulas can be found,
for instance, in
[\bght, \casevi, \geho, \te].
\vskip 0.1in
We can now describe the scheme for
recovering $H$ from $d\rho$, or equivalently,
from $m(z)
= \int d\rho(\lambda) \,
(\lambda - z)^{-1}$:
\roster
\item"\rom{(i)}" Use the trace formulas
(via (3.1) or (3.11), (3.12)) to recover
$b_1$ and
$a^2_1$.
\item"\rom{(ii)}" Use (2.15), viz.,
$$
m_+(z,1) = a^{-2}_1 \biggl(b_1 - z -
\frac{1}{m(z)}\biggr)
$$
to find $m_+(z,1)$ which is the
$m$-function for $H_{[2,\infty)}$.
\item"\rom{(iii)}" Use the trace
formulas to find $b_2, a^2_2$ and
then (2.15)
to find $m_+
(z,2), \dots$, etc.
\endroster
This clearly shows a given $d\rho$ can
come from at most one $H$, since we have
just
described how to compute the $b_j$ and
$a^2_j$ from $d\rho$. We want to prove
existence via
this method, that is, given any $d\rho$
of compact support, this method yields
an $H$ which is
bounded and whose spectral measure is
precisely $d\rho$.
\proclaim{Lemma 3.4} Suppose that
$m(z)=\int d\rho(\lambda) \,
(\lambda-z)^{-1}$, where
$d\rho$ is a probability measure on
$[-C,C]$ whose support contains more than
one point.
Define
$$
b_1 = \int \lambda\, d\rho(\lambda),
\qquad a^2_1 = \int \lambda^2 \,
d\rho(\lambda) -
b^2_1 \tag 3.14
$$
\rom($a^2_1$ is always strictly positive
by the support hypothesis on
$d\rho$\rom). Define
$m_1 (z)$ by
$$
m_1(z) = a^{-2}_1 \biggl[ b_1 - z -
\frac{1}{m(z)}\biggr].
$$
Then
$$
m_1 (z) = \int \frac{d\rho_1 (\lambda)}
{\lambda- z}\, , \tag 3.15
$$
where $d\rho_1$ is a probability measure
also supported on $[-C, C]$. Moreover,
$\rho$ is
supported on exactly $N$ points if and
only if $\rho_1$ is supported on exactly
$(N-1)$ points.
\endproclaim
\demo{Proof} By (3.14) and an expansion
of a geometric series, (3.1) holds, so
$$
\widehat m(z) := (-m(z))^{-1} = z -
b_1 - \frac{a^2_1}{z} + O(z^{-2}).
\tag 3.16
$$
Since $m(z)$ has $\text{Im} \,
(m(z)) > 0$
when $\text{Im}\, (z) >0$ (we recall that
$m$ is a
Herglotz function), $\widehat m(z)$=
$(-m(z))^{-1}$ has the same property.
Moreover,
$\widehat m(z)$ is analytic on $\Bbb C
\backslash [-C,C]$ since $m(\lambda) >0$
for
$\lambda <-C$ and $m(\lambda) <0$ for
$\lambda > C$. Thus, by the Herglotz
representation
theorem,
$$
\widehat m(z) = \hat {c} + \hat {d} z
+ \int\frac{d\hat \rho (\lambda)}{\lambda
- z}
$$
for a measure $d\hat\rho$. By (3.16),
$\hat c = -b_1$, $\hat d = 1$, and $\int
d\hat\rho
(\lambda) = a^2_1$. Thus,
$$
m_1 (z) = \int \frac{d\rho_1(\lambda)}
{\lambda -z }
$$
and $d\rho_1 = a^{-2}_1 d\hat\rho$ is
also a probability measure.
Since $d\rho$ is supported on $N$ points
if and only if $m(z)$ is a ratio
$P_{N-1}(z)/
Q_N (z)$ of polynomials with $\deg
(P_{N-1} (z)) =N-1$, $\deg (Q_N (z))
= N$, we
obtain
the last assertion. \qed
\enddemo
\proclaim{Theorem 3.5} {\rom{($\equiv$
Theorem A.6) \, }} Every $N$-point
probability
measure arises as the spectral measure
of a unique $N\times N$ Jacobi matrix.
Every
probability measure of bounded and
infinite support arises as the spectral
measure of a
unique semi-infinite bounded Jacobi
matrix.
\endproclaim
\demo{Proof} By iterating the $\rho\to
\rho_1$ procedure of the lemma, we can
find suitable
$a^2_j, b_j$ inductively. If $d\rho$ has
$N$-point support, the process
terminates after $N-1$
steps where $d\rho_N$ has a single point,
and we define $b_N$ to be that point.
If $d\rho$ has
infinite support, the process continues
indefinitely. Because
$\text{supp}(d\rho)\subseteq
[-C,C]$, we infer $\text{supp}(d\rho_1)
\subseteq [-C,C]$, $|a_1|$ and $|b_1|$
are bounded by
$C$, and so $H$ is bounded.
