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\topmatter
\title Absolute Summability of the Trace Relation \\
for Certain Schr\"odinger Operators
\endtitle
\rightheadtext{Absolute Summability of the Trace Relation}
\author F. Gesztesy$^{1}$, H. Holden$^{2}$, and B. Simon$^{3}$
\endauthor
\leftheadtext{F. Gesztesy, H. Holden, and B. Simon}
\thanks $^{1}$ Department of Mathematics, University of Missouri, Columbia, 
MO 65211. E-mail:mathfg\@mizzou1.\linebreak missouri.edu
\endthanks
\thanks $^{2}$ Department of Mathematical Sciences, The Norwegian 
Institute of Technology, University of Trondheim, N-7034 Trondheim, 
Norway. E-mail: holden\@imf.unit.no
\endthanks
\thanks $^{3}$ 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 Commun.~Math.~Phys.}
\endthanks
\abstract A recently established general trace formula for 
one-dimensional Schr\"odinger operators is systematically studied in 
the context of short-range potentials, potentials which approach 
different spatial asymptotes sufficiently fast, and appropriate impurity 
(defect) interactions in one-dimensional solids.  We prove the 
absolute summability of the trace formula and establish its 
connections with scattering quantities, such as reflection 
coefficients, in each case.
\endabstract
\endtopmatter

\document
\bigpagebreak
\flushpar {\bf \S 1. Introduction}

This paper is the third in a series on a general trace formula and its 
ramifications in (inverse) spectral theory for one-dimensional 
Schr\"odinger operators started in [16] and continued in [19].  The 
main theme in [16] concentrates around a general trace formula for 
self-adjoint Schr\"odinger operators $H$ in $L^{2}(\Bbb R)$ of the 
type
$$
H=-\frac{d^2}{dx^2}+V, \tag 1.1 
$$
where we assume that the real-valued potential $V(x)$ is continuous 
and bounded from below. In order to gain some information on $V(y)$, we 
shall compare $H$ with the associated self-ajoint Dirichlet operator 
$H^{D}_{y}$ obtained from $H$ by imposing an additional Dirichlet 
boundary condition $\lim\limits_{\epsilon\downarrow 0}\psi(y\pm\epsilon)=0$ 
at the point $y\in\Bbb R$.  Since the resolvent difference $[(H^{D}_{y}-z)^
{-1}-(H-z)^{-1}]$ is rank one (cf.~(2.37)), Krein's spectral shift function 
$\xi (\lambda, y)$ [25], [33] for the pair $(H^{D}_{y}, H)$ exists for all 
$y\in\Bbb R$ and a.e.~$\lambda\in\Bbb R$ (with respect to Lebesgue 
measure) and one obtains for all $y\in\Bbb R$
$$
\text{Tr}[f(H^{D}_{y})-f(H)]=\int\limits_{\Bbb R} d\lambda 
f'(\lambda)\xi(\lambda, y), \tag 1.2 
$$
$$
0\leq\xi(\lambda, y)\leq 1\quad\text{a.e.~$\lambda\in\Bbb R$}, \tag 1.3 
$$
$$
\xi (\lambda, y)=0,\quad\lambda<\inf\,\sigma(H) \tag 1.4 
$$
for any $f\in C^{2}(\Bbb R)$ with 
$(1+\lambda^{2})f^{(j)}\in L^{2}((0,\infty))$, $j=1, 2$  
and $f(\lambda)=(\lambda - z)^{-1}$, $z\in\Bbb C\backslash [\inf\,
\sigma(H), \infty)$.
(Here $\sigma(\cdot)$ denotes the spectrum.) A closer look at the 
rank-one resolvent difference of $H^{D}_{y}$ and $H$ reveals the additional 
result that for each $y\in\Bbb R$ and a.e.~$\lambda\in\Bbb R$,
$$
\xi (\lambda, y)=\lim_{\epsilon\downarrow 0} \pi^{-1}\text{Im}
\{\ln [G(\lambda +i\epsilon , y, y)]\}, \tag 1.5 
$$
where $G(z, x, x')$ denotes the Green's function of $H$ (i.e., the 
integral kernel of $(H-z)^{-1}$).  The main results proven in [16], [19] 
then revolve around the following general trace formula.

\proclaim{Theorem 1.1} \rom{[16,19]} Let $V$ be a measurable function on 
$\Bbb R$ satisfying
\roster
\item"\rom{(i)}" $\sup\limits_{n\in\Bbb N}\int\limits^{n+1}_{n}
dx |V_{-}(x)|<\infty$,
\item"\rom{(ii)}" $\int\limits^{n+1}_{n} dx |V_{+}(x)|<\infty$ for all 
$n\in\Bbb N$,
\endroster
where $V_{\pm}(x)=[|V(x)|\pm V(x)]/2$ and suppose $E_{o}\leq\inf\,\sigma(H)$.  
If $x$ is a point of Lebesgue continuity for $V$, then
$$
V(x)=E_{o}+\lim_{\epsilon\downarrow 0} \int\limits^{\infty}_{E_{o}}
d\lambda e^{-\epsilon\lambda}[1-2\xi (\lambda, x)]. \tag 1.6 
$$
\endproclaim

The proof of (1.6) combines (1.2) for $f(\lambda)=e^{-\epsilon\lambda}$, 
$\epsilon >0$ with path integral arguments to control the trace of 
the heat kernel difference as $\epsilon\downarrow 0$.

In the particularly simple case $V(x)\equiv 0$, $G(\lambda, x, 
x)=i/\lambda^{1/2}$, Im$(\lambda^{1/2})\geq 0$ for $\lambda\geq 0$ and 
hence $\xi(\lambda, x)=\cases 1/2, \lambda >0 \\0, \lambda <0\endcases$. 
Further explicit examples can be found in Remark 2.5 in the context of 
reflectionless ($N$-soliton) potentials and in (4.18)--(4.20) in 
connection with periodic potentials.  In fact, historically, after the 
pioneering work by Gel'fand and Levitan [12] on regularized traces for 
Schr\"odinger operators on a compact interval, the trace formula 
(4.19) for periodic (and certain classes of almost periodic) 
potentials was one of the two previously systematically studied trace 
formulae of the type (1.6) for Schr\"odinger operators on the whole 
real line (see, e.g., [8], [11], [22], [30], [34] and more recently [5], 
[24], [27], [28]). The other case studied in detail by Deift and Trubowitz 
[7] in 1979 was concerned with short-range potentials $V(x)$ decaying 
sufficiently fast as $|x|\to\infty$ under the assumption that $H=-\frac
{d^2}{dx^2}+V$ has no eigenvalues.  They proved that
$$
V(x)=\frac{2i}{\pi}\int\limits^{\infty}_{-\infty}dk\,k\,\ln\biggl[
1+R(k)\frac{f_{+}(k, x)}{f_{-}(k, x)}\biggr] \tag 1.7
$$
(where $f_{\pm}(k, x)$ are the Jost functions at energy $E=k^2$ and
$R(k)$ is a reflection coefficient) which is an analog of (1.6). In the 
special case of positive $C^\infty$-potentials of compact support, a 
trace formula of the type
$$
V(x)=\int\limits^{\infty}_{0} d\lambda[1-2\xi(\lambda, x)], \quad 
x\in\Bbb R \tag 1.8 
$$
has recently been established by Venakides [35].  However, the equivalence 
of (1.7) and (1.8) was not established in [35].  Moreover, $\xi(\lambda, x)$ 
was not identified as Krein's spectral shift function for the pair 
$(H^{D}_{x}, H)$ and also the connection (1.5) between $\xi(\lambda, 
x)$ and the Green's function of $H$ was not made.

