\input amstex
\documentstyle{amsppt}
\magnification=1200
\baselineskip=1.5 pt
\TagsOnRight
\NoBlackBoxes
\topmatter
\title The Xi Function
\endtitle
\author F.~Gesztesy$^{1}$ and B.~Simon$^{2}$
\endauthor
\leftheadtext{F.~Gesztesy and B.~Simon}
\thanks $^{1}$ Department of Mathematics, University of Missouri,
Columbia, MO 65211. E-mail: mathfg\@mizzou1.\linebreak missouri.edu
\endthanks
\thanks $^{2}$ Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, 253-37,\linebreak Pasadena, CA
91125. This material is based upon work supported by the National
Science Foundation under Grant No.~DMS-9101715. The Government has
certain rights in this material.
\endthanks
\thanks To be submitted to {\it Ann.~Math.}
\endthanks
\endtopmatter
\vskip 0.5in
\document
\flushpar {\bf \S1. Introduction}
Despite the fact that spectral and inverse spectral properties of
one-dimensional Schr\"o-dinger operators $H=-\frac{d^2}{dx^{2}}+V$
have been extensively studied for seventy-five years, there remain
large areas where our knowledge is limited. For example, while the
inverse theory for operators on $L^{2}(-\infty, \infty)$ is well
understood in case $V$ is periodic [12,24,25,35,39--42,49], it is not
understood in case $\lim\limits_{|x|\to\infty}\,V(x)=\infty$ and $H$
has discrete spectrum.
Our goal here is to introduce a special function $\xi(x, \lambda)$ on
$\Bbb R\times\Bbb R$ associated to $H$ which we believe will be a
valuable tool in the spectral and inverse spectral theory. In a sense
we'll make precise, it complements the Weyl $m$-functions, $m_{\pm}(x,
\lambda)$.
A main application of $\xi$ which we will make here concerns a
generalization of the trace formula for Schr\"odinger operators to
general $V$'s.
Recall the well-known trace formula for periodic potentials: Let
$V(x)=V(x+1)$. Then, by Floquet theory (see, e.g., [10,37,44])
$$
\text{spec}(H)=[E_{0}, E_{1}]\cup [E_{2}, E_{3}]\cup\dots
$$
a set of bands. If $V$ is $C^1$, one can show that the sum of the gap
sizes is finite, that is,
$$
\sum^{\infty}_{n=1}|E_{2n}-E_{2n-1}|<\infty. \tag 1.1
$$
For fixed $y$, let $H_y$ be the operator on $-\frac{d^2}{dx^2}+V$
on $L^{2}([y, y+1])$ with $u(y)=u(y+1)=0$ boundary conditions. Its
spectrum is discrete, that is, there are eigenvalues
$\{\mu_{n}(y)\}^{\infty}_{n=1}$ with
$$
E_{2n-1}\leq\mu_{n}(y)\leq E_{2n}. \tag 1.2
$$
The trace formula says
$$
V(y)=E_{0}+\sum^{\infty}_{n=1} [E_{2n}+E_{2n-1}-2\mu_{n}(y)]. \tag 1.3
$$
By (1.2),
$$
|E_{2n}+E_{2n-1}-2\mu_{n}(y)|\leq |E_{2n}-E_{2n-1}|
$$
so (1.1) implies the convergence of the sum in (1.3).
The earliest trace formula for Schr\"odinger operators was found on a
finite interval in 1953 by Gel'fand and Levitan [15] with later
contributions by Dikii [8], Gel'fand [13], Halberg-Kramer [23],
and Gilbert-Kramer [22]. The first trace formula for periodic $V$ was
obtained in 1965 by Hochstadt [24], who showed that for finite-gap
potentials
$$
V(x)-V(0)=2\sum^{g}_{n=1}[\mu_{n}(0)-\mu_{n}(x)].
$$
Dubrovin [9] then proved (1.3) for finite-gap potentials. The general
formula (1.3) under the hypothesis that $V$ is periodic and
$C^{\infty}$ was proven in 1975 by McKean-van Moerbeke [41], and Flaschka
[12], and later for general $C^3$ potentials by Trubowitz [49]. Formula
(1.3) is a key element of the solution of inverse spectral problems
for periodic potentials [9,12,24,35,39,41,42,49].
There have been two classes of potentials for which (1.3) has been
extended. Certain almost periodic potentials are studied in Levitan
[34,35], Kotani-Krishna [31], and Craig [5].
In 1979, Deift-Trubowitz [7] proved that if $V(x)$ decays sufficiently
rapidly at infinity and $-\frac{d^2}{dx^2}+V$ has no negative
eigenvalues, then
$$
V(x)=\frac{2i}{\pi}\int\limits^{\infty}_{-\infty}\,dk\,k\,
\ln\biggl[1+R(k)\frac{f_{+}(x, k)}{f_{-}(x, k)}\biggr] \tag 1.4
$$
(where $f_{\pm}(x, k)$ are the Jost functions at energy $E=k^{2}$ and
$R(k)$ is a reflection coefficient) which, as we will see, is an analog
of (1.3). Recently, Venakides [49] studied a trace formula for $V$, a
positive smooth potential of compact support, by writing (1.3) for the
periodic potential
$$
V_{L}(x)=\sum^{\infty}_{n=-\infty}V(x+nL)
$$
and then taking $L$ to $\infty$. He found an integral formula which,
although he didn't realize it, is precisely (1.4)!
The basic definition of $\xi$ depends on the theory of the Krein
spectral shift [32]. If $A$ and $B$ are self-adjoint operators with
$A\geq\eta$, $B\geq\eta$ for some real $\eta$ and so that $[(A+i)^{-1}-
(B+i)^{-1}]$ is trace class, then there exists a measurable function
$\xi(\lambda)$ associated with the pair $(B, A)$ so that
$$
\text{Tr}[f(A)-f(B)]=-\int\limits_{\Bbb R}f'(\lambda)\xi(\lambda)\, d\lambda
\tag 1.5
$$
for $f$'s which are sufficiently smooth and which decay sufficiently
rapidly at infinity, and, in particular for $f(\lambda)=e^{-t\lambda}$
for any $t>0$; and so that
$$
\xi(\lambda)=0\qquad \text{if } \lambda<\eta. \tag 1.6
$$
Moreover, (1.5), (1.6) uniquely determine $\xi(\lambda)$ for
a.e.~$\lambda$, and if $[(A+i)^{-1}-(B+i)^{-1}]$ is rank $n$, then
$$
|\xi(\lambda)|\leq n
$$
and if $B\geq A$, then $\xi(\lambda)\geq 0$.
For the rank one case of importance in this paper, an extensive study
of $\xi$ can be found in [48] and a brief discussion in the appendix to
this paper.
Let $V$ be a continuous function on $\Bbb R$ which is bounded from below.