Let $d\tilde\rho$ be the spectral
measure for the $H$ that has just been
constructed. We will
show $d\rho = d\tilde\rho$, thereby
completing the proof.
Let $\tilde m(z)=\int d\tilde\rho(\lambda)
\, (\lambda -z)^{-1}$. Then by
construction,
$$
\tilde m(z) = \frac{-1}{z-b_1 + a^2_1
\left[ \frac{-1}{z-b_2 + a^2_2
\dots}\right]}\, .
$$
That is, $m$ and $\tilde m$ have identical
partial fraction expansions although
a priori the
remainders could be different. This means
that the Taylor series for $\tilde
m(z)$ near $z=
\infty$ agrees with that for $m$ near
$z=\infty$ so $m(z) = \tilde m(z)$, and
hence $d\rho
=d\tilde\rho$. \qed
\enddemo
\remark{Remark} The Taylor series for
$m(z)$ only converges in the region $|z| >
C$, where
$C=\max(\text{supp}(d\rho))$. But
$H_{[1,N]} \to H$ strongly, so $(\delta_1,
(H_{[1,N]}
-z)^{-1} \delta_1) := m_N (z) \to m(z)$
as $N\to\infty$ for all
$z\notin\text{supp}(d\rho)$,
where $m_N (z) = P_{N-1}(z)/Q_{N} (z)$
is a ratio of polynomials of degree $N-1$
and $N$.
By the above argument, the Taylor series
agree near infinity to order $2N$, that
is, $m_N$ is
the $[N-1, N]$ Pad\'e approximant, which
we see converges everywhere outside of
$\text{supp}
(d\rho)$. See Baker and Graves-Morris
[\bak] for further discussion of the
Pad\'e approximants.
\endremark
\vskip 0.1in
The continuum analog of the orthogonal
polynomial approach of the appendix is
the
Gel'fand-Levitan [\gl] inverse spectral
theory which is a kind of continuum
orthonormalization.
It would be very interesting to find a
continuum analog of the $m$-function
approach to inverse
problems that we discussed in this section.
\vskip 0.1in
As an application of the $m$-function
approach to inverse problems, we prove the
following
(which can also be obtained via
orthogonal polynomials):
\proclaim{Theorem 3.6} Fix $N \in \Bbb N$.
Consider the following
parametrizations of $N
\times N$ Jacobi matrices:
$$\alignat2
\text{\rom{(i)}} & \quad \{a_n\}^{N-1}_{n=1}
\cup\{b_n\}^N_{n=1}, &&\qquad (a_n
> 0). \\
\text{\rom{(ii)}} & \quad
\{\lambda_j\}^N_{j=1} \cup
\{\nu_\ell\}^{N-1}_{\ell=1},
&&\qquad (\lambda_1 < \nu_1 < \lambda_2
< \cdots < \nu_{N-1} < \lambda_N). \\
\text{\rom{(iii)}} & \quad
\{\lambda_j\}^N_{j=1}
\cup \{\alpha_j\}^N_{j=1},
&&\qquad
\biggl(\lambda_1 < \cdots < \lambda_N,
\, \, \alpha_j > 0, \, \,
\sum^N_{k=1}\alpha_k=1 \biggr).
\endalignat
$$
Here $\lambda_j$ are the eigenvalues
of $H$, $\nu_\ell$ are the eigenvalues of
$H_{[2,n]}$,
and the $\alpha$'s are the residues of
the poles in $m$ so $m(z) = \sum^N_{j=1}
\alpha_j
(\lambda_j -z)^{-1}$ \rom(or $d\rho(\lambda)
= \sum \alpha_j \delta(\lambda -
\lambda_j) \,
d\lambda$\rom). The maps between these
parameters are real bianalytic
diffeomorphisms.
\endproclaim
%\proclaim{Theorem 3.6} Fix $N \in
%\Bbb N$. Consider the following
% parametrizations of $N
%\times N$ Jacobi matrices:
%$$\alignat2
%\text{\rom{(i)}} & \quad \{a_n\}^{N-1}
%_{n=1}\cup\{b_n\}^N_{n=1}, &&\qquad (a_n
% > 0). \\
%\text{\rom{(ii)}} & \quad \
%{\lambda_j\}^N_{j=1} \cup
% \{\nu_\ell\}^{N-1}_{\ell=1},
% \alpha_j > 0, \, \,
%\sum^N_{k=1}\alpha_k=1 \biggr).
%\endalignat
%$$
%Here $\lambda_j$ are the eigenvalues of
%$H$, $\nu_\ell$ are the eigenvalues of
% $H_{[2,n]}$,
%and the $\alpha$'s are the residue of
%the poles in $m$ so $m(z) = \sum^N_{j=1}
% \alpha_j
%(\lambda_j -z)^{-1}$ \rom(or $d\rho(\lambda)
%= \sum \alpha_j \delta(\lambda -
% \lambda_j) \,
%d\lambda$\rom). The maps between these
% parameters are real bianalytic
% diffeomorphisms.