The analog of the trace formula (1.6) and the associated formalism for 
second-order finite-difference (Jacobi) operators, a summability result 
for operators $H$ with purely discrete spectrum along with a powerful 
new characterization of the absolutely continuous spectrum 
$\sigma_{\text{ac}}(H)$ of $H$ as
$$
\sigma_{\text{ac}}(H)=\overline{\{\lambda\in\Bbb R\mid 0<\xi(\lambda, 
x)<1\}}^{\text{ ess}} \tag 1.9 
$$
for each fixed $x\in\Bbb R$ and some of its applications to the almost 
Mathieu equation or Harper's model (here $^{\text{---ess}}$ denotes the 
essential closure) are presented in the first paper [16] of our series.

The case of general self-adjoint boundary conditions of the type 
$\psi'(x)+\beta\psi(x)=0$, $\beta\in\Bbb R\cup\{\infty\}$ together with 
trace formulas for all higher-order KdV invariants (expressed as 
differential polynomials in $V$) are studied in detail in the second 
paper [19] of our series.

In the present third paper of our series, we shall give a systematic 
study of short-range perturbations in which the regularization 
(Abelian limit) $\epsilon\downarrow 0$ in (1.6) can be removed and 
prove the absolute summability of the trace formula (1.6).  
Specifically, we shall study the following three situations:
\roster
\item"\rom{(i)}" Sufficiently short-range potentials with certain 
regularity properties (typically, $V\in H^{2,1}(\Bbb R)$, see (2.1)) 
in \S 2.
\item"\rom{(ii)}" Potentials which tend to different asymptotes as 
$x\to\pm\infty$ sufficiently fast in \S 3.
\item"\rom{(iii)}"  Impurity (defect) scattering in connection with 
potentials of the type $V=V^{o}+W$, where $V^{o}(x+a)=V^{o}(x)$ for 
some $a>0$ represents the periodic background and the short-range 
perturbation $W$ models impurities (defects) in a one-dimensional 
crystal, are treated in \S 4.
\endroster

In each of these three situations, we establish the connection between 
$\xi(\lambda, x)$ and appropriate scattering quantities, such as 
reflection coefficients and Jost functions, and prove the absolute 
summability of the trace formula (1.6),
$$
\int\limits^{\infty}_{R}d\lambda |1-2\xi(\lambda, x)|<\infty, 
\quad R\in\Bbb R \tag 1.10 
$$
removing the Abelian limit $\epsilon\downarrow 0$ in (1.6).

It should be pointed out at this occasion that the Abelian limit 
$\epsilon\downarrow 0$ in (1.6) cannot be removed in general if 
$V(x)\to\infty$ as $x\to\infty$ or $x\to -\infty$ irrespective of the 
regularity properties of $V(x)$. This is particularly plain in the 
case where $V(x)\to\infty$ as $x\to\pm\infty$ since then for each 
$x\in\Bbb R$, $|1-2\xi(\lambda, x)|=1$ for a.e.~$\lambda\in\Bbb R$.  
But even if $V(x)$ tends to a constant sufficiently fast as $x\to -
\infty$ and $V(x)\operatornamewithlimits{\longrightarrow}
\limits_{x\to\infty}\infty$, explicit examples (such as, e.g., 
$V(x)=e^x$) in [26] show that $[1-2\xi(., x)]\notin L^{1}
((R, \infty); d\lambda), R, x\in\Bbb R$. In these situations, the 
Abelian limit $\epsilon\downarrow 0$ in (1.6) represents a genuine 
summability method.

The fourth paper [17] in our series is devoted to various multidimensional
trace formulas in terms of heat kernel asymptotics.  A brief announcement of 
our results appeared in [18], expository accounts of this circle of ideas 
can be found in [14], [33].  Papers exploring several new solutions of
inverse spectral problems are in preparation.

\bigpagebreak
\flushpar {\bf \S 2. Short-range Potentials}

In this section we illustrate the trace formula (1.6) in the 
particular case of short-range potentials satisfying
$$
V\in H^{2,1}(\Bbb R), \quad\text{$V$ real-valued.} \tag 2.1 
$$
Here $H^{m, p}(\Bbb R)$, $m, p\in\Bbb N$ denotes the usual Sobolev 
space whose elements have up to $m$ distributional derivatives in 
$L^{p}(\Bbb R)$.  The regularity condition on $V$ in (2.1) is 
essential in connection with our main result in Theorem 2.3, the 
removal of a regularization procedure (Abelian limit) in our trace 
formula (1.6) (cf. (2.39)--(2.41)).

The associated self-adjoint Schr\"odinger operator $H$ in $L^{2}(\Bbb 
R)$ is then defined by
$$
H=-\frac {d^2}{dx^2}+V, \quad \Cal D(H)=H^{2, 2}(\Bbb R) \tag 2.2 
$$
and the spectrum $\sigma (H)$ of $H$ is of the type 
$$
\sigma (H)=\sigma_{d}(H)\cup [0, \infty), \quad 
\sigma_{\text{ess}}(H)=[0, \infty). \tag 2.3 
$$
Here $\sigma_{\text{ess}}(H)$ is the essential spectrum of $H$ and the 
discrete spectrum $\sigma_{d}(H)$ of $H$ is a 
bounded subset of $(-\infty, 0)$ which may be empty, finite, or 
countably infinite.  We denote the latter by
$$
\sigma_{d}(H)=\{e_{j}\}_{j\in J} \ ,\quad e_{j}<e_{j+1} \ ,  \tag 2.4 
$$
where
$$
J=\cases
\emptyset \\ \{0,1,2,\dots,N\} \\ \Bbb N_{0}=\Bbb N\cup\{0\}
\endcases  \tag 2.5 
$$
is an appropriate index set.  For later purposes we will also need
the notation
$$
J_{+}=\cases
\emptyset \\ \{1,2,\dots, N, N+1\} \ ,\quad e_{N+1}=0 \\
\Bbb N \endcases  \tag 2.6 
$$
depending on whether $J$ is empty, finite, or infinite.  We also 
remark that each eigenvalue of $H$ is simple, $H$ has no eigenvalues 
embedded into $(0, \infty)$, and the spectrum of $H$ in $(0, \infty)$ 
is purely absolutely continuous and of uniform multiplicity two under 
hypothesis (2.1).  It should perhaps be noted that the weak 
falloff condition on $V(x)$ as $|x|\to\infty$ in (2.1), in principle, 
admits situations where zero is a (necessarily simple) eigenvalue of 
$H$ though these cases can easily be excluded by adding the assumption 
$V\in L^{1}(\Bbb R; (1+|x|)dx)$ in (2.1).  For details in connection 
with these spectral properties of $H$, see, for example, [4], [6], 
[7], [13], and the references therein.