Let $H=-\frac{d^2}{dx^2}+V$ which is self-adjoint on $C^{\infty}_{0}
(\Bbb R)$ (see, e.g., [43]) and let $H_{D; x}$ be the operator on
$L^{2}(-\infty, x)\oplus L^{2}(x, \infty)$ with $u(x)=0$ Dirichlet boundary
conditions. Then $[(H_{D;x}+i)^{-1}-(H+i)^{-1}]$ is rank one, so there
results a Krein spectral shift $\xi(x, \lambda)$ for the pair
$(H_{D; x}, H)$ which in particular
obeys:
$$
\text{Tr}(e^{-tH}-e^{-tH_{D; x}})=t\int\limits^{\infty}_{0}
e^{-t\lambda}\xi(x, \lambda)\, d\lambda. \tag 1.7
$$
While $\xi$ is defined in terms of $H$ and $H_{D; x}$, there is a
formula that only involves $H$, or more precisely, the Green's
function $G(x,y; z)$ defined by
$$
((H-z)^{-1}f)(x)=\int\limits_{\Bbb R} G(x, y, z)f(y)\, dy \tag 1.8
$$
for $\text{Im }z\neq 0$. Then by general principles, $\lim\limits_
{\epsilon\downarrow 0}\, G(x, y; \lambda+i\epsilon)$ exists for
a.e.~$\lambda\in\Bbb R$, and
\proclaim{Theorem 1.1}
$$
\xi(x, \lambda)=\frac{1}{\pi}\,\text{\rom{Arg}}\bigl(\lim\limits_
{\epsilon\downarrow 0}\, G(x, x, \lambda+i\epsilon)\bigr).
$$
\endproclaim
This is {\it formally} equivalent to formulae that Krein [32] has for
$\xi$ but in a singular setting (i.e., corresponding to an infinite coupling
constant). It follows from equations (A.8)--(A.10) in the appendix. With
this definition out of the way, we can state the general trace formula:
$$
V(x)=\lim\limits_{\alpha\downarrow 0}\, \biggl[E_{0}+\int\limits
^{\infty}_{E_0} e^{-\alpha\lambda}[1-2\xi(x, \lambda)]\, d\lambda
\biggr], \tag 1.9
$$
where $E_{0}\leq\inf\,\text{spec}(H)$. In particular, if
$\int\limits^{\infty}_{E_0}|1-2\xi(x, \lambda)|\,d\lambda<\infty$, then
$$
V(x)=E_{0}+\int\limits^{\infty}_{E_0}[1-2\xi(x, \lambda)]\, d\lambda.
\tag 1.10
$$
For certain almost periodic potentials, Craig [5] used a
regularization similar to the $\alpha$-regularization in (1.9).
We will prove (1.9) in \S3 if $V$ is continuous, bounded below, and
obeys a bound
$$
|V(x)|\leq C_{1}e^{C_{2}x^{2}}. \tag 1.11
$$
In a subsequent paper [17], we'll allow any $V$ which is bounded below
and even drop the continuity property ((1.9) will then hold at points,
$x$, of Lebesgue continuity for $V$). That paper will also discuss
``higher order trace formulae,'' familiar from the context of the
Korteweg-de Vries hierarchy, that is, formulae where the left side
has suitable polynomials in derivatives of $V$ at $x$. Basically,
(1.9) will follow from (1.7) and an asymptotic formula,
$$
\text{Tr}(e^{-tH}-e^{-tH_{D;x}})=\frac{1}{2}[1-tV(x)+o(t)]. \tag 1.12
$$
Examples of the trace formula can be found in \S3 including the case
$V(x)\to\infty$ as $|x|\to\infty$. In [16], we'll prove that (1.4) is
a special case of (1.9).
The proof in \S3 depends on technical preliminaries in \S2. We
discuss the case of Jacobi matrices (discrete Schr\"odinger operators)
in \S4 including a result for $\Bbb Z^n$. A general $\Bbb R^n$ result
that is a kind of analog of (1.12) can be found in [18].
In \S5 we turn to some continuity properties of $\xi(x, \lambda)$ in
the potential $V$ and use them to find a new proof (and generalization)
of a recent striking result of Last [33]. In particular, we establish
$\xi(x, \lambda)$ as a new tool in spectral theory and derive a novel
criterion for the essential support of the absolutely continuous
spectrum of one-dimensional Schr\"odinger operators and
(multi-dimensional) Jacobi matrices. In \S6, we discuss an
overview of the connection of the function $\xi$ to inverse problems,
including a generalized trace formula that shows how to recover
the diagonal Green's function $g(x, z):=G(x, x, z)$ (a Herglotz function
w.r.t.~$z$) from $\xi(x, \lambda)$.
\bigpagebreak
\flushpar {\bf \S2. First Order Asymptotics of the Heat Kernel Trace}
As we have seen, a basic role is played by the asymptotics of
$$
\text{Tr}(e^{-tH}-e^{-tH_{D; x}})\qquad\text{as } t\downarrow 0.
$$
In this section we'll prove:
\proclaim{Theorem 2.1} Let $V$ be a continuous function which is
bounded from below and which obeys
$$
|V(x)|\leq C_{1}e^{C_{2}x^{2}}. \tag 2.1
$$
Then
$$
\text{\rom{Tr}}(e^{-tH}-e^{-tH_{D; x}})=\frac12 [1-tV(x)+o(t)].
\tag 2.2
$$
\endproclaim
Under hypothesis (2.1), one can prove this result using the method of
images and a DuHamel expansion of $e^{-tH}$ in terms of $e^{-tH_0}$. We will
instead use a path integral expansion. The advantage of this approach
is that by a more detailed analysis of the path space measure, one can
automate higher order expansions in $t$ as $t\downarrow 0$ and can
drop the growth condition (2.1). This will be described in a
subsequent paper [17]. By translation invariance, we henceforth set
the point $x$ in (2.2) to $x=0$.
So, our first step in proving Theorem 2.1 is to define a process we will
call the xi process, that is, a probablity measure on paths $\omega:[0,
1]\to\Bbb R$. Recall [46] that the Brownian bridge is the Gaussian
process $\{\alpha(s)\mid 0\leq s\leq 1\}$ with $E(\alpha(s))=0$,
$E(\alpha(s)\alpha(t))=s(1-t)$ if $0\leq s\leq t\leq 1$. The Brownian
bridge is of interest because of the following Feynman-Kac formula
[46]:
$$
e^{-tH}(x, x) =(4\pi t)^{-1/2} E\biggl(\exp\biggl(-
t\int\limits^{1}_{0} V(x+\sqrt{2t}\, \alpha(s))\, ds\biggr)\biggr),
\tag 2.3
$$
$$
e^{-tH_{D; 0}}(x, x) =(4\pi t)^{-1/2} E\biggl(\exp\biggl(-
t\int\limits^{1}_{0} V(x+\sqrt{2t}\, \alpha(s))\, ds\biggr)
\chi(\alpha\mid x+\sqrt{2t}\,\alpha(s)\neq 0 \text{ all } s)\biggr),
\tag 2.4
$$
where $\chi(\alpha\mid x+\sqrt{2t}\,\alpha(s)\neq 0 \text{ all } s)$ is the
characteristic function of those $\alpha$ for which $x+\sqrt{2t}\,\alpha(s)$
is non-vanishing for all $s$ in $[0, 1]$, so since paths are continuous,
those $\alpha$ with $x+\sqrt{2t}\,\alpha(s)>0$ for all $s$ if $x>0$. There
is a $\sqrt{2t}$ in (2.3/2.4) rather than the $\sqrt{t}$ in [46] because [46]
considers $-\frac12\, \frac{d^2}{dx^2}$ where we consider $-\frac{d^2}{dx^2}$.
Consider the measure $d\kappa$ on $\Omega=\Bbb R\times C([0, 1])$ given by
$(4\pi)^{-1/2}\, dx\otimes \Cal D\alpha$ where $dx$ is Lebesgue measure, and
define $\omega$ on $\Omega$ by $\omega(s)=x+\alpha(s)$. Since
$\alpha(0)=\alpha(1)\equiv 0$, we have $\omega(0)=\omega(1)=x$.