%\endproclaim
\remark{Remark} There are $2N-1$
parameters. These appear to be $2N$
in (iii)
but the fact
that $\sum^N_{j=1} \alpha_j =1$ eliminates
one parameter.
\endremark
\demo{Proof} It is well known and
elementary (the determinant of the Jacobian
matrix is just
$\pm \prod _{j0$. Every such
sum arises as the $11$ Green's function
of an
$H$ and of exactly one such $H$.
\endproclaim
For general $n$, define $\tilde n =
\min (n, N+1 -n)$. Then we will prove the
following theorems:
\proclaim{Theorem 6.2} $(\delta_n,
(H-z)^{-1}\delta_n)$ has the form
$\sum^k_{j=1}
\alpha_j (\lambda_j -z)^{-1}$ with
$k$ one of $N, N-1, \dots, N-\tilde n + 1$
and $\lambda_1
< \cdots < \lambda_k, \, \sum^k_{j=1}
\alpha_j =1$ and each $\alpha_j >0$.
Every such
sum arises
as the $nn$ Green's function of at
least one $H$.
\endproclaim
\proclaim{Theorem 6.3} If $k=N$, then
precisely $\binom{N-1}{n-1}$ $H$'s yield
the given
$nn$ Green's function.
\endproclaim
\proclaim{Theorem 6.4} If $k 0$ are determined by the
$\alpha$'s and
$\lambda$'s. By (2.18),
$$
-G(z;n,n)^{-1} = z - b_n + a^2_n m_+(z,n)
+ a^2_{n-1} m_-(z,n). \tag 6.2
$$
$m_-(z,n) = (\delta_{n-1}, (H_{[1, n-1]}
-z)^{-1}\delta_{n-1})$ determines
$H_{[1,n-1]}$
uniquely (by Theorem~3.5) and has the
form
$$
m_-(z,n) = \sum^{n-1}_{j=1}\frac{\gamma_j}
{e_j -z} \, , \qquad \gamma_j > 0,
\tag 6.3
$$
where $\sum^{n-1}_{j=1} \gamma_j =1$ and
the $e_j$'s are the eigenvalues of
$H_{[1, n-1]}$.
Similarly, $m_+(z,n)$ $= (\delta_{n+1},
(H_{[n+1, N]} -z)^{-1} \delta_{n+1})$
determines
$H_{[n+1, N]}$ uniquely and has the form
$$
m_+(z,n) = \sum^{N-n}_{j=1}
\frac{\kappa_j}{f_j -z}\, , \qquad
\kappa_j > 0,
\tag 6.4
$$
where $\sum^{N-n}_{j=1} \kappa_j =1$
and the $f_j$'s are the eigenvalues of
$H_{[n+1,N]}$.
Comparing (6.1)--(6.4), we see that
$\{\mu_\ell\}^{N-1}_{\ell=1} =
\{e_j\}^{n-1}_{j=1} \cup
\{f_j\}^{N-n}_{j=1}$. We can choose
which $\mu_\ell$ are to be $e_j$ in
$\binom{N-1}{n-1}$
ways. Once we make the choice,
$$
a^2_{n-1} = \sum_{\text{$\ell$ so
$\mu_\ell$ is an $e_j$}} \beta_\ell
\qquad\text{and}\qquad
a^2_n = \sum_{\text{$\ell$ so $\mu_\ell$
is an $f_j$}} \beta_\ell
$$
and $m_\pm(z,n)$ are determined. But
$H_{[1,n-1]}$, $H_{[n+1, N]}$, and
$a_{n-1}, b_n,
a_n$ determine $H$. Thus for each choice,
we can uniquely determine $H$.
Moreover, since
any sums of the form (6.3), (6.4) are
legal for $m_\pm(z,n)$, we have existence
for each of the
$\binom{N-1}{n-1}$ choices.
$k=N$ if and only if all the
eigenfunctions $\varphi_j (n)$ are
non-vanishing at
$n$.
Eigenfunctions obey the boundary conditions
at both ends, so if $\varphi_j (n)$
vanishes, so do
$P(z,n)$ and $\psi_+(z,n)$, which are
polynomials of degree $n-1$ and $N-n$; so
at most
$\min(n-1, N-n) := \tilde n -1$ eigenvalues
of $H$ can fail to contribute to
$G(z,n,n)$, that is,
at least $N-\tilde n + 1$ eigenvalues
must contribute, that is, $k$ is one of
$N, N-1, \dots,
N-\tilde n +1$. Eigenvalues that don't
contribute are zeros of $G(z,n,n)$ and
simultaneously
eigenvalues of $H_{[1, n-1]}$ and
$H_{[n+1, N]}$.
Thus if $kk\geq N-\tilde n +1$ implies
$n_0 >0$, $n_1
\geq 0$,
$n_2 \geq 0$ and that $n_0 + n_1 + n_2
= k-1$, $n_0 + n_1 =n-1$, and $n_0 + n_2
= N-n$.