In addition to $H$ we also need to introduce the closely associated 
self-adjoint Dirichlet operator $H^{D}_{y}$ in $L^{2}(\Bbb R)$ defined 
by
$$
H^{D}_{y}=-\frac {d^2}{dx^2}+V, \ \Cal D(H^{D}_{y})=
\left\{g\in H^{1, 2}(\Bbb R)\cap H^{2, 2}(\Bbb R\backslash \{y\})\mid
g(y)=0\right\}, \quad y\in\Bbb R. \tag 2.7 
$$
The spectrum of $H^{D}_{y}$ is then given by
$$
\sigma(H^{D}_{y})=\sigma_{d}(H^{D}_{y})\cup [0, \infty), \quad 
\sigma_{\text{ess}}(H^{D}_{y})=[0, \infty), \tag 2.8 
$$
where
$$
\gathered
\sigma_{d}(H^{D}_{y})=\{\mu_{j}(y)\}_{j\in J_{+}}\cap (e_{0}, 0), \\
e_{0}<\mu_{1}(y)\leq e_{1}, \quad e_{j-1}\leq\mu_{j}(y)\leq 
e_{j}, \quad j\in J_{+}\backslash \{1\}. 
\endgathered \tag 2.9 
$$
(In the special case where $J_{+}=\{1, 2,\dots, N+1\}$ is finite and 
$\mu_{N+1}(y)=e_{N+1}=0$, our notation in (2.9) indicates that 
$\mu_{N+1}(y)=0$ is not a discrete Dirichlet eigenvalue though it may 
be a (non-isolated) eigenvalue of $H^{D}_{y}$ due to our weak falloff 
conditions on $V$ as $|x|\to\infty$.) In particular,
$$
H^{D}_{y}\geq H. \tag 2.10 
$$
Since the resolvent of $H^{D}_{y}$ is a rank-one perturbation of that 
of $H$ (see (2.37)) and
$$
H^{D}_{y}=H^{D}_{-, y}\oplus H^{D}_{+, y}, \tag 2.11 
$$
where $H^{D}_{\pm, y}$ denote the corresponding half-line Dirichlet 
operators in $L^{2}((y, \pm\infty))$, the spectrum of $H^{D}_{y}$ in
$(0, \infty)$ is purely absolutely continuous and of uniform multiplicity 
two.  The discrete spectrum $\sigma_{d}(H^{D}_{y})$ however, in contrast 
to that of $H$, is not necessarily simple.  More precisely, $\mu_{j}(y)$ 
is a simple eigenvalue of $H^{D}_{y}$ if and only if $e_{j-1}<\mu_{j}(y)
<e_{j}$.  In this case, $\mu_{j}(y)$ is a (simple) eigenvalue of 
either $H^{D}_{-, y}$ or $H^{D}_{+, y}$, but not of both. Whenever $\mu_{j}(y)
\in\{e_{j-1}, e_{j}\}$ (possibly excluding the case $\mu_{N+1}(y)=0$ 
as explained after (2.9)), the multiplicity of $\mu_{j}(y)$ is two and 
$\mu_{j}(y)$ is a (simple) eigenvalue of both $H^{D}_{-, y}$ and 
$H^{D}_{+, y}$.

As a final preparation for the main results of this section, we briefly 
recall a few basic formulas in connection with scattering theory for the 
pair $(H, H_o)$, where $H_{o}=-\frac{d^2}{dx^2}$, $\Cal 
D(H_{o})=H^{2, 2}(\Bbb R)$.  Details can be found, for example, in [2]--[4], 
[7], [13], [28], Ch.~6, [29], Sect.~3.5, and the references therein.
The Jost solutions $f_{\pm}(z, x)$ of $H$ are defined by
$$
\gathered 
f_{\pm}(z, x)=e^{\pm iz^{1/2}x} -\int\limits^{\pm\infty}_{x} 
dx'z^{-1/2}\sin[z^{1/2}(x-x')]V(x')f_{\pm}(z, x'), \\
z\in\Bbb C\backslash \{0\}, \text{Im}(z^{1/2})\geq 0, x\in\Bbb R 
\endgathered \tag 2.12 
$$
such that
$$
Hf_{\pm}(z, x)=zf_{\pm}(z, x), \quad z\in\Bbb C\backslash\{0\} \tag 2.13 
$$
in the distributional sense.  The unitary scattering matrix 
$S(\lambda)$, $\lambda> 0$ in $\Bbb C^2$ associated with the pair $(H, 
H_o)$ then explicitly reads in terms of transmission and reflection 
coefficients from left and right incidence
$$
S(\lambda)=\pmatrix T(\lambda) & R^{r}(\lambda) \\
R^{\ell}(\lambda) & T(\lambda)
\endpmatrix , \quad \lambda >0, \tag 2.14 
$$
where
$$
T(\lambda)=\frac {2i\lambda^{1/2}}{W(f_{-}(\lambda), 
f_{+}(\lambda))} = \left[ 1-\frac 
{1}{2i\lambda^{1/2}}\int\limits_{\Bbb R}dxV(x)e^{\pm i\lambda^{1/2}x}
f_{\mp}(\lambda, x)\right]^{-1}, \tag 2.15
$$
$$
R^{\ell}(\lambda)=-\frac {W(\overline{f_{-}(\lambda)}, 
f_{+}(\lambda))}{W(f_{-}(\lambda), f_{+}(\lambda))} =\frac 
{T(\lambda)}{2i\lambda^{1/2}} \int\limits_{\Bbb R} 
dxV(x)e^{i\lambda^{1/2}x}f_{+}(\lambda, x), \tag 2.16 
$$
$$
R^{r}(\lambda)=-\frac {W(f_{-}(\lambda), \overline{f_{+}(\lambda)})}
{W(f_{-}(\lambda), f_{+}(\lambda))} = \frac 
{T(\lambda)}{2i\lambda^{1/2}} \int\limits_{\Bbb R} dxV(x)
e^{-i\lambda^{1/2}x}f_{-}(\lambda, x), \tag 2.17 
$$
where
$$
f_{\pm}(\lambda, x)=\lim_{\epsilon\downarrow 0} f_{\pm}(\lambda 
+i\epsilon, x), \quad \lambda > 0 \tag 2.18 
$$
and $W(f, g)(x)=f(x)g'(x)-f'(x)g(x)$ denotes the Wronskian of $f$ and 
$g$.  In addition, we recall that
$$
T(\lambda)f_{\pm}(\lambda, x)=R\Sp \ell \\ r\endSp(\lambda)f_{\mp}
(\lambda, x) + \overline{f_{\mp}(\lambda, x)}, \quad \lambda >0 \tag 2.19
$$
and that the Green's function of $H$ (the integral kernel of $(H-z)^
{-1}$) is given by
$$
G(z, x, x')=\frac {f_{+}(z, x)f_{-}(z, x')}{W(f_{+}(z), f_{-}(z))},
\quad x'\leq x, z\in\Bbb C\backslash\{0\}, \text{Im}(z^{1/2})\geq 0.
\tag 2.20
$$
Specializing to $x=x', G(z, x, x)$ is well known to be a Herglotz 
function in $z\in\Bbb C_{+}=\mathbreak\{z\in\Bbb C\mid\text{Im}(z)>0\}$ for all 
$x\in\Bbb R$, that is, $G(z, x, x)$ is analytic in $\Bbb C_{+}$ and 
$$
\text{Im}[G(z, x, x)]>0,\quad\overline{G(z, x, x)}=G(\bar{z}, x, x), 
\quad z\in\Bbb C_{+}, x\in\Bbb R. \tag 2.21
$$
As a consequence (see, e.g., [1] or, for a different approach, [16], 
[33]) $G(z, x, x)$ admits the exponential representation
$$
G(z, x, x)=\exp\left[c+\int\limits_{\Bbb R}\left[\frac{1}{\lambda-z}
-\frac{\lambda}{1+\lambda^2}\right]\, \xi(\lambda, x)d\lambda\right],
\quad z\in\Bbb C_{+}, x\in\Bbb R, \tag 2.22
$$
where
$$
c\in\Bbb R,\quad 0\leq\xi(\lambda, x)\leq 1 \quad\text{for 
a.e.~$\lambda\in\Bbb R$}. \tag 2.23
$$
Fatou's lemma then implies that
$$
\xi(\lambda, x)=\lim_{\epsilon\downarrow 0}\pi^{-1}
\text{Im}\{\ln[G(\lambda +i\epsilon, x, x)]\} \tag 2.24
$$
exists for all $x\in\Bbb R$ and a.e.~$\lambda\in\Bbb R$.  The 
normalization
$$
\xi(\lambda, x)=0,\quad\lambda <E_{o}=\inf\,\sigma(H) \tag 2.25
$$
is then consistent with (2.23), (2.10) and the fact that
$$
G(\lambda +i0, x, x)>0, \quad\lambda<E_{o}. \tag 2.26
$$
As pointed out in Theorem 1.1 of the introduction, the general trace 
formula for $V\in C(\Bbb R)$, $V$ bounded from below, proven in [16] 
then reads
$$
V(x)=E_{o}+\lim_{\epsilon\downarrow 0}\int\limits^{\infty}_{E_{o}} 
d\lambda e^{-\epsilon\lambda}[1-2\xi(\lambda, x)], \quad x\in\Bbb R.
\tag 2.27
$$
Before we discuss how to remove the Abelian limit in (2.27) in the 
present short-range case, it should be pointed out that $\xi(\lambda, 
x)$ is Krein's spectral shift function [25] for the pair $(H^{D}_{x}, 
H)$.  In particular, for $\lambda\in\sigma_{\text{ac}}(H)^{o}$ ($A^o$ 
denotes the interior of a subset $A\subset\Bbb R$), $\xi(\lambda, x)$ 
is essentially the scattering phase shift for the pair $(H^{D}_{x}, H)$ 
since one verifies that
$$
\det\left[S(\lambda, H^{D}_{x}, H)\right]=\frac 
{\overline{G(\lambda + i0, x, x)}}{G(\lambda +i0, x, x)}=
e^{-2\pi i\xi(\lambda, x)}, \quad \lambda\in\sigma_{\text{ac}}(H)^{o},
\tag 2.28
$$
where $S(\lambda, H^{D}_{x}, H)$ denotes the unitary scattering matrix 
in $\Bbb C^2$ for the pair $(H^{D}_{x}, H)$.