$\omega$ defines a natural map of $\Omega$ to $C([0, 1])$ and we
henceforth view everything on that space. Let $\Omega_{0}=
\{\omega\mid\omega(s)=0 \text{ for some $s$ in } [0, 1]\}$.
\proclaim{Proposition 2.2} The $\kappa$ measure of $\Omega_0$ is
$\frac12$.
\endproclaim
\demo{Proof} Let $\Delta=\frac{d^2}{dx^2}$ and $\Delta_D$ be the same
operator with a Dirichlet boundary condition at $x=0$. By (2.3--4)
with $V=0$,
$$\align
\kappa(\Omega_0) &=\int\limits_{\Bbb R} dx\, [e^{\Delta/2}(x, x)-
e^{\Delta_{D}/2}(x, x)] \\
&=\int\limits_{\Bbb R} dx\, e^{\Delta/2}(x, -x) \\
&=\int\limits_{\Bbb R} dx\, e^{\Delta/2}(2x, 0) \\
&=\frac12\int\limits_{\Bbb R} dx\, e^{\Delta/2}(0, x)=\frac12 ,
\endalign
$$
where the second equality is by the method of images. \qed
\enddemo
We will call the probability measure $2\chi_{\Omega_{0}}\, d\kappa$ on
$C([0, 1])$ the xi process and denote its expectations as $E_\omega$.
(2.3--4) and the regularity of the integral kernel immediately imply
that
\proclaim{Proposition 2.3} For any $V$ which is bounded from below:
$$
\text{\rom{Tr}}(e^{-tH}-e^{-tH_{D; 0}})=\frac12 E_{\omega} \biggl(\exp
\biggl(-t\int\limits^{1}_{0} V(\sqrt{2t}\, \omega(s))\, ds\biggr)\biggr).
$$
\endproclaim
We note the $t^{-1/2}$ in front of (2.3--4) is absorbed in the change
of variables from $x+\sqrt{2t}\, \alpha(s)$ to $\sqrt{2t}\,
(x+\alpha(s))$.
\proclaim{Lemma 2.4} If $C<\frac12$, then $E_{\omega}(e^{C\omega(s)^{2}})
<\infty$ for each $s$ with a bound uniform in $s$.
\endproclaim
\demo{Proof} Let $f$ be a bounded even function on $\Bbb R$. Then
$$\split
E_{\omega}(f(\omega(s))&=2\int\limits \Sb x>0 \\ y>0 \endSb \left[e^{(1-
s)\Delta/2}(x, y) f(y)\, e^{s\Delta/2}(y, x)\right. \\
&\qquad \left. - e^{(1-s)\Delta_{D}/2}
(x, y)f(y)\, e^{s\Delta_{D}/2}(y, x)\right]\, dx\, dy
\endsplit
$$
and it easy to see using the method of images that for $f(y)=\min(R,
e^{Cy^2})$ the integral remains finite as $R\to\infty$. \qed
\enddemo
\demo{Proof of Theorem {\rom{2.1}}} It is easy to see if we prove the formula
for $V(x)$, it follows for $V$ replaced by $V(x)+C$. Thus, without
loss we suppose $V\geq 0$. Let $a\leq 0$. Then by Taylor's theorem
with remainder:
$$
|e^{a}-1-a|\leq\frac12\, a^2.
$$
Thus, with $a=-t\int\limits^{1}_{0}V(\sqrt{2t}\, \omega(s))\, ds$
$$
\left|\frac12 E_{\omega}(e^{a})-\frac12-\frac12\, E_{\omega}(a)\right|
\leq\frac14\, E_{\omega}(a^{2}).
$$
Using (2.1) and Lemma 2.4, it is easy to see that
$$
E_{\omega}(a^{2})=O(t^{2})
$$
so it suffices to show that
$$
E_{\omega}\biggl(\int\limits^{1}_{0} V(\sqrt{2t}\, \omega(s))\,\biggr)
=V(0)+o(1).
$$
This follows from Lemma 2.4, (2.1), continuity of $V$ at $x=0$, and
dominated convergence. \qed
\enddemo
\remark{Remark} This is crude analysis compared with the detailed
path space argument in [17] but it is elementary and beyond the
argument of previous authors who supposed that $V$ is bounded.
\endremark
\bigpagebreak
\flushpar {\bf \S3. The Trace Formula: Schr\"odinger Case}
Our main goal in this section is to prove:
\proclaim{Theorem 3.1} Suppose $V$ is a continuous function bounded
from below on $\Bbb R$. Let $\xi(x, \lambda)$ be the Krein spectral
shift for the pair $(H_{D;x}, H)$ with $H_{D;x}$ the operator on
$L^{2}(-\infty, x)\oplus L^{2}(x, \infty)$ obtained from $H=-\frac{d^2}
{dx^2}+V$ with a Dirichlet boundary condition at $x$. Let $E_{0}\leq
\inf\,\text{\rom{spec}}(H)$. Then
$$
V(x)=\lim_{\alpha\downarrow 0} \biggl[E_{0}+\int\limits^{\infty}
_{E_0} e^{-\alpha\lambda} [1-2\xi(x, \lambda)]\, d\lambda\biggr].
\tag 3.1
$$
\endproclaim
\demo{Proof} Let $E_{1}=\inf\,\text{spec}(H)$. Then for
$E_{0}\leq E_{1}$
$$
E_{0}+\int\limits^{\infty}_{E_0} e^{-\alpha\lambda}(1-2\xi)\,
d\lambda=E_{0}+\int\limits^{E_1}_{E_0} e^{-\alpha\lambda}\,d\lambda +
\int\limits^{\infty}_{E_1} e^{-\alpha\lambda}(1-2\xi)
$$
and $\lim\limits_{\alpha\downarrow 0}\,\int\limits^{E_1}_{E_0} e^{-
\alpha\lambda}\,d\lambda=E_{1}-E_{0}$ so the formula for $E_1$ implies
it for $E_0$; that is, without loss we suppose $E_{0}=E_{1}$. By
Theorems 2.1 and eq.~(1.7),
$$
\alpha\int\limits^{\infty}_{E_0} e^{-\alpha\lambda}\xi(x, \lambda)\,
d\lambda=\frac12\, [1-\alpha V(x)+o(\alpha)].
$$
Moreover,
$$
\frac12\,\alpha\int\limits^{\infty}_{0} e^{-\alpha\lambda}\,
d\lambda=\frac12 \tag 3.2
$$
so
$$
\frac12\, \alpha\int\limits^{\infty}_{E_0}e^{-\alpha\lambda}\,
d\lambda=\frac12\, [1-\alpha E_{0}+o(\alpha)]
$$
and hence
$$
\alpha\int\limits^{\infty}_{E_0} e^{-\alpha\lambda}\biggl(\xi-\frac12
\biggr)\, d\lambda=-\frac12\, \alpha[V(x)-E_{0}]+o(\alpha)
$$
which is (3.1). \qed
\enddemo
\example{Example 3.2} $V=0$. Then $g(x, \lambda)=\frac12 \,(-
\lambda)^{-1/2}$ and so $\arg\,g(x, \lambda)=0$ (resp.~$\pi/2$) if
$\lambda<0$ (resp.~$\lambda>0$). Thus, by Theorem 1.1, $\xi(x,
\lambda)\equiv\frac12$ on $[0, \infty)$ and (3.2) is just Theorem 2.1
for $V=0$. When $\xi=\frac12$ on a subset of $\text{spec}(H)$, that set
drops out of (3.1).
\endexample
\example{Example 3.3} Suppose that $V(x)\to\infty$ as $|x|\to\infty$.