To reconstruct $m_\pm(z,n)$ given
$-G (z, n,n)^{-1}$, we have to make two sets
of choices:
\roster
\item"\rom{(i)}" Figure out which of
$\mu_1, \dots, \mu_{k-1}$ lie in each of
the three sets.
This yields $\binom{k-1}{n_0}
\binom{k-1-n_0}{n_1} = \frac{(k-1)!}
{n_0! n_1! n_2!}$ discrete choices.
\item"\rom{(ii)}" For each of the $n_0$
$\mu_\ell$'s in the set of common
eigenvalues, we
must pick a decomposition
$$
\beta_\ell = \beta^{(1)}_\ell +
\beta^{(2)}_\ell, \, \, \beta^{(j)}_{\ell} > 0
$$
and then take
$$
a^2_{n} m_+(z,n) = \sum \Sb \text
{$\ell$ so that} \\ \text{$\mu_\ell$ is solely
an} \\
H_{[n+1, N]} \text{ eigenvalue}
\endSb \frac{\beta_\ell}{\mu_\ell - z}
+ \sum \Sb \text{$\ell$ so that} \\
\text{$\mu_\ell$ is a} \\ \text{common
eigenvalue}\endSb
\frac{\beta^{(1)}_\ell}{\mu_\ell -z}
$$
and
$$
a^2_{n-1} m_-(z,n) = \sum \Sb
\text{$\ell$ so that} \\ \text{$\mu_\ell$ is
solely an} \\
H_{[1, n-1]} \text{ eigenvalue}
\endSb \frac{\beta_\ell}{\mu_\ell - z} +
\sum \Sb \text{$\ell$ so that} \\
\text{$\mu_\ell$ is a} \\ \text{common
eigenvalue}\endSb
\frac{\beta^{(2)}_\ell}{\mu_\ell -z}\, .
$$
\endroster
Every such choice yields an acceptable
$H$. Since the map from poles and
residues to matrices
is a diffeomorphism (Theorem~3.6),
the $\frac{(k-1)!}{n_0! n_1! n_2!}$ disjoint
sets of poles
and $\operatornamewithlimits{\times}
\limits_{\text{$n_0$ $\ell$'s}} (0,
\beta_\ell)$ residues
lead to that number of manifolds
diffeomorphic to the $n_0$-dimensional
open
ball. \qed
\enddemo
\vskip 0.3in
\flushpar{\bf {\S 7. The Discrete
Schr\"odinger Inverse Spectral Problem}}
\vskip 0.1in
A Jacobi matrix with all $a_n = 1$ is
called a discrete Schr\"odinger operator.
The inverse
problem for such operators is open,
that is, there are no effective conditions
on a spectral
measure $d\rho$ that tell us that its
associated Jacobi matrix has all $a_n =1$.
(The isospectral
manifold of general Jacobi matrices
with $a_n \in \Bbb R $ is discussed in
[\tom].)
Consider the finite case, $N \in
\Bbb N$. The number $N$ of free parameters
$\{b_n\}^N_{n=1}$ equals exactly the
number of eigenvalues
$\{\lambda_j\}^N_{j=1}$. The
natural inverse problem is from
$\lambda$'s to $b$'s. We do not have
a complete
solution, but
have a number of conjectures and
comments which we make in this section.
$\lambda_1 <
\lambda_2 < \cdots < \lambda_N$ are
the eigenvalues of $H$. For any $\bold b =
(b_1, \dots
b_N) \in \Bbb R^N$, define $\Lambda
(\bold b) = (\lambda_1, \dots, \lambda_N)
\in \Bbb R^N$
as the eigenvalues. Let $S_N =
\text{Ran}(\Lambda)$.
\example{Main Conjecture 7.1} $S_N$
is a closed set in $\Bbb R^N$ whose
interior
$S^{\text{\rom{int}}}_N$ is dense in
$S_N$. For any $\boldsymbol \lambda \in
S^{\text{\rom{int}}}_N$, $\Lambda^{-1}
(\boldsymbol \lambda)$ contains $N!$
points. For
any $\boldsymbol \lambda \in\partial
S_N$, $\Lambda^{-1}(\boldsymbol \lambda)$
contains
fewer than $N!$ points.
Thus, we believe that $\Lambda^{-1}
[S^{\text{\rom{int}}}_N]$ is an $N!$-fold
cover of
$S^{\text{\rom{int}}}_N$, but it is
likely an uninteresting one.
\endexample
\example{Conjecture 7.2} $\Lambda^{-1}
[S^{\text{\rom{int}}}_N]$ is a union of
$N!$
disjoint sets. On each of them,
$\Lambda$ is a diffeomorphism to
$S^{\text{\rom{int}}}_N$.
In the complex domain, things are
more interesting. There is a small
neighborhood, $D$, of
$\Bbb R^N$ in $\Bbb C^N$ to which
$\Lambda$ can be analytically
continued and on
which
$\lambda_j\neq \lambda_k$ still holds.