We start our analysis with the following lemma.

\proclaim{Lemma 2.1}  Suppose $V\in H^{2, 1}(\Bbb R)$ is real-valued.  
Then for all $x\in\Bbb R$,
$$
\xi(\lambda, x)=0, \quad \lambda <E_{o}=\inf\,\sigma (H), \tag 2.29
$$
$$
\xi(\lambda, x)=\frac12 +\pi^{-1}\text{\rom{Im}}
\left\{\ln\left[1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}
(\lambda, x)^{2}}{|f_{\pm}(\lambda, x)|^{2}}\right]\right\}, 
\quad \lambda >0. \tag 2.30
$$
In particular, $\xi(\lambda, x)$ is continuous for $\lambda>0$. 
Moreover,
$$
|1-2\xi(\lambda, x)|\leq |R^{r(\ell)}(\lambda)|, \quad \lambda >0
\tag 2.31
$$
and
$$
[1-2\xi(\lambda, x)]\operatornamewithlimits{=}_{\lambda\to\infty}
o(\lambda^{-3/2}) \tag 2.32
$$
uniformly with respect to $x\in\Bbb R$.  In addition,
$$\align
&\xi(\lambda, x)=\cases 0, &\text{\rom{if} $\lambda<e_{0}$\rom{ or if} 
$\mu_{j}(x)<\lambda <e_{j}$}\\
1, &\text{\rom{if} $e_{j-1}<\lambda$}\endcases\quad 
\text{\rom{if} $\mu_{j}(x)\in (e_{j-1}, e_{j})$}, \tag 2.33 \\
&\xi(\lambda, x)= \cases 0, &\text{\rom{if} $e_{j-1}<\lambda <e_{j}$
\rom{and} $\mu_{j}(x)=e_{j}$} \\
1, &\text{\rom{if} $e_{j-1}<\lambda <e_{j}$ \rom{and} $\mu_{j}(x)=e_{j-1}$}
\endcases \tag 2.34
\endalign
$$
whenever $\sigma_{d}(H)\neq \emptyset$.
\endproclaim

\demo{Proof}  Equation (2.30) follows from (2.24) and
$$\aligned
G(\lambda +i0, x, x) &=-\frac{T(\lambda)}{2i\lambda^{1/2}}
f_{+}(\lambda, x)f_{-}(\lambda, x) \\
&=(i/2\lambda^{1/2})
|f_{\pm}(\lambda, x)|^{2}\left[1+R\Sp r\\ \ell\endSp(\lambda)
\frac{f_{\pm}(\lambda, x)^{2}}{|f_{\pm}(\lambda, x)|^{2}}\right], 
\quad \lambda > 0 \endaligned
\tag 2.35
$$
which in turn is implied by (2.15), (2.18)--(2.20). Continuity of 
$\xi(\lambda, x)$ for $\lambda >0$ follows from the fact that 
$G(\lambda +i0, x, x)$ is continuous and zero-free for $\lambda\in (0, 
\infty)$.  Inequality (2.31) follows from (2.30) and an elementary 
geometrical argument.  (In fact, $|\arg(1+Re^{i\varphi})|\leq \arcsin 
(|R|)\leq\pi/2$ for $|R|\leq 1, \varphi\in\Bbb R$ and $\sin(x)\geq 
2x/\pi$ for $0\leq x\leq\pi/2$ imply $|\arg(1+Re^{i\varphi})|\leq\pi
|R|/2$.)  Relation (2.32) is then implied by (2.31) and
$$
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty}
o(\lambda^{-3/2}) \tag 2.36
$$
which is a consequence of (2.15)--(2.17) and two integrations by parts 
applying the \linebreak Riemann-Lebesgue lemma.  Relation (2.33) 
directly follows from (2.24) and the fact that $G(z, x, x)$ is 
real-valued for $z<0$ with zeros precisely at the Dirichlet eigenvalues 
$\mu_{j}(x)$ of $H^{D}_{x}$ since
$$
(H^{D}_{x}-z)^{-1}=(H-z)^{-1}-G(z, x, x)^{-1}(\overline{G(z, x, .)}, .)
G(z, ., x), \, z\in\Bbb C\backslash\{\sigma(H^{D}_{x})\cup\sigma(H)\}. 
\tag 2.37
$$
Here $(., .)$ denotes the scalar product in $L^{2}(\Bbb R)$.  Relation 
(2.34) then follows from (2.33) by a continuity argument. \qed
\enddemo

\remark{Remark \rom{2.2}} (i)  Inequality (2.31) holds for any real-valued 
potential satisfying $V\in L^{1}(\Bbb R)$ and hence 
$$
[1-2\xi(., x)]\in L^{1}((0, \infty); d\lambda) \ \text{if} \ R^{r(\ell)}
\in L^{1}((0, \infty); d\lambda). \tag 2.38
$$
In particular, $R^{r(\ell)}\in L^{1}((0, \infty); d\lambda)$ 
together with (2.31) are all that's needed to remove the Abelian limit 
in (2.27) (see Theorem 2.3).  We also note that (2.36) (and hence (2.32)) 
holds if $V'$ is merely piecewise absolutely continuous admitting 
finitely-many jump discontinuities (i.e., there exists a finite partition 
of $\Bbb R, -\infty =x_{0}<x_{1}<\cdots <x_{M}<x_{M+1}=\infty$ such that 
$V'$ is (locally) absolutely continuous on each interval $(x_{m}, x_{m+1})$, 
$0\leq m\leq M$).

(ii)  Obviously, (2.33) and (2.34), in contrast to (2.30), are 
generally valid in spectral gaps of $H$ and by no means linked to the 
short-range nature of $V$ subject to (2.1).  In particular, (2.33) is 
a general property of Krein's spectral function as long as $e_j$ and 
$\mu_{j}(x)$ are simple eigenvalues of $H$ and $H^{D}_{x}$, 
respectively.
\endremark

Given Lemma 2.1, we can now remove the Abelian limit 
$\epsilon\downarrow 0$ in the trace formula (2.27) for $V(x)$ and 
state the principal result of this section.  (We recall our notational 
conventions in (2.4)--(2.6), (2.8), (2.9), and the paragraph following 
(2.9).)