Then $H$ has eigenvalues $E_{0}-\infty$
($\Bbb R$ case) or $\sup\limits_{(n, j)\in\Bbb N\times\Bbb Z}
|V_{n}(j)|<\infty$ ($\Bbb Z$ case).
\item"{(ii)}" For each $R<\infty$, $\sup\limits_{|x|\leq R} |V_{n}(x)
-V(x)|\to 0$ as $n\to\infty$.
\endroster
\enddefinition
\proclaim{Lemma 5.3} If $V_{n}\to V$ locally as $n\to\infty$ and
$H_n, H$ are the corresponding Schr\"odinger operators \rom(resp.~Jacobi
matrices\rom), then $(H_{n}-z)^{-1}\to (H-z)^{-1}$ strongly for
$\text{Im}\,z\neq 0$ as $n\to\infty$.
\endproclaim
\demo{Proof} Let $\varphi\in C^{\infty}_{0}(\Bbb R)$ or a finite
sequence in $\ell^{2}(\Bbb Z)$. Then $[(H_{n}-z)^{-1}-(H-z)^{-1}]
(H-z)\varphi=(H_{n}-z)^{-1}(V-V_{n})\varphi\to 0$ as $n\to\infty$. But $\{(H-
z)\varphi\}$ is a dense set (since $H$ is essentially self-adjoint on
$C^{\infty}_{0}(\Bbb R)$ resp.~on finite sequences). \qed
\enddemo
\proclaim{Theorem 5.4} If $V_{n}\to V$ locally as $n\to\infty$ and
$\xi_{n}(x,\lambda)$, $\xi(x, \lambda)$ are the corresponding xi functions
for fixed $x$, then $\xi_{n}(x, \lambda)\, d\lambda$ converges to
$\xi(x, \lambda)\, d\lambda$ weakly in the sense that
$$
\int\limits_{\Bbb R} f(\lambda)\xi_{n}(x, \lambda)\, d\lambda \to
\int\limits_{\Bbb R} f(\lambda)\xi(x, \lambda)\, d\lambda \qquad
\text{as } n\to\infty \tag 5.2
$$
for any $f\in L^{1}(\Bbb R; d\lambda)$.
\endproclaim
\demo{Proof} By a simple density argument (using $|\xi(x,
\lambda)|\leq 1$), it suffices to prove this for $f(\lambda)=(\lambda-
z)^{-2}$ and all $z\in\Bbb C\backslash\Bbb R$. But by (A.$7'$):
$$
\int\limits_{\Bbb R} (\lambda-z)^{-2}\xi_{n}(x, \lambda)\,d\lambda
=\frac{d}{dz}\,F_{n}(x, z),
$$
where $F_{n}(x, z)=\ln\,g_{n}(x, z)$. Since the $F$'s are analytic
and uniformly bounded, pointwise convergence of the $F$'s implies
convergence of the derivatives $dF_{n}/dz$. Thus we need only show
$$
g_{n}(x, z)\operatornamewithlimits{\to}\limits_{n\to\infty} g(x, z).
$$
This follows from Lemma 5.3 (and, in the Schr\"odinger case, some
elliptic estimates to turn convergence of the operators to pointwise
convergence of the integral kernels). \qed
\enddemo
\definition{Definition} For any $H$, let $|S_{\text{\rom{ac}}}(H)|$
denote the Lebesgue measure of the essential support of the absolutely
continuous spectrum of $H$.
\enddefinition
\proclaim{Theorem 5.5} \rom{(For one-dimensional Schr\"odinger
operators or Jacobi matrices)} Suppose $V_{n}\to V$ locally as
$n\to\infty$ and each $V_n$ is periodic. Then for any interval
$(a, b)\subset\Bbb R$:
$$
|(a, b)\cap S_{\text{\rom{ac}}}|\geq \varlimsup\limits_{n\to\infty}
|(a, b)\cap S_{\text{\rom{ac}}} (H_{n})|.
$$
\endproclaim
\remark{Remark} The periods of $V_{n}$ need {\it not} be fixed;
indeed, almost periodic $V$'s are allowed.
\endremark
\demo{Proof} By periodicity, $\xi_{n}(x, \lambda)$ is $0$, $\frac12$,
$1$ for a.e.~$\lambda\in\Bbb R$. Let $A_{n}=\{\lambda\in (a, b)\mid\xi_{n}(x,
\lambda)=0\}$ and $A=\{\lambda\in (a, b)\mid\xi(x, \lambda)=0\}$.
Then, $\xi_{n}(x, \lambda)\geq\frac12$ on $A\backslash A_{n}$, so for
any $a, b$:
$$
\int\limits_{A} \xi_{n}(x, \lambda)\, d\lambda \geq\frac12
|(A\backslash A_{n})|.
$$
But by Theorem 5.4, $\int\limits_{A}\xi_{n}(x, \lambda)\, d\lambda
\operatornamewithlimits{\to}\limits_{n\to\infty}
\int\limits_{A}\xi(x, \lambda)\, d\lambda=0$. Thus, $\frac12
|A\backslash A_{n}|\to 0$, so $|A|\leq\varliminf\limits_{n\to\infty}
|A_{n}|$. Similarly,
using $1-\xi$, we get an inequality on
$$
|\{\lambda\in (a, b)|\xi (x,\lambda)=1\}|\leq\varliminf\limits_{n\to\infty}
|\{\lambda\in (a, b)|\xi (x, \lambda)=1\}|.
$$
This implies the result. \qed
\enddemo
\example{Example 5.6} Let $\alpha_n$ be a sequence of rationals and
$\alpha=\lim\limits_{n\to\infty}\,\alpha_n$. Let $H_n$ be the Jacobi
matrix with potential $\lambda\cos (2\pi\alpha_{n}+\theta)$ for $\lambda,
\theta$ fixed. Then [2] have shown for $|\lambda|\leq 2$, $|S_{n}|
\geq 4-2|\lambda|$. It follows from the last theorem that $|S|\geq
4-2|\lambda|$. This provides a new proof (and a strengthening)
of an important result of Last [33].
\endexample
\example{Example 5.7} Let $\{a_{m}\}_{m\in\Bbb N}$ be a sequence with
$s=\sum\limits^{\infty}_{m=1}2^{m}|a_{m}|<2$. Let
$V(n)=\sum\limits^{\infty}_{m=1}a_{m}\cos(2\pi n/2^{m})$, a limit periodic
potential on $\Bbb Z$. Let $h$ be the corresponding Jacobi matrix.
We claim that
$$
|\sigma_{\text{ac}}(h)|\geq 2(2-s). \tag 5.31
$$
For let $V_{M}(n)=\sum\limits^{M}_{m=1} a_{m}\cos(2\pi n/2^{m})$ with $h_{M}$
the associated Jacobi matrix. Then the external edges of the spectrum move in
at most by $\|V_{M}\|_{\infty}\leq\sum\limits^{M}_{m=1}|a_{m}|$. $V_{M-1}$
has at most $2^{M-1}-1$ gaps. They increase in size in going from $V_{M-1}$
to $V_{M}$ by $2|a_{M}|$. In addition, $V_{M}$ has $2^{M-1}$ new gaps. Thus,
$\sigma(h_{M})\geq 4-2\|V\|_{\infty}-\sum\limits^{M}_{m=1} (2^{m}-1)
|a_{m}|\geq 4-2s$, which yields (5.31) on account of Theorem 5.5.