Introduce
$$
\tilde S_N = \Lambda [D]\, \quad\text{and}
\quad B=\{\boldsymbol \lambda \in
\tilde S_N
\mid \Lambda^{-1}[\boldsymbol \lambda]
\text{ has ordinality less than $N!$}\}.
$$
\endexample
\example{Conjecture 7.3} $B$ has real
codimension 2. $\Lambda^{-1} [\tilde S_N
\backslash
B]$ is connected and is an $N!$-cover
of $\tilde S_N\backslash B$.
Thus, $\Lambda^{-1}$ is a ramified
cover of $\tilde S_N$. We begin with an
analysis of the
case $N=2$, so $H=\left( \smallmatrix
b_1& 1 \\ 1 & b_2 \endsmallmatrix
\right)$. Then
$$
\Lambda(\bold b) = \biggl( \frac{b_1+ b_2}
{2} - \sqrt{\left(\frac{b_1 - b_2}{2}
\right)^2 + 1}\, ,\, \frac{b_1 + b_2}{2}
+ \sqrt{\left(\frac{b_1-
b_2}{2}\right)^2 + 1}\, \biggr).
\tag 7.1
$$
Then $S_2 = \{(\lambda_1, \lambda_2)
\in \Bbb R^2 \mid \lambda_2 \geq \lambda_1
+ 2\}$.
$\partial S_2 = \{(\lambda_1, \lambda_2)
\in \Bbb R^2 \mid\lambda_2 = \lambda_1
+ 2 \}$
and $\Lambda^{-1}(\alpha - 1, \alpha + 1)
= \left\{\left(\smallmatrix \alpha & 1
\\ 1 & \alpha
\endsmallmatrix\right)\right\}$,
otherwise $\Lambda^{-1} ((\lambda_1,
\lambda_2))$ has two
points $\left( \smallmatrix x & 1 \\
1 & y\endsmallmatrix \right)$ and $\left(
\smallmatrix
y & 1 \\ 1 & x \endsmallmatrix \right)$.
$\Lambda^{-1} (S^{\text{\rom{int}}}_2)$
has two
connected components where $b_1 > b_2$
and where $b_2 > b_1$. If one continues
into the
complex domain, $\Lambda^{-1} [\tilde
S_2 \backslash B]$ is connected.
Thus, our conjectures are true in the
not quite trivial case $N=2$.
\endexample
At first sight, it may seem surprising
that $S_N$ is closed. After all, the
eigenvalue image of
all Jacobi matrices $\{ \boldsymbol\lambda
\in \Bbb R^N \mid \lambda_1<
\lambda_2 < \cdots
< \lambda_N\}$ is open and not closed.
The existence of strict inequalities is a
reflection of the
condition $a_n >0$. Once $a_n \equiv 1$,
they disappear.
\proclaim{Theorem 7.4} $S_N$ is closed.
\endproclaim
\demo{Proof} Let $\boldsymbol\lambda_{m}
\in S_N$ and pick $\bold b_{m} \in
\Bbb R^N$ so that $\Lambda(\bold b_{m})
= \boldsymbol \lambda_{m}$. Suppose
$\boldsymbol \lambda_{m} \to
\boldsymbol \lambda_{\infty} \in
\Bbb R^N$ as $m
\to \infty$.
Let $H(\bold b)$ be the $N\times N$
Schr\"odinger matrix with the components of
$\bold b$
along the diagonal. Then
$$
|\Lambda (\bold b)|^2 = \text{Tr}
(H(\bold b)^2) = 2(N-1) + \|\bold b\|^2,
$$
so $\{\bold b_{m}\}$ is a bounded subset
of $\Bbb R^N$. Thus, we can find a
subsequence
$\{m_p\}$ such that $\bold b_{m_{p}} \to
\bold b_{\infty}$ as $p \to \infty$. By
continuity
of $\Lambda$, $\Lambda (\bold b_{\infty})
= \boldsymbol \lambda_{\infty}$, that
is,
$\boldsymbol \lambda_{\infty} \in S_n$.
\qed
\enddemo
This theorem implies that if $\| \bold b\|
\leq R$, then there is a minimum
distance between
eigenvalues. One might think there are
global bounds on eigenvalue splittings
(i.e.,
$N$-dependent but independent of $R$), but
that is false if $N\geq 3$, as is
seen by the
following example motivated by tunneling
considerations. Let $H(\beta)$ be the
$N\times N$
Schr\"odinger matrix with $b_1 = b_N =
\beta$ and $b_2 = \cdots = b_{N-1} =0$.
Then for
$\beta$ large, the two largest eigenvalues
$E_\pm (\beta)$ satisfy
$$
E_\pm (\beta) = \beta \pm O
(\beta^{-(N-2)}) \tag 7.2
$$
and if $N\geq 3$, $|E_+ (\beta) -
E_- (\beta)| \to 0$ as $\beta \to \infty$.