\proclaim{Theorem 2.3}  Suppose $V\in H^{2, 1}(\Bbb R)$ is real-valued 
and denote $E_{o}=\inf\,\sigma(H)$.  Then $[1-2\xi(., x)]\in L^{1} 
((E_{o}, \infty); d\lambda), x\in\Bbb R$ and 
$$\align 
V(x) &= E_{o}+\int\limits^{\infty}_{E_o}d\lambda[1-2\xi(\lambda, x)]  
\tag 2.39\\
&= 2\{ e_{0}+\sum_{j\in J_{+}} [e_{j}-\mu_{j}(x)]\}+
\int\limits^{\infty}_{0}d\lambda [1-2\xi(\lambda, x)] 
\tag 2.40\\
&= 2\{ e_{0}+\sum_{j\in J_{+}} [e_{j}-\mu_{j}(x)]\} 
-(2/\pi)\int\limits^{\infty}_{0} d\lambda\text{\rom{ Im}}\left\{ 
\ln\left[1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}
{|f_{\pm}(\lambda, x)|^{2}}\right] \right\}, \, x\in\Bbb R.  \tag 2.41
\endalign
$$
If $\sigma_{d}(H)=\emptyset$, the discrete spectrum part $2\{\dots\}$ 
in \rom{(2.40)} and \rom{(2.41)} is to be deleted.
\endproclaim

\demo{Proof}  The trace formula (2.39) follows from (2.27), (2.31), 
and (2.32) applying the \linebreak Lebesgue dominated convergence theorem.  
Equalities (2.40) and (2.41) are then obvious from (2.39), (2.30), 
(2.33), and (2.34). \qed
\enddemo

\remark{Remark \rom{2.4}} Formula (2.41), in the special case 
$\sigma_{d}(H)=\emptyset$, is due to Deift and Trubowitz [7].  Formula
(2.39), on the other hand, again in the special case 
$\sigma_{p}(H)=\emptyset$ (more precisly, for $0\leq V\in 
C^{\infty}_{0}(\Bbb R)$), appeared in a paper by Venakides [35].  
However, the connection (2.24) between $\xi(\lambda, x)$ and the 
Green's function $G(z, x, x)$ of $H$, and hence the connection between 
(2.39) and the earlier result (2.41) in [7], was not established in 
[35]. Moreover, $\xi(\lambda, x)$ was not identified as Krein's 
spectral shift function for the pair $(H^{D}_{x}, H)$ in [35].
\endremark

\remark{Remark \rom{2.5}}  It seems worthwhile to point out the particularly 
simple step function-like structure of $\xi(\lambda, x)$ in connection 
with reflectionless potentials characterized by 
$R^{r(\ell)}(\lambda)\equiv 0, \lambda >0$.  In this case 
$\xi(\lambda, x)$ is given by (2.33), (2.34) for $\lambda <0$ and by
$$
\xi(\lambda, x)=\frac12 , \quad \lambda >0 \tag 2.42
$$
on the (interior of the) absolutely continuous spectrum of $H$.  This 
applies, in particular, to all $N$-soliton potentials (including 
$V\equiv 0$) and to a class of $\infty$-soliton potentials (having 
infinitely-many negative eigenvalues accumulating at zero) introduced 
in [20], [21].
\endremark

We conclude this section with a few remarks on the low-energy behavior 
of $\xi(\lambda, x)$ as $\lambda\downarrow 0$.  Assuming, in addition 
to (2.1), that $V$ satisfies
$$
V\in L^{1}(\Bbb R; (1+x^{2})dx), \tag 2.43
$$
we need to consider the following case distinctions:
\example{Case I}  $W(f_{-}(0), f_{+}(0))\neq 0$ and $f_{-}(0, x) f_{+}(0, 
x)\neq 0.$
\endexample
\flushpar (The first requirement can be expressed as 
$\int\limits_{\Bbb R} dx V(x)f_{\pm}(0, x)\neq 0$ and is equivalent to 
the fact that $H$ has no threshold resonance; see, e.g. [2], [3].  The 
second requirement says $\lim\limits_{x'\to x}\mu_{N+1}(x')\neq 0$.)  
Then
$$
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow 0}
-1+O(\lambda^{1/2}), \tag 2.44
$$
$$T(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow 0} 
\frac{-2i\lambda^{1/2}}{\int\limits_{\Bbb R}dx'V(x')f_{\mp}(0, x')}
[1+O(\lambda^{1/2})], \tag 2.45
$$
$$
G(\lambda +i0, x, x)\operatornamewithlimits{=}_{\lambda\downarrow 0}
\frac{-f_{+}(0, x)f_{-}(0, x)}{\int\limits_{\Bbb R}dx'V(x')f_{\pm}(0, 
x')} +O(\lambda^{1/2}) \tag 2.46
$$
and hence
$$
\xi(\lambda, x)=\pi^{-1}\arg[G(\lambda+i0, x, x)]
\operatornamewithlimits{=}_{\lambda\downarrow 0} O(\lambda^{1/2}) \
\text{in case I} \tag 2.47
$$
since $f_{\pm}(0, x)$ are real-valued.

\example{Case II} $W(f_{-}(0), f_{+}(0))=0$ and $f_{-}(0, x)f_{+}(0, 
x)\neq 0$.
\endexample
\flushpar (The first requirement can be written as $\int\limits_{\Bbb 
R} dxV(x)f_{\pm}(0, x)=0$ and is equivalent to the fact that $H$ has a 
threshold resonance, see, e.g., [2], [3].)  Then [2], [3]
$$
R\Sp r\\ \ell\endSp(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow 0}
\mp \frac{2c_{1}\overline{c_{2}}}{|c_{1}|^{2}+|c_{2}|^{2}}
+O(\lambda^{1/2}), \tag 2.48
$$
$$
T(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow 0}
\frac{|c_{1}|^{2}-|c_{2}|^{2}}{|c_{1}|^{2}+|c_{2}|^{2}} 
+O(\lambda^{1/2}), \tag 2.49
$$
with $|c_{1}|\neq |c_{2}|$ and hence $R^{r(\ell)}(0)\neq -1$.  Thus
$$
G(\lambda+i0, x, x)\operatornamewithlimits{=}_{\lambda\downarrow 0}
(i/2\lambda^{1/2})|f_{\pm}(0, x)|^{2}[1+R\Sp r\\ \ell\endSp(0)]
+O(1) \tag 2.50
$$
and hence
$$
\xi(\lambda, x)=\pi^{-1}\arg[G(\lambda+i0, x, x)]
\operatornamewithlimits{=}_{\lambda\downarrow 0} \frac12 
+O(\lambda^{1/2}) \ \text {in case II}. \tag 2.51
$$
We emphasize that $V\equiv 0$ and more generally, all $N$-soliton 
potentials $V_N$ mentioned in Remark 2.5 have a zero-energy resonance 
and hence belong to case II.

The case where $f_{-}(0, x)f_{+}(0, x)=0$ can be dealt with 
analogously, but requires higher-order computations.

\bigpagebreak
\flushpar {\bf \S 3. Cascades}

As the the title of this section suggests, we shall now indicate how 
to extend the results of \S 2 to potentials with non-trivial spatial 
asymptotics. More precisely, we shall assume that $V$ satisfies 
$$\gathered
V, V'\in AC_{\text{loc}}(\Bbb R), \ V \text{real-valued},\ V', V''\in 
L^{1}(\Bbb R),\\ \int\limits^{0}_{-\infty}dx|V(x)|+
\int\limits^{\infty}_{0}dx|V(x)-V_{+}|<\infty \ \text{for some 
$V_{+}>0$}
\endgathered \tag 3.1
$$
throughout the major part of this section.  (By reflection, $x\to -x$, 
it suffices to consider $V_{+}>0$.)

Since most of the details will be similar to those in the previous 
section, we shall mostly refer to \S2 for notations and basic facts 
and dwell only on situations markedly different in the present context 
of (3.1).

Introducing $H$, $H^{D}_{y}$, $J$, $J_{+}$, etc.~as in \S 2, the 
absolutely continuous spectrum of $H$ and $H^{D}_{y}$ now equals $[0, 
\infty)$ with uniform spectral multiplicity one on $(0, V_{+})$ and 
two on $(V_{+}, \infty)$.  While $H$ has no embedded eigenvalues in 
$(0, \infty)$, $H^{D}_{y}$ may have (countably infinitely-many) 
eigenvalues in $[0, V_{+}]$ as briefly discussed in Remark 3.4.