Knill-Last [29] have shown how to use our Theorem 5.5 to treat more general
limit periodic potentials, including Schr\"odinger operators of the form
studied by Chulaevsky [4], and have also treated quasi-periodic potentials
of the form $V(n)=\sum\limits^{\infty}_{m=1}\lambda_{m}\cos([2\pi\alpha n
+\theta]m)$ where they show $|\sigma_{\text{ac}}|\geq 4-6\sum
\limits^{\infty}_{m=1}m|\lambda_{m}|$.
\endexample
\bigpagebreak
\flushpar {\bf \S6. Inverse Problems}
We want to give an overview of how we believe $\xi(x, \lambda)$ can be
an important tool in the study of inverse problems and apply the
philosophy in a few cases. Roles are played by $\xi(x_{0}, \lambda)$, the
diagonal Green's function $g(x_{0}, \lambda)$, and the Weyl $m$-functions
$m_{\pm}(x_{0}, z)$ (corresponding to the Dirichlet boundary condition
at $x=x_{0}$). The relationship is that $\xi$ is closest to
spectral and scattering information and it, under proper
circumstances, determines $g(x_{0}, \lambda)$ and the derivative $g'(x_{0},
\lambda)$. They determine $m_{\pm}(x_{0}, \lambda)$, which in turn
determine $V(x)$ for a.e.~$x\in\Bbb R$ by the Gel'fand-Levitan method
[14,36]. The scheme underlying our philosophy is illustrated in
Fig.~1.
That $m_{\pm}(y, \lambda)$ for all $\lambda$ and a single $y$
determine $V(x)$ on $(-\infty, y)$ and $(y, \infty)$ is well known [38].
That $g(x, \lambda)$ and $\frac{d}{dx}\,g(x, \lambda)$ at a single point
$x$ determine $m_{\pm}(x, \lambda)$ follows from the pair of formulae:
$$\align
g(x, \lambda) &= -[m_{+}(x, \lambda)+m_{-}(x, \lambda)]^{-1},
\tag 6.1 \\
g'(x, \lambda) &= -[m_{+}(x, \lambda)-m_{-}(x, \lambda)]\big/
[m_{+}(x, \lambda)+m_{-}(x, \lambda)]. \tag 6.2
\endalign
$$
(6.2) follows from (6.1) and the Riccati equations
$$
m'_{\pm}(x, \lambda)=\mp[m^{2}_{\pm}(x, \lambda)-V(x)+\lambda].
$$
(6.2) is not new; it can be found, for example, in Johnson-Moser [26].
Thus, to recover $V(x)$ for all $x\in\Bbb R$ from $\xi(x_{0}, \lambda)$
for a fixed $x_{0}$ and all $\lambda$, we only need a method to compute
$g(x_{0}, \lambda)$ and $g'(x_{0}, \lambda)$ from $\xi(x_{0}, \lambda)$.
One can get $g$ in general from the following formula which follows from
Theorem A.2 in the appendix and the proposition below:
$$
g(x, z)=(-z)^{-1/2}\lim\limits_{\gamma\to\infty}\, \exp\biggl(
\int\limits^{\infty}_{E_{0}}\biggl[\frac{\xi(x, \lambda)-\frac{1}{2}}
{z-\lambda}\biggr]\,\biggl[\frac{\gamma}{\gamma+\lambda}\biggr]\,
d\lambda\biggr), \qquad E_{0}=\inf\,\text{spec}(H). \tag 6.3
$$
The proposition we need is
\proclaim{Proposition 6.1} Let $V$ be continuous and bounded from
below and let $g(x, z)=\mathbreak G(x, x, z)$ be the diagonal Green's
function for $H=-\frac{d^2}{dx^2}+V$. Then
$$
\lim\limits_{\lambda\to\infty}\, g(x, -\lambda)\big/(\lambda)^{-1/2}
=1. \tag 6.4
$$
\endproclaim
\demo{Proof} Let $p(x, t)$ be the diagonal heat kernel for $H$. By
the Feynman-Kac formula [46]
$$
p(x, t)=(4\pi t)^{-1/2} E\,\biggl(\exp\biggl(-t\int\limits^{1}_{0}
V(x+\sqrt{2t}\,\alpha(s))\, ds\biggr)\biggr),
$$
where $\alpha$ is the Brownian bridge. It follows by the dominated
convergence theorem that
$$
p(x, t)\big/(4\pi t)^{-1/2}\to 1 \tag 6.5
$$
as $t\downarrow 0$. Since
$$
g(x, -\lambda)=\int\limits^{\infty}_{0} e^{-\lambda t}p(x, t)\, dt
$$
we obtain (6.4). \qed
\enddemo
\remark{Remarks} (i) (6.4) can also be read off of asymptotics of
$m_{\pm}$ found in [1,11].
(ii) (6.5) can be used to prove the following stronger version of
(6.3):
$$
g(x, z)=(-z)^{-1/2}\lim\limits_{\alpha\downarrow 0}\,\exp\biggl(
\int\limits^{\infty}_{E_{0}}\biggl[\frac{\xi(x, \lambda)-\frac{1}{2}}
{z-\lambda}\biggr]\,e^{-\alpha\lambda}\, d\lambda\biggr).
$$
Thus, the solution of the inverse problem for going from
$\xi(x_{0}, \cdot)$ at a single $x_{0}$ to $V(x)$ for all $x\in\Bbb
R$ is connected to finding $g'(x_{0}, z)$ from $\xi(x_{0}, \lambda)$.
In absolute generality, we are unsure how to proceed with this because we
have no general theory for a differential equation that $\xi(x, \lambda)$
obeys for $\lambda$ in the essential spectrum of $H$. Indeed, for random
$V$'s where typically $\text{spec}(H)=[\alpha, \infty)$ for some $\alpha$,
$\xi(x, \lambda)=1$ or $0$ on $\Bbb R$ and $\overline{\{\lambda\in\Bbb R\mid
\xi(x, \lambda)=1\}}=[\alpha, \infty)$, $\overline{\{\lambda\in\Bbb R\mid\xi
(x, \lambda)=0\}}=\Bbb R$ and the $x$ dependence must be very complex.
However, one class of potentials does allow some progress:
\endremark
\definition{Definition} We say that $V$ is discretely dominated if
for all $x\in\Bbb R$, $\xi(x, \lambda)=\frac12$ for a.e.~$\lambda\in
\sigma_{\text{ess}}(H)$.
\enddefinition
Examples include reflectionless (soliton) potentials in the short-range
case, the periodic case, algebro-geometric finite-gap potentials and
limiting cases thereof (such as solitons relative to finite-gap
backgrounds), certain almost periodic potentials, and potentials with
$V(x)\to\infty$ as $|x|\to\infty$. In this case if $E_{0}=\inf\,\text{spec}
(H)$, $[E_{0}, \alpha)\backslash\text{spec}(H)=\operatornamewithlimits{\cup}
\limits^{N}_{n=1} (\alpha_{n}, \beta_{n})$ where $N$ is finite or infinite.