\vskip 0.1in
An important open question is finding
some kind of effective description of
$S_N$. We note
that if $\varphi_+ = (\frac{1}{\sqrt N},
\dots, \frac{1}{\sqrt N})$ and
$\varphi_- = (\frac{1}
{\sqrt N}, - \frac{1}{\sqrt N}, \frac{1}
{\sqrt N}, \dots,
\frac{(-1)^{N+1}}{\sqrt N})$, then
$(\varphi_+, H\varphi_+) - (\varphi_-,
H\varphi_-) = 4 (1-\frac{1}{N})$ so
$\lambda_N -
\lambda_1 \geq 4(1-\frac{1}{N})$.
\vskip 0.1in
The $N!$ in our main conjecture comes
from the following
\proclaim{Theorem 7.5} For $\beta$ large,
$\boldsymbol \lambda_{\beta} :=
(\beta, 2\beta,
3\beta, \dots, N\beta) \in S_N$ and
$\Lambda^{-1} (\boldsymbol \lambda_{\beta})$
has $N!$
points.
\endproclaim
\demo{Proof} Consider the $N!$ Hamiltonians
$$
H_\pi (\beta) = \beta \pmatrix
\pi (1) & {} & 0 \\
{} & \ddots & {} \\
0 & {} & \pi (N)
\endpmatrix +
\pmatrix
0 & 1 & {} & 0 \\
1 & \ddots & \ddots & {} \\
{} & \ddots & {} & 1 \\
0 & {} & 1 & 0 \endpmatrix , \tag 7.3
$$
where $\pi$ is an arbitrary permutation
on $\{ 1, \dots, N\}$. Then $A(\beta) =
\beta^{-1}
H_{\pi}(\beta)$ at $\beta =0$ has $N$
eigenvalues $(1,2, \dots, N)$ and it is
easy to see that
for $\beta$ small, the Jacobian of
$\Lambda$ is invertible. It follows by the
inverse function
theorem that for $\beta$ large, there is
a unique $\tilde H_\pi (\beta) = H_\pi
(\beta) +
O(\beta)^{-1})$ so that the eigenvalues
of $\tilde H_\pi (\beta)$ are precisely
$(\beta, 2\beta,
\dots, N\beta)$.
A separate and easy argument shows that
for $\beta$ large, any Schr\"odinger
matrix with
eigenvalues $(\beta, \dots, N\beta)$ must
have $b_n= \beta\pi (n) +
O(\beta^{-1})$ for some
permutation $\pi$, and so must be one of
the $\tilde H_\pi (\beta)$. \qed
\enddemo
The evidence for the strong forms of the
conjectures here is not overwhelming.
We make them
as much to stimulate further research as
because we are certain they are true.
\vskip 0.3in
\example{Acknowledgments} We would like to
thank R.~del Rio and G.~Teschl for
discussions and
pertinent
hints to the literature. F.G.~is indebted
to A.~Kechris and C.W.~Peck for a kind
invitation to
Caltech during the summer of 1996 where
some of this work was done. The
extraordinary
hospitality and support by the Department
of Mathematics at Caltech are
gratefully
acknowledged. B.S.~would like to thank
M.~Ben-Artzi of the Hebrew University
where some
of this work was done.
\vskip 0.2in
\flushpar{\bf {Appendix: Orthogonal
Polynomials and the Inverse Problem }}
\vskip 0.1in
Let $H$ be a Jacobi matrix on $\Bbb N$ or
on $\{1, \dots, N\}$, that is,
$$\alignedat2
(Hu)(n) &= a_n u(n+1) + b_n u(n) + a_{n-1}
u(n-1), &&\qquad n\neq 1, (N) \\
&= a_1 u(2) + b_1 u(1), && \qquad n=1 \\
&= b_N u(N) + a_{N-1} u(N-1), && \qquad n=N
\endalignedat \tag A.1
$$
where the $(N)$ refers to the finite matrix
case. Here $a_n >0$, $b_n \in \Bbb
R$, and we
will suppose $a_n, b_n$ are bounded.
It will sometimes be useful to refer to a
single sequence $c_1, c_2, \ldots :=
b_1, a_1, b_2,
a_2$, $\dots$, that is,
$$
a_n = c_{2n}, \, \, b_n = c_{2n-1}, \qquad
n=1,2, \dots \, .
$$
Let $d\rho$ denote the spectral measure for
the vector $\delta_1 =
(1,0,\dots)$, that is,
$$
(\delta_1, e^{-itH}\delta_1) = \int
e^{-it\lambda}\, d\rho(\lambda).
$$
In the finite case, $H$ has $N$ eigenvectors
$\varphi_1, \dots, \varphi_N$ with
$$
(\varphi_j, \varphi_k) = \delta_{j,k} \, ,
\qquad\ H\varphi_j = \lambda_j
\varphi_j \tag A.2
$$
and
$$
d\rho(\lambda) = \sum^N_{j=1} |\varphi_j
(1)|^2 \delta(\lambda- \lambda_j) \,
d\lambda.
\tag A.3
$$
Obviously, given $\{a_n\},\{b_n\}$,
$d\rho$ is uniquely determined ($H$ is
bounded by the
hypotheses on $a_n$, $b_n$ and so
self-adjoint). It is an important fact that
$d\rho$ determines
$\{a_n \}, \{b_n \}$, that any $d\rho$ of
bounded support is allowed (in the
finite case, any
$N$-point measure is allowed; in the
semi-infinite case, $\text{supp}(d\rho)$
must be infinite).