Concerning the stationary scattering theory for $H$, one has to 
replace the Jost solutions (2.12) by
$$\gathered
f_{\pm}(z, x)=e^{\pm ik_{\pm}x}-\int\limits^{\pm\infty}_{x}
dx'k^{-1}_{\pm}\sin[k_{\pm}(x-x')][V(x')-V_{\pm}]f_{\pm}(z, x'), \\
k_{+}(z)=(z-V_{+})^{1/2}, \ k_{-}(z)=z^{1/2}, \ V_{-}=0, \ \text{Im}
[k_{\pm}(z)]\geq 0, \ z\in\Bbb C\backslash\{0, V_{+}\}, \ x\in\Bbb R
\endgathered \tag 3.2
$$
to obtain
$$
Hf_{\pm}(z, x)=zf_{\pm}(z, x), \quad z\in\Bbb C\backslash\{0, V_{+}\}
\tag 3.3
$$
in the distributional sense.  The unitary scattering matrix 
$S(\lambda)$, $\lambda >0$ in $\Bbb C$ resp. $\Bbb C^2$ now reads as 
follows:
$$
S(\lambda)=R^{\ell}(\lambda)=-\frac{\overline{W(f_{-}(\lambda),
f_{+}(\lambda))}}{W(f_{-}(\lambda), f_{+}(\lambda))}, \quad 0<\lambda <V_{+}
\tag 3.4
$$
(note that $S(\lambda)$ is unimodular in this case since 
$f_{+}(\lambda, x)$ is real-valued for $0<\lambda < V_{+}$) and
$$
S(\lambda)= \pmatrix T(\lambda) & R^{r}(\lambda) \\
R^{\ell}(\lambda) & T(\lambda) \endpmatrix , \quad \lambda > V_{+},
\tag 3.5
$$
where
$$
T(\lambda)=\frac{2i[k_{+}(\lambda)k_{-}(\lambda)]^{1/2}}
{W(f_{-}(\lambda), f_{+}(\lambda))}, \tag 3.6
$$
$$
R^{\ell}(\lambda)=-\frac{W(\overline{f_{-}(\lambda)}, f_{+}(\lambda))}
{W(f_{-}(\lambda), f_{+}(\lambda))}, \tag 3.7
$$
$$
R^{r}(\lambda)=-\frac{W(f_{-}(\lambda), \overline{f_{+}(\lambda)})}
{W(f_{-}(\lambda), f_{+}(\lambda))}. \tag 3.8
$$
Here $f_{\pm}(\lambda, x)=\lim\limits_{\epsilon\downarrow 0} 
f_{\pm}(\lambda+i\epsilon, x)$ and the Green's function $G(z, x, x)$ 
of $H$ now satisfies
$$\gathered
G(\lambda +i0, x, x)=\frac{f_{+}(\lambda, x)f_{-}(\lambda, x)}
{W(f_{+}(\lambda), f_{-}(\lambda))} \\
=[i/2k_{\pm}(\lambda)]|f_{\pm}(\lambda, x)|^{2}\left[1+R\Sp r\\ \ell\endSp
(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}{|f_{\pm}(\lambda, 
x)|^{2}}\right], \quad \lambda > \cases V_{+} \\ 0 \endcases , \quad
x\in\Bbb R.
\endgathered \tag 3.9
$$
The above expression for $G(\lambda+i0, x, x)$ involving 
$R^{r}(\lambda)$ (as opposed to that involving $R^{\ell}(\lambda)$) 
appears to be singular at $\lambda = V_+$ since $k_{+}(\lambda)^{-
1}=(\lambda-V_{+})^{-1/2}$ (whereas $k_{-}(V_{+})^{-1}=V^{-1/2}_{+}$).  
However, this apparent contradiction is easily resolved by observing 
that $R^{r}(\lambda)\operatornamewithlimits{=}\limits_{\lambda\downarrow 
V_{+}}-1+o(1)$ (see also (3.33) for more details).

\remark{Remark \rom{3.1}} Scattering theory for potentials with 
different spatial asymptotics has been studied in detail, for example, in 
[4], [6], [13], and we have freely used these results in (3.2)--(3.9).  
That $S(\lambda)$ for $0,\lambda <V_{+}$ is unimodular in (3.4) 
illustrates the fact that total reflection occurs from left incidence 
in this energy regime as explored in detail in [6] (see also [13]).
\endremark

Since relations (2.20)--(2.27) are independent of the short-range 
nature of $V$, they apply in the present case.  In particular, the 
definition of $\xi(\lambda, x)$ in (2.24) and the trace formula (2.27) 
remain valid.  Similarly, Lemma 2.1 extends to potentials subject to 
the hypothesis (3.1) with only one minor change.

\proclaim{Lemma 3.2}  Suppose $V$ satisfies the conditions 
\rom{(3.1)}. Then \rom{(2.29)}, \rom{(2.31)--(2.34)} are valid in the 
present case. Equation \rom{(2.30)} turns into 
$$
\xi(\lambda, x)=\frac12+\pi^{-1}\text{\rom{Im}}\left\{ \ln\left[
1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}
{|f_{\pm}(\lambda, x)|^{2}}\right]\right\}, \ 
\lambda > \cases V_{+} \\  0 \endcases \tag 3.10
$$
and $\xi(\lambda, x)$ is continuous in $\lambda >V_{+}$.
\endproclaim

\demo{Proof}  Except for the analog of (2.32), which follows again 
from 
$$
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty}
o(\lambda^{-3/2}), \tag 3.11
$$
everything else is proven as in Lemma 2.1. The actual proof of (3.11), 
however, is now more cumbersome since no simple formulas such as 
the right-hand sides in (2.15)--(2.17) appear to be available in the 
present case. Hence, we briefly sketch a different (though 
straightforward) approach to (3.11) (following Lemma 2.3 in [13]).  
>From the outset it is readily verified that
$$\gather
T(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty} 1+O(\lambda^{-
1/2}), \tag 3.12 \\
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty} 
O(\lambda^{-1/2}). \tag 3.13
\endgather
$$
In order to improve on (3.13), using the additional smoothness 
conditions on $V$, we explicitly compute the Wronskian of 
$\overline{f_{-}(\lambda, x)}$ and $f_{+}(\lambda, x)$ (for 
simplicity, at $x=0$).
$$\gather
W(\overline{f_{-}(\lambda)}, f_{+}(\lambda))(0)=i(k_{+}-k_{-})-
\int\limits^{\infty}_{0} dx \cos(k_{+}x)[V(x)-V_{+}] 
f_{+}(\lambda, x) \\
-\int\limits^{0}_{-\infty}dx \cos(k_{-}x)V(x)\overline{f_{-}(\lambda, 
x)} -(ik_{+}/k_{-})\int\limits^{0}_{-\infty}dx\sin(k_{-}x)V(x)
\overline{f_{-}(\lambda, x)} \\
-(ik_{-}/k_{+})\int\limits^{\infty}_{0}dx\sin(k_{+}x)[V(x)-
V_{+}]f_{+}(\lambda, x) \\
+k^{-1}_{-} \int\limits^{0}_{-\infty}dx\sin(k_{-}x/V(x)
\overline{f_{-}(\lambda, x)} \int\limits^{\infty}_{0} 
dx'\cos(k_{+}x')[V(x')-V_{+}] f_{+}(\lambda, x')\\
-k^{-1}_{+} \int\limits^{0}_{-\infty} dx\cos(k_{-}x)V(x)
\overline{f_{-}(\lambda, x)} \int\limits^{\infty}_{0} dx' \sin(k_{+}x')
[V(x')-V_{+}] f_{+}(\lambda, x'), \quad \lambda > V_{+}.  \tag 3.14
\endgather
$$
Next, one observes
$$
i[k_{+}(\lambda)-k_{-}(\lambda)]\operatornamewithlimits{=}
_{\lambda\to\infty}(-i/2\lambda^{1/2})V_{+}+O(\lambda^{-3/2}), \tag 3.15
$$
$$[ik_{\pm}(\lambda)/k_{\mp}(\lambda)]\operatornamewithlimits{=}
_{\lambda\to\infty} i\mp (i/2\lambda)V_{+}+O(\lambda^{-2}), \tag 3.16
$$
$$
k_{\pm}(\lambda)^{-1}\operatornamewithlimits{=}_{\lambda\to\infty}
\lambda^{-1/2}+O(\lambda^{-3/2}), \tag 3.17
$$
and
$$
|g_{\pm}(\lambda, x)|+|g'_{\pm}(\lambda, x)|+|g''_{\pm}(\lambda, x)|
\leq C, \quad x\lesseqgtr 0, \lambda\geq V_{+}+1, \tag 3.18
$$
where
$$
g_{\pm}(\lambda, x)=e^{\mp ik_{\pm}(\lambda)x} f_{\pm}(\lambda, x).
\tag 3.19
$$
(The estimate (3.18) immediately follows from (3.2).)  Employing 
(3.15)--(3.19), one arrives at
$$
W(\overline{f_{-}(\lambda)}, f_{+}(\lambda))\operatornamewithlimits{=}
_{\lambda\to\infty} o(\lambda^{-1}) \tag 3.20
$$
after two integrations by parts in (3.14) (and a few tears) using the 
Riemann-Lebesgue lemma and
$$\align
g_{\pm}(\lambda, 0) &\operatornamewithlimits{=}_{\lambda\to\infty}
1\mp (1/2i\lambda^{1/2})\int\limits^{\pm\infty}_{0} dxe^{\mp 
ik_{\pm}x} [V(x)-V_{\pm}] f_{\pm}(\lambda, x)+O(\lambda^{-1}), 
\tag 3.21 \\
g'_{\pm}(\lambda, 0) &\operatornamewithlimits{=}_{\lambda\to\infty}
O(\lambda^{-1/2}). \tag 3.22
\endalign
$$       
Combining (3.6)--(3.8), (3.12), and (3.20) then proves (3.11) \qed
\enddemo