For each $x$, there is at most one eigenvalue for $H_{D; x}$ in each
$(\alpha_{n}, \beta_{n})$; call it $\mu_{n}(x)$. If there is no eigenvalue
in $(\alpha_{n}, \beta_{n})$, then $\xi(x, \lambda)$ is either $1$ on
$(\alpha_{n}, \beta_{n})$ or $0$, and then we set $\mu_{n}(x)$ equal to
$\beta_{n}$ or to $\alpha_{n}$. Thus
$$
\xi(x, \lambda) = \cases \frac12, &\quad \lambda\in\text{spec}(H) \\
0, &\quad \mu_{n}(x)<\lambda<\beta_{n} \\
1, &\quad \alpha_{n}<\lambda<\mu_{n}(x)
\endcases
$$
and the inverse formulae at a fixed $x$ say that
$$
V(x)=\lim\limits_{t\downarrow 0}\sum^{N}_{n=1} [2e^{-t\mu_{n}(x)}
-e^{-t\alpha_{n}}-e^{-t\beta_{n}}]\big/t, \tag 6.6
$$
$$
g(x, z)=(E_{0}-z)^{-1/2}\lim\limits_{\gamma\to\infty}\,
\biggl[\prod^{N}_{n=1}\biggl\{\frac{[z-\mu_{n}(x)]^{2}}{(z-\alpha_{n})
(z-\beta_{n})}\, \frac{(\gamma+\alpha_{n})(\gamma+\beta_{n})}
{[\gamma-\mu_{n}(x)]^{2}}\biggr\}\biggr]^{1/2}. \tag 6.7
$$
If $\sum\limits_{n}|\beta_{n}-\alpha_{n}|<\infty$, then
$$
g(x, z)=(E_{0}-z)^{-1/2}\biggl[\prod^{N}_{n=1}\,\frac{[z-
\mu_{n}(x)]^{2}}{(z-\alpha_{n})(z-\beta_{n})}\biggr]^{1/2} \tag 6.8
$$
(with an absolutely convergent product if $N=\infty$).
The $\mu$'s obey a differential equation essentially that was found by
Dubrovin [9] in 1975 for the finite-gap periodic case and extended
later by McKean-Trubowitz [42], Trubowitz [49], Levitan [34,35],
Kotani-Krishna [31], and Craig [5]. The form we give is the one in
Kotani-Krishna [31]. Previous authors only considered the periodic or
almost periodic case, so, in particular, our result is new in the case
$|V(x)|\to\infty$ where the regulariziation (6.6) is needed since
$\sum\limits_{n\in\Bbb N}|\beta_{n}-\alpha_{n}|=\infty$:
\proclaim{Theorem 6.2} Let $\alpha_{n}<\mu_{n}(x_{0})<\beta_{n}$.
Then $\mu_n$ is $C^1$ near $x_0$ and
$$
\left.\frac{d}{dx}\,\mu_{n}(x)\right|_{x=x_{0}} =\pm
1\bigg/\left.\frac{\partial g}{\partial\lambda}\,(x_{0}, \lambda)\right|_
{\lambda=\mu_{n}(x_{0})}, \tag 6.9
$$
where $g$ is given by \rom{(6.7)} or \rom{(6.8)}. In \rom{(6.9)},
the $\pm 1$ is $+1$ \rom(resp.~$-1$\rom) if $\mu_{n}(x_{0})$ is an eigenvalue
of $H_{x_{0}; D}$ on $(x_{0}, \infty)$ \rom(resp.~$(-\infty, x_{0})$\rom).
\endproclaim
\demo{Proof} The number $\mu_{n}(x)$ obeys
$$
g(x, \mu_{n}(x))=0.
$$
It is easy to see that $g$ is strictly monotone; indeed,
$\frac{\partial g}{\partial\lambda}>0$ on each $(\alpha_{n},
\beta_{n})$ and so by the implicit function theorem, $\mu_{n}(x)$ is
$C^1$ and
$$
\frac{d\mu_{n}}{dx}=-\frac{\partial g/\partial x}{\partial
g/\partial\lambda}
$$
so (6.9) is equivalent to $\frac{\partial g}{\partial x}=\mp 1$ if the
eigenvalue corresponds to the half-line $(x, \infty)$ (resp.~$(-\infty, x)$).
But the associated eigenvector lies in $L^{2}(x_{0}, \infty)$,
(resp.~$L^{2}(-\infty, x_{0})$) if and only if $m_{+}(x, \lambda)$
(resp.~$m_{-}(x, \lambda)$) is $\infty$ at $x=x_{0}$, $\lambda=\mu_{n}(x_{0})$.
By (6.2), $\frac{\partial g}{\partial x}=\mp 1$ if $m_{\pm}=\infty$. \qed
\enddemo
The simple example of the unique discretely dominated potential with
$\sigma(H)=\{-1\}\cup [0, \infty)$ (the one-soliton potential) is
discussed in [48]. (6.8)/(6.9) become an elementary differential equation
and $V$ is then given by (6.6). We will further explore this approach in
future papers [20,21].
Analogs of $\xi$ in the related inverse cases are also useful. For
example, we have shown that the $\xi$ function relating to half-line
problems on $[0, \infty)$ with different boundary conditions at $0$
determines the potential uniquely a.e. This result was previously
obtained independently by Borg [3] and Marchenko [38] in 1952 under the
strong additional hypothesis that the corresponding spectra were purely
discrete. Our approach allows us to dispense with the discrete spectrum
hypothesis and applies to arbitrary spectra.
\bigpagebreak
\flushpar {\bf Appendix: Rank One Perturbations and the Krein Spectral
Shift}
In this appendix, we will give a self-contained approach to the Krein
spectral shift in a slightly more general setting than usual and using
more streamlined calculations. The lecture notes [48] contain more
about this approach. Let $A\geq 0$ be a positive self-adjoint
operator in some complex separable Hilbert space $\Cal H$ and let
$\Cal H_{k}(A)(-\infty 0$
we have $0\leq\text{Arg}(\cdot)\leq\pi$ (and $\text{Im }F(z)>0$ if
$\text{Im }z>0$) and thus
$$
0\leq\xi_{\alpha}(\lambda)\leq 1
$$
in this case.
Since $\text{Arg}(F(\lambda+i0))=\text{Im }\ln(F(\lambda+i0))$, an
elementary contour integral argument ([48]) shows that (A.7) becomes
$$
\text{Tr}[(A-z)^{-1}-(A_{\alpha}-z)^{-1}]=\int\limits^{\infty}_
{E_{\alpha}}\frac{\xi_{\alpha}(\lambda)\, d\lambda}{(\lambda-z)^{2}},
\qquad E_{\alpha}=\inf\,\text{spec}(A_{\alpha}). \tag A.9
$$
(A.9) is a special case of
$$
\text{Tr}[f(A)-f(A_{\alpha})]=-\int\limits^{\infty}_{E_{\alpha}}
f'(\lambda)\xi_{\alpha}(\lambda)\,
d\lambda \tag A.10
$$
for the functions $f_{z}(\lambda)=(\lambda-z)^{-1}$. By analyticity
in $z$, one sees immediately that $[(A-z)^{-n}-(A_{\alpha}-z)^{-n}]$ is
trace class and (A.10) holds for $f_{z, n}(\lambda)=(\lambda-z)^{-n}$.