Indeed, there is an elegant formalism for
finding the $a$'s and $b$'s given
$d\rho$. This
formalism involves orthogonal polynomials.
It has been discussed,
for instance, in [\akh, \ber, \casi--\ck,
\la, \mr]. We summarize
it here for the reader's convenience
and to fix
notation.
We begin by analyzing the direct problem,
that is, we suppose the $a$'s and
$b$'s are given.
Define functions $\{P(z,n)\}^N_{n=1}$ in
the finite case and
$\{P(z,n)\}^\infty_{n=1}$ in
the $\Bbb N$ case by requiring
$$\aligned
P(z,n+1) &= a^{-1}_n [(z-b_n) P(z,n) -
a_{n-1} P(z,n-1)], \qquad n\geq 2, \\
P(z,1) &= 1, \, \, P(z,2) = a^{-1}_1 (z-b_1).
\endaligned \tag A.4
$$
The $P(z,n)$ for $z$ fixed and $n$ variable
satisfy $(H-z)P(z)=0$ in the sense
that if $\psi$
has finite support in the $\Bbb N$ case
and if $\psi$ is supported on
$[1,2,\dots, N-1]$ in
the finite case, then
$$
\sum^{N\text{ or }\infty}_{n=1} P(z,n)
((H-z)\psi)(n)=0.
$$
Clearly, by induction $P(z,n)$ is a
polynomial of degree $n-1$. Moreover,
$$
\tilde P(z,n) = a_1 \dots a_{n-1}
P(z,n) \tag A.5
$$
are monic polynomials.
\proclaim{Proposition A.1} Define
$P(H,n)$ using the functional calculus. Then
$$
P(H,n) \delta_1 = \delta_n. \tag A.6
$$
\endproclaim
\demo{Proof} Clearly, (A.6) holds for
$n=1$. Suppose it holds for $n=1, \dots,
n_0$. Then
$$\align
\delta_{n+1} &= a^{-1}_n [H\delta_n -
b_n \delta_n - a_{n-1} \delta_{n-1}] \\
&= a^{-1}_n [(H-b_n) P(H,n) \delta_1 -
a_{n-1} P(H,n-1) \delta_1 ] \\
&= P(H,n+1)\delta_1
\endalign
$$
by the definition of $P$. Here we used
the induction hypothesis in the second
equality. \qed
\enddemo
\proclaim{Corollary A.2} The spectral
measure for $\delta_n$ is $P(\lambda, n)^2
\, d\rho
(\lambda)$.
\endproclaim
\proclaim{Corollary A.3}
$$
\int P(\lambda,m) P(\lambda,n) \,
d\rho (\lambda) = \delta_{m,n}. \tag A.7
$$
\endproclaim
\demo{Proof} This just says that
$(\delta_m, \delta_n) = \delta_{m,n}$. \qed
\enddemo
Thus, the $P$'s are the orthonormal
polynomials defined by $d\rho$ via
Gram-Schmidt on the
functions $1, \lambda,\lambda^2, \dots$.
The $\tilde P$ are the more
conventional orthogonal
polynomials [\akh, \sze] (orthogonal and
monic rather than orthogonal and
normalized). The
defining relation of the $P$'s,
$$
zP(z,n) = a_n P(z,n+1) + b_n P(z,n) +
a_{n-1} P(z,n-1), \tag A.8
$$
or equivalently,
$$
z\tilde P(z,n) = \tilde P(z,n+1) +
b_n \tilde P(z,n) + a^2_{n-1}\tilde P(z,n-1),
\tag A.9
$$
are the standard three-term recursion
relations for orthogonal polynomials.
\proclaim{Proposition A.4} The following
determine each other
\roster
\item"\rom{(i)}" $\{P(z,n) \}
^{n_0 +1}_{n=1}$.
\item"\rom{(ii)}" $\int \lambda^j \,
d\rho (\lambda),\quad j=1,2,\dots, 2n_0$.
\item"\rom{(iii)}" $\{ c_j\}^{2n_0}_{j=1}$.
\endroster
\endproclaim
\proclaim{Proposition A.5} The
following determine each other
\roster
\item"\rom{(i)}" $\{\tilde P(z,n) \}
^{n_0+1}_{n=1}$.
\item"\rom{(ii)}" $\int \lambda^j \,
d\rho (\lambda), \quad j=1,2,\dots, 2n_0
-1$.
\item"\rom{(iii)}" $\{c_j \}^{2n_0-1}_{j=1}$.
\endroster
\endproclaim
\demo{Proofs} (A.8), (A.9) show that the
claimed $c$'s determine the $P$'s (or
$\tilde P$'s)
and vice-versa.