The asymptotic behavior (3.11) slightly improves Lemma 1.4 (iv) in [4] 
since we arrive at the conclusion $o(\lambda^{-3/2})$ instead of 
$O(\lambda^{-3/2})$ and their extra hypothesis $\int\limits^{0}_{-
\infty} dx(1+|x|)|V(x)|+\int\limits^{\infty}_{0}dx(1+|x|)|V(x)-
V_{+}|<\infty$ is not needed in our proof.

Remark 2.2 and the paragraph following it clearly apply in the present 
context.

Lemma 3.2 enables one to again remove the Abelian limit in the trace 
formula (1.6) for $V(x)$.  (We recall our notational conventions 
in (2.4)--(2.6), (2.8), (2.9), and the paragraph following (2.9).)

\proclaim{Theorem 3.3}  Suppose $V, V'\in AC_{\text{\rom{loc}}}(\Bbb R)$, 
$V$ is real-valued, $V', V''\in L^{1}(\Bbb R)$, $\int\limits^{0}_{-
\infty} dx |V(x)|\mathbreak+\int\limits^{\infty}_{0}dx|V(x)-V_{+}|<\infty$ for 
some $V_{+}>0$.  Let $E_{o}=\inf\,\sigma(H)$.  Then $[1-2\xi(., x)]\in 
L^{1}((E_{o}, \infty); d\lambda)$, $x\in\Bbb R$ and
$$\align
V(x)&=E_{o}+\int\limits^{\infty}_{E_o}d\lambda[1-2\xi(\lambda, x)]
\tag 3.23 \\
&=2\{e_{0}+\sum_{j\in J_{+}}[e_{j}-\mu_{j}(x)]\} +
\int\limits^{\infty}_{0}d\lambda[1-2\xi(\lambda, x)] \tag 3.24 \\
&=2\{e_{0}+\sum_{j\in J_{+}}[e_{j}-\mu_{j}(x)]\}
-(2/\pi)\int\limits^{\infty}_{0}d\lambda\text{\rom{ Im}}\left\{\ln
\left[1+R^{\ell}(\lambda)\frac{f_{-}(\lambda, x)^{2}}{|f_{-}(\lambda, 
x)|^{2}}\right]\right\}, \, x\in\Bbb R. \tag 3.25
\endalign
$$
If $\sigma_{p}(H)=\emptyset$, the discrete spectrum part $2\{\dots\}$ 
in \rom{(3.24)} and \rom{(3.25)} is to be deleted.
\endproclaim

Given Lemma 3.2, the proof of Theorem 3.3 is identical to that of 
Theorem 2.3.

\remark{Remark \rom{3.4}} While $\xi(\lambda, x)$ is continuous in 
$\lambda >V_{+}$, $\xi(\lambda, x)$ may have (countably 
infinitely-many) jump discontinuities of size one in $[0, V_{+}]$.  
These discontinuities occur at those special energies 
$\mu_{j}(x)\in [0, V_{+}]$ which are eigenvalues of the Dirichlet 
operator $H^{D}_{+, x}$ (the restriction of $H^{D}_{x}$ to $(x, 
\infty)$).  The Green's function $G(z, x, x)$ of $H$ is of the type
$$
G(z, x, x)=[m^{-}_{x}(z)-m^{+}_{x}(z)]^{-1}, \tag 3.26
$$
where $m^{\pm}_{x}(z)$ are the Weyl $m$-functions associated with 
$H^{D}_{\pm, x}$ in $L^{2}((x, \pm\infty))(H^{D}_{x}=H^{D}_{-, x}
\oplus H^{D}_{+, x})$ with $m^{-}_{x}(z)$ being continuous near 
$\mu_{j}(x)$ while $m^{+}_{x}(z)$ has a first-order pole with a 
negative residue at $\mu_{j}(x)$.  Thus
$$
G(z, x, x)\operatornamewithlimits{=}_{z\to\mu_{j}(x)} c[z-\mu_{j}(x)]
+O([z-\mu(x)]^{2}), \quad \mu_{j}(x)\in (0, V_{+}) \tag 3.27
$$
for some $c>0$ and hence $\xi(\lambda, x)$ has the jump discontinuity
$$
\lim_{\epsilon\downarrow 0} \xi(\mu_{j}(x)\pm\epsilon)=\cases 0 \\ 1
\endcases \tag 3.28
$$
at $\mu_{j}(x)\in (0, V_{+})$. A comparison of formulas (3.10) for 
$\xi(\lambda, x), \lambda >V_{-}$ and (3.7) for $R_{-}(\lambda)$ then 
shows that
$$
1+R^{\ell}(\mu_{j}(x))\frac{f_{-}(\mu_{j}(x), x)^{2}}
{|f_{-}(\mu_{j}(x), x)|^{2}}=0 \tag 3.29
$$
since $f_{+}(\mu_{j}(x), x)=0$.  Clearly $\xi(\lambda, x)$ is 
continuous in $\lambda >0$ away from these special energies 
$\mu_{j}(x)\in (0, V_{+}]$.
\endremark

Next, we shall briefly consider the behavior of $\xi(\lambda, x)$ as 
$\lambda\downarrow 0$ and as $\lambda\downarrow V_{+}$ (since $H$ 
changes spectral multiplicity at $V_{+}$) similarly to the discussion 
at the end of \S 2.  We shall assume
$$
\int\limits^{0}_{-\infty}dx (1+x^{2})|V(x)|+\int\limits^{\infty}
_{0} dx (1+x^{2})|V(x)-V_{+}|<\infty \tag 3.30
$$
in addition to (3.1) and consider the following case distinctions 
depending on whether or not $H$ has a threshold resonance at $\lambda 
=0$.