A straightforward limiting argument lets one prove ([48]) that if $f$
is $C^2$ on $\Bbb R$ with $(1+|x|)^{2}\,\frac{d^{j}f}{dx^{j}}\in
L^{2}(0, \infty)$ for $j=1,2$, then $[f(A)-f(A_{\alpha})]$ is trace
class and (A.10) holds. In particular,
$$
\text{Tr}(e^{-At}-e^{-tA_{\alpha}})=t\int\limits^{\infty}_{E_{\alpha}}
e^{-t\lambda}\xi_{\alpha}(\lambda)\, d\lambda. \tag A.11
$$
For the case where $\alpha=\infty$ and $\varphi\in\Cal H$,
$f(A_{\alpha})$ is interpreted as the operator on $\Cal H(A_{\infty})$
extended to $\Cal H$ by setting it equal to zero on $\Cal
H(A_{\infty})^{\perp}$. This follows from the approximation argument
since that is the meaning of $(A_{\infty}-z)^{-1}$. In particular,
\proclaim{Theorem A.1} Let $A$ be a bounded operator and $\varphi$ a
unit vector in $\Cal H$. Let $Q=\Bbb I-(\varphi, \cdot)\varphi$. Then
$A-QAQ$ is finite rank and
$$
\text{Tr}(A-QAQ)=-\int\limits^{\infty}_{-\infty} f'(\lambda)\xi_{\infty}
(\lambda)\, d\lambda,
$$
where $\xi_{\infty}(\lambda)=\frac{1}{\pi}\text{Arg}(\varphi, (A-
\lambda-i0)^{-1}\varphi)$ and $f$ is any function in $C^{\infty}_{0}$
with $f(x)=x$ for $x\in [-\|A\|_{\infty}, \|A\|_{\infty}]$.
\endproclaim
One cannot recover $F(z)$ from $\xi_{\infty}(A)$ without some
additional information. For by (A.$7'$), $\xi_{\infty}$ determines
$\frac{d}{dz}\, \ln\,F(z)$. There is then a constant needed to get
$F$ by integration. However, asymptotics of $F$ at $-\infty$ are
often enough to recover $F$ from $\xi_{\infty}$. This is what is
needed in \S6. For generalizations, see [48].
\proclaim{Theorem A.2} Let $A\geq 0$. Suppose $(-z)^{1/2}F(z)\to 1$
as $z\to -\infty$ along the real axis. Then
$$
F(z)=(-z)^{-1/2}\lim\limits_{\gamma\to\infty}\,\exp\biggl[
\int\limits^{\infty}_{0}\biggl[\frac{\xi_{\infty}(\lambda)-\frac{1}{2}}
{z-\lambda}\biggr]\,\biggl[\frac{\gamma}{\lambda+\gamma}\biggr]\,
d\lambda\biggr].
$$
\endproclaim
\demo{Proof} Let $F^{(0)}(z)=(-z)^{-1/2}$. Then
$$
\frac{d}{dz}\,\ln\, F^{(0)}(z)=\frac{1}{2}\int\limits^{\infty}_{0}
\frac{d\lambda}{(z-\lambda)^{2}}
$$
so by (A.$7'$):
$$
\frac{d}{dz}\,\biggl(\biggl[\frac{F(z)}{F^{(0)}(z)}\biggr]\biggr)=
\int\limits^{\infty}_{0}\frac{[\xi_{\infty}(\lambda)-\frac{1}{2}]}
{(z-\lambda)^{2}}\, d\lambda
$$
hence integrating,
$$
\ln\,\biggl[\frac{F(z)}{F^{(0)}(z)}\biggr] - \ln\,\biggl[\frac{F(-
\gamma)}{F^{(0)}(-\gamma)}\biggr] =\int\limits^{\infty}_{0}\,
\frac{[\xi_{\infty}(\lambda)-\frac{1}{2}]}{(\lambda-z)(\lambda+\gamma)}\,
(\gamma+z)\,d\lambda.
$$
By hypothesis, $\lim\limits_{\gamma\to\infty}\,\ln\,\bigl[\frac{F(-\gamma)}
{F^{(0)}(-\gamma)}\bigr]=0$ and by dominated convergence for any fixed $z$,
\break $\lim\limits_{\gamma\to\infty}\bigl[\int\limits^{\infty}_{0}
\frac{[\xi_{\infty}(\lambda)-\frac{1}{2}]}{(\lambda-z)(\lambda+\gamma)}\,
d\lambda\bigr]=0$, proving the theorem. \qed
\enddemo
As an example of the abstract theory, fix $V$, a continuous function
on $\Bbb R$ which is bounded from below, and $x_{0}\in\Bbb R$. Let
$A=-\frac{d^2}{dx^2}+V$. Let $F:Q(A)\to\Bbb C$ by $F(f)=f(x_{0})$.
By a Sobolev estimate and using $\Cal H_{1}(A)\subset\Cal H_{1}(-
\frac{d^2}{dx^2})$, $F$ is a functional in $\Cal H_{-1}$, so we write
$F(f)=\langle\varphi, f\rangle$ with $\varphi(x)=\delta(x-x_{0})$.
The form domain of $A_{\infty}$ is thus $f\in\Cal H_{1}(A)$ with
$f(x_{0})=0$; thus $A_{\infty}$ is exactly the operator $H_{x_{0}; D}$
with a Dirichlet boundary condition at $x_{0}$ that we discuss in the
body of the paper.
\vskip 0.3in
\example{Acknowledgments} We would like to thank H.~Holden, Y.~Last,
and Z.~Zhao for discussions on this subject. F.G.~is indebted to the
Department of Mathematics at Caltech for its hospitality and support
during the summers of 1992 and 1993 where some of this work was done.
\endexample
\vskip 0.3in
\Refs
\endRefs
\item{[1]} F.V.~Atkinson, {\it On the location of the Weyl circles},
Proc.~Roy.~Soc.~Edinburgh {\bf 88A} (1981), 345--356.
\item{[2]} J.~Avron, B.~Simon, and P.~van Mouche, {\it On the measure
of the spectrum for the almost Mathieu operator},
Commun.~Math.~Physics {\bf 132} (1990), 103--118.
\item{[3]} G.~Borg, {\it Uniqueness theorems in the spectral theory of
$y''+(\lambda-q(x))y=0$}, Proc.~11th Scandinavian Congress of
Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, pp.~276--287.
\item{[4]} V.~Chulaevsky, {\it On perturbations of a Schr\"odinger
operator with periodic potential}, Russian Math.~Surveys {\bf 36:5}
(1981), 143--144.
\item{[5]} W.~Craig, {\it The trace formula for Schr\"odinger
operators on the line}, Commun.~Math. Phys. {\bf 126} (1989), 379-407.
\item{[6]} P.~Deift and B.~Simon, {\it Almost periodic Schr\"odinger
operators, III.~The absolutely continuous spectrum in one dimension},
Commun.~Math.~Phys. {\bf 90} (1983), 389--411.
\item{[7]} P.~Deift and E.~Trubowitz, {\it Inverse scattering on the
line}, Commun.~Pure Appl.~Math. {\bf 32} (1979), 121--251.
\item{[8]} L.A.~Dikii, {\it Trace formulas for Sturm-Liouville
differential operators}, Amer.~Math.~Soc. Trans.~Ser. (2) {\bf 18}
(1961), 81--115.
\item{[9]} B.A.~Dubrovin, {\it Periodic problems for the Korteweg-de
Vries equation in the class of finite band potentials},
Funct.~Anal.~Appl. {\bf 9} (1975), 215--223.
\item{[10]} M.S.P.~Eastham, {\it The Spectral Theory of Periodic
Differential Equations}, Scottish Academic Press, Edinburgh, 1973.
\item{[11]} W.N.~Everitt, {\it On a property of the $m$-coefficient of
a second-order linear differential equation}, J.~London Math.~Soc.
{\bf 4} (1972), 443--457.
\item{[12]} H.~Flaschka, {\it On the inverse problem for Hill's
operator}, Arch.~Rat.~Mech.~Anal. {\bf 59} (1975), 293--309.
\item{[13]} I.M.~Gel'fand, {\it On identities for eigenvalues of a
second order differential operator}, Usp.~Mat.~Nauk {\bf 11:1} (1956),
191--198 (Russian); Engl.~ transl.~in Izrail M.~Gelfand, Collected
Papers Vol.~I (S.G.~Gindikin, V.W.~Guillemin, A.A.~Kirillov,
B.~Kostant, S.~Sternberg, eds.) Springer-Berlin, 1987, pp.~510--517.