The $P$'s and $\tilde P$'s are a basis
for polynomials up to $z^{n_{0}}$, so one
can write
$\{z^j\}_{j=0}^{n_{0}}$ in terms of the
$P$'s or $\tilde P$'s. In the $P$ case,
the orthogonality
relations (A.7) then determine the
integral $\int \lambda^j \lambda^k\, d\rho
(\lambda)$ for
$j,k=0,1,\dots, n_0$ and so the stated
moments. Conversely, by Gram-Schmidt,
these moments
determine the $P$'s.
In the $\tilde P$ case, we argue as
follows. The $\tilde P$ determine $a_1,
\dots, a_{n_0 -1}$,
and so by (A.5), $\{P (z,j)\}^{n_0}_{j=1}$.
The orthonormality relations then
determine
the moments up to order $2n_0 -2$. The
orthogonality relation $\int \lambda^{n_0
-1}
P(\lambda, n_0 +1)\, d\rho (\lambda)$
determines the final $\int \lambda^{2n_0
-1} d\rho
(\lambda)$ moment. Conversely, Gram-Schmidt
and the moments determine the
$\tilde P$'s.
Since we don't normalize $\tilde P(z,n_0 +1)$,
we don't need the $\int
\lambda^{2n_0}\, d\rho
(\lambda)$ moment. \qed
\enddemo
\proclaim{Theorem A.6} Every $N$-point
probability measure $d\rho (\lambda)$
arises as
the spectral measure of a unique
$N\times N$ Jacobi matrix. Every probability
measure of
bounded and infinite support arises as
the spectral measure of a unique Jacobi
matrix on
$\Bbb N$.
\endproclaim
\demo{Proof} Because a polynomial of
degree $k$ has at most $k$ zeros, if
$d\rho$ is
supported on $N$ points, the monomials
$\{\lambda^n\}^{N-1}_{n=0}$ are linearly
independent in $L^2 (d\rho)$. If $d\rho$
has infinite support,
$\{\lambda^n\}^\infty_{n=0}$
are linearly independent in $L^2 (d\rho)$.
Thus given $d\rho$, we can use
unnormalized Gram-Schmidt to construct monic
polynomials
$\{\tilde P(z,n)\}^{N-1}_{n=0}$
(with $N=\infty$ if $d\rho$ has infinite
support). By
construction, $\{\tilde P(z,n)\}^p_{n=0}$
is a basis for the polynomials of
degree $p-1$. Thus,
$$
z\tilde P(z,n) = \tilde P(z,n+1) +
\text{ linear combination of } \tilde P(z,1),
\dots,\tilde P(z,n).
$$
The coefficient in front of $\tilde P(z,n+1)$
is one because $\tilde P(z,n)$ and
$\tilde P(z,n+1)$
are monic. Since $z\tilde P(z, \ell)$ is a
linear combination of $\{\tilde
P(z,m)\}^{\ell + 1}_{m=0}$,
orthogonality implies $\int \tilde
P(\lambda, k) [\lambda \tilde
P(\lambda,\ell)]\,
d\rho(\lambda) =0$ if $\ell+1 0$).
Given $d\rho$ of bounded support, define
$a_n$, $b_n$ by (A.15). By (A.14), the
Jacobi
matrix is bounded. Let $d\tilde\rho$ be
its spectral measure. By construction,
the orthogonal
polynomials for $d\rho$ and $d\tilde\rho$
are the same. Thus, their moments are
the same by
Propositions~A.4 and A.5. But the moments
uniquely determine a measure of finite
support by
Weierstrass' theorem on the density of
polynomials in the continuous functions.
Thus, $d\rho
= d\tilde\rho$. \qed
\enddemo
As an application of the formalism, we can
translate Hochstadt's theorem
[\hmixed]
{\it and proof} into this language:
\proclaim{Theorem A.7}{\rom{ (Hochstadt
[\hmixed])}} Given the
eigenvalues $\lambda_1 <
\cdots <
\lambda_N$ and the numbers $\{c_j\}
^{N-1}_{j=1}$ of an $N\times N$ Jacobi
matrix, there
is at most one set of $\{c_j\}^{2N-1}_{j=N}$
consistent with the data.
\endproclaim
\demo{Proof {\rom{([\hmixed])}}} Let
$d\rho(\lambda) = \sum^N_{j=1} \alpha_j
\delta
(\lambda -\lambda_j) \, d\lambda$. We have
to show the $\alpha_j$ are uniquely
determined,
since by Theorem~A.6, they uniquely determine
$\{c_j\}^{2N-1}_{j=1}$. By
Propositions~A.4
and A.5, the $\{c_j\}^{N-1}_{j=1}$ determine
the moments $m_k = \int \lambda^k\,
d\rho (\lambda)$ for $k=1, \dots, N-1$ and,
of course, $m_0 = 1$. Thus we have
$$
\sum^N_{j=1} \lambda^k_j \alpha_j =
m_k \tag A.16
$$
for $k=0,1, \dots, N-1$. By the
non-vanishing of van der Monde determinants,
$$
\det((\lambda^k_j)_{0\leq k \leq N-1,
1\leq j \leq N}) = \prod_{r