\item{[14]} I.M.~Gel'fand and B.M.~Levitan, {\it On the determination
of a differential equation from its spectral function},
Izv.~Akad.~Nauk SSSR {\bf 15} (1951), 309--360 (Russian); Engl.~transl.~in
Amer.~Math.~Soc.~Transl.~Ser. 2, {\bf 1} (1955), 253--304.
\item{[15]} I.M.~Gel'fand and B.M.~Levitan, {\it On a simple identity
for eigenvalues of a second order differential operator},
Dokl.~Akad.~Nauk SSSR {\bf 88} (1953), 593--596 (Russian); English
transl.~in Izrail M.~Gelfand, Collected Papers Vol.~I (S.G.~Gindikin,
V.W.~Guillemin, A.A.~Kirillov, B.~Kostant, S.~Sternberg, eds.)
Springer, Berlin, 1987, pp.~457--461.
\item{[16]} F.~Gesztesy, H.~Holden, and B.~Simon, {\it Absolute
summability of the trace relation for certain Schr\"odinger
operators}, to be submitted to Commun.~Math.~Phys.
\item{[17]} F.~Gesztesy, H.~Holden, B.~Simon, and Z.~Zhao, {\it
Higher order trace relations for Schr\"o-dinger operators}, to be
submitted to Commun.~Pure Appl.~Math.
\item{[18]} F.~Gesztesy, H.~Holden, B.~Simon, and Z.~Zhao, {\it A trace
formula for multidimensional Schr\"odinger operators}, preprint.
\item{[19]} F.~Gesztesy and B.~Simon, {\it Rank one perturbations at
infinite coupling}, to appear in J.~Funct.~Anal.
\item{[20]} F.~Gesztesy and B.~Simon, {\it Inverse spectral theory for
one-dimensional Schr\"odinger operators}, in preparation.
\item{[21]} F.~Gesztesy and B.~Simon, {\it Inverse spectral theory for
one-dimensional Jacobi matrices}, in preparation.
\item{[22]} R.C.~Gilbert and V.A.~Kramer, {\it Trace formulas for powers
of a Sturm-Liouville operator}, Canad.~J.~Math. {\bf 16} (1964),
412--422.
\item{[23]} C.J.A.~Halberg and V.A.~Kramer, {\it A generalization of
the trace concept}, Duke Math.~J. {\bf 27} (1960), 607--617.
\item{[24]} H.~Hochstadt, {\it On the determination of a Hill's
equation from its spectrum}, Arch.~Rat. Mech.~Anal. {\bf 19} (1965),
353--362.
\item{[25]} K.~Iwasaki, {\it Inverse problem for Sturm-Liouville and
Hill equation}, Ann.~Mat.~Pura Appl.~Ser.~4, {\bf 149} (1987),
185--206.
\item{[26]} R.~Johnson and J.~Moser, {\it The rotation number for
almost periodic potentials}, Commun.~Math.~Phys. {\bf 84} (1982),
403--438.
\item{[27]} T.~Kato, {\it Perturbation Theory for Linear Operators},
2nd ed., Springer, Berlin, 1980.
\item{[28]} Y.~Katznelson, {\it An Introduction to Harmonic Analysis},
Dover, New York, 1976.
\item{[29]} O.~Knill and Y.~Last, {\it Spectral properties of Jacobi
matrices arising from twist maps}, in preparation.
\item{[30]} S.~Kotani, {\it Ljapunov indices determine absolutely
continuous spectra of stationary random one-dimensional Schr\"odinger
operators}, Stochastic Analysis (K.~Ito, ed.), North-Holland,
Amsterdam, 1984, pp.~225--247.
\item{[31]} S.~Kotani and M.~Krishna, {\it Almost periodicity of some
random potentials}, J.~Funct.~Anal. {\bf 78} (1988), 390--405.
\item{[32]} M.~G.~Krein, {\it Perturbation determinants and a formula
for the traces of unitary and self-adjoint operators},
Sov.~Math.~Dokl. {\bf 3} (1962), 707--710.
\item{[33]} Y.~Last, {\it A relation between a.c.~spectrum of ergodic
Jacobi matrices and the spectra of periodic approximants},
Commun.~Math.~Phys. {\bf 151} (1993), 183--192.
\item{[34]} B.M.~Levitan, {\it On the closure of the set of
finite-zone potentials}, Math.~USSR Sbornik {\bf 51} (1985), 67--89.
\item{[35]} B.M.~Levitan, {\it Inverse Sturm-Liouville Problems},
VNU Science Press, Utrecht, 1987.
\item{[36]} B.M.~Levitan and M.G.~Gasymov, {\it Determination of a
differential equation by two of its spectra}, Russian Math.~Surveys
{\bf 19:2} (1964), 1--63.
\item{[37]} W.~Magnus and S.~Winkler, {\it Hill's Equation}, Dover,
New York, 1979.
\item{[38]} V.A.~Marchenko, {\it Some questions in the theory of
one-dimensional linear differential operators of the second order,
I.}, Trudy Moskov.~Mat.~Ob\u s\u c. {\bf 1} (1952), 327--420
(Russian); Engl.~transl.~in Amer.~Math.~Soc.~Transl.~Ser. 2, {\bf 101}
(1973), 1--104.
\item{[39]} V.A.~Marchenko, {\it Sturm-Liouville Operators and
Applications}, Birkh\"auser, Basel, 1986.
\item{[40]} V.A.~Marchenko and I.V. Ostrovskii, {\it A
characterization of the spectrum of Hill's operator}, Math.~USSR
Sbornik {\bf 26} (1975), 493--554.
\item{[41]} H.P.~McKean and P.~van Moerbeke, {\it The spectrum of
Hill's equation}, Invent.~Math. {\bf 30} (1975), 217--274.
\item{[42]} H.P.~McKean and E.~Trubowitz, {\it Hill's operator and
hyperelliptic function theory in the presence of infinitely many
branch points}, Commun.~Pure Appl.~Math. {\bf 29} (1976), 143--226.
\item{[43]} M.~Reed and B.~Simon, {\it Methods of Modern Mathematical
Physics, II.~Fourier Analysis, Self-Adjointness}, Academic Press, New
York, 1975.
\item{[44]} M.~Reed and B.~Simon, {\it Methods of Modern Mathematical
Physics, IV.~Analysis of Operators}, Academic Press, New York, 1978.
\item{[45]} B.~Simon, {\it A canonical decomposition for quadratic
forms with applications to monotone convergence theorems},
J.~Funct.~Anal. {\bf 28} (1978), 377--385.
\item{[46]} B.~Simon, {\it Functional Integration and Quantum
Physics}, Academic Press, New York, 1979.
\item{[47]} B.~Simon, {\it Trace Ideals and their Applications},
Cambridge Univ.~Press, Cambridge, 1979.
\item{[48]} B.~Simon, {\it Spectral analysis of rank one perturbations
and applications}, Lecture given at the 1993 Vancouver Summer School,
preprint.
\item{[49]} E.~Trubowitz, {\it The inverse problem for periodic
potentials}, Commun.~Pure Appl.~Math. {\bf 30} (1977), 321--337.
\item{[50]} S.~Venakides, {\it The infinite period limit of the
inverse formalism for periodic potentials}, Commun.~Pure Appl.~Math.
{\bf 41} (1988), 3--17.
\enddocument