\example{Case I} $W(f_{-}(0), f_{+}(0))\neq 0$ and $f_{-}(0, 
x)f_{+}(0, x)\neq 0$.
\endexample
\flushpar Then, since $f_{\pm}(0, x)$ are real-valued, one infers
$$
R^{\ell}(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow 0}
-1+O(\lambda^{1/2}), \tag 3.31
$$
$$
G(\lambda +i0, x, x)\operatornamewithlimits{=}_{\lambda\downarrow 0}
\frac{f_{+}(0, x)f_{-}(0, x)}{W(f_{+}(0), f_{-}(0))} + 
O(\lambda^{1/2}) \tag 3.32
$$
and hence
$$
\xi(\lambda, x)=\pi^{-1}\arg[G(\lambda +i0, x, x)]
\operatornamewithlimits{=}_{\lambda\downarrow 0} O(\lambda^{1/2}) \ 
\text{in case I}. \tag 3.33
$$

\example{Case II}  $W(f_{-}(0), f_{+}(0))=0$ and $f_{-}(0, x)f_{+}(0, 
x)\neq 0$.
\endexample
\flushpar  In this case one infers (see, e.g., Proposition 2.4 in [4] 
or Lemma 2.5 in [13]) that
$$
W(f_{-}(\lambda), f_{+}(\lambda))\operatornamewithlimits{=}_
{\lambda\downarrow 0} i\gamma\lambda^{1/2}+O(\lambda), \quad 
\gamma\in\Bbb R\backslash\{0\}, \tag 3.34
$$
$$
G(z+i0, x, x)\operatornamewithlimits{=}_{\lambda\downarrow 0}
(i/\gamma)f_{+}(0, x)f_{-}(0, x)[1+O(\lambda^{1/2})]. \tag 3.35
$$
Thus we get
$$
\xi(\lambda, x)=\pi^{-1}\arg[G(\lambda +i0, x, x)]
\operatornamewithlimits{=}_{\lambda\downarrow 0}\frac12 +
O(\lambda^{1/2}) \ \text{in case II}. \tag 3.36
$$

Discussion of the case $\lambda\downarrow V_{+}$ remains.  Since 
$f_{+}(V_{+}, x)$ is real-valued and $W(f_{-}(\lambda),\mathbreak 
f_{+}(\lambda))\neq 0$ for $\lambda > 0$ (see, e.g., Lemma 1.2 in 
[4] or Lemma 2.1 in [13]), one infers that
$$
R^{r}(\lambda)\operatornamewithlimits{=}_{\lambda\downarrow V_{+}}
-1+O((\lambda -V_{+})^{1/2}), \tag 3.37
$$
$$
G(\lambda +i0, x, x)\operatornamewithlimits{=}_{\lambda\downarrow 
V_{+}} \frac{f_{+}(V_{+}, x)f_{-}(V_{+}, x)}{W(f_{+}(V_{+}),
f_{-}(V_{+}))} + O((\lambda - V_{+})^{1/2}), \tag 3.38
$$
and hence
$$
\xi(\lambda, x)\operatornamewithlimits{=}_{\lambda\downarrow V_{+}}
\frac12 + \pi^{-1}\text{Im}\left\{\ln\left[1+R^{\ell}(V_{+})
\frac{f_{-}(V_{+}, x)^{2}}{|f_{-}(V_{+}, x)|^{2}}\right]\right\} 
+ O((\lambda -V_{+})^{1/2}). \tag 3.39
$$
This serves as an illustration that $\xi(\lambda, x)$ is insensitive 
to the fact that $H$ changes its spectral multiplicity at $V_{+}$.

We conclude this section with a brief discussion of the case where 
$V(x)\operatornamewithlimits{\longrightarrow}\limits_{x\to\infty}\infty$, 
that is, we now assume
$$
V\in C(\Bbb R), \quad \int\limits^{0}_{-\infty}dx|V(x)|<\infty, \quad
\lim_{x\to\infty}V(x)=\infty. \tag 3.40
$$
$H$ is then defined as the form sum of $H_{o}=-\frac{d^2}{dx^2}$ and 
$V$ in $L^{2}(\Bbb R)$.  Then $f_{-}(z, x)$ can be defined as in (3.2) 
(or (2.13)) and, since $V(x)\to\infty$ as $x\to\infty$, the Weyl $m$-function 
associated with $H^{D}_{+, 0}$ (the restriction of $H$ to $(0, 
\infty)$ with a Dirichlet boundary condition at $x=0$) is meromorphic. 
Hence, there exists an entire function $f_{+}(z, x)$ satisfying
$$
f_{+}(z, .)\in L^{2}((0, \infty)), \quad z\in\Bbb C \tag 3.41
$$
and (3.3).  $f_{+}(z, x)$ can be chosen to be real-valued for 
$\lambda\in\Bbb R$ (see, e.g., [26] for further details).  $H$ now has 
simple spectrum which is purely absolutely continuous on $(0, 
\infty)$.  The reflection coefficient from left incidence is then 
defined as in (3.4), that is,
$$
R^{\ell}(\lambda)=-\frac{\overline{W(f_{-}(\lambda), f_{+}(\lambda))}}
{W(f_{-}(\lambda), f_{+}(\lambda))}, \quad \lambda >0 \tag 3.42
$$
and hence
$$
|R^{\ell}(\lambda)|=1, \quad \lambda > 0 \tag 3.43
$$
proves total reflection from left incidence at all positive energies 
$\lambda > 0$.  The Green's function $G(z, x, x)$ of $H$ now satisfies
$$\align
G(\lambda +i0, x, x)&=\frac{f_{+}(\lambda, x)f_{-}(\lambda, x)}
{W(f_{+}(\lambda), f_{-}(\lambda))}\\
&=(i/2\lambda^{1/2})|f_{-}(\lambda, x)|^{2}\left[1+R^{\ell}(\lambda)
\frac{f_{-}(\lambda, x)^{2}}{|f_{-}(\lambda, x)|^{2}}\right], \quad
\lambda > 0 \tag 3.44
\endalign
$$
and one infers
$$
\xi(\lambda, x)=\frac12 +\pi^{-1}\text{Im}\left\{\ln\left[1+R^{\ell}
(\lambda)\frac{f_{-}(\lambda, x)^{2}}{|f_{-}(\lambda, x)|^{2}}\right]
\right\}, \quad \lambda >0 \tag 3.45
$$
as in (3.10). However, due to the total reflection at all positive 
energies $\lambda >0$,
$$
R^{\ell}(\lambda)=e^{ir(\lambda)}\operatornamewithlimits{\not\longrightarrow}
_{\lambda\to\infty} 0 \tag 3.46
$$
for some real-valued function $r$ and hence
$$
1-2\xi(\lambda, x)\operatornamewithlimits{\not\longrightarrow}_
{\lambda\to\infty} 0. \tag 3.47
$$
In fact, in the explicit example $V(x)=e^{x}$ discussed, for instance 
in [26], where $R^{\ell}(\lambda)=-
\exp\{2i\arg[\Gamma(1+2i\lambda^{1/2})]\}, \lambda\geq 0$ (here 
$\Gamma(.)$ denotes the gamma function), one can verify that $[1-
2\xi(., x)]\notin L^{1}((0, \infty); d\lambda), x\in\Bbb R$.  As a 
consequence, the Abelian limit in the trace formula (1.6) for $V(x)$, 
in general, cannot be removed in the case (3.36) and hence (1.6) 
represents a genuine summability method in this situation.  As 
mentioned briefly in the introduction, this becomes even more 
transparent in the case where 
$V(x)\operatornamewithlimits{\longrightarrow}\limits_{x\to\pm\infty}\infty$ 
since then for all $x\in\Bbb R$ and a.e.~$\lambda\in\Bbb R$, $|1-2\xi
(\lambda, x)|=1$.

\bigpagebreak
\flushpar {\bf \S 4. Hill Operators with Impurities}

In our final section, we shall consider short-range perturbations $W$ 
of Hill operators $H^{o}=\frac{d^2}{dx^2}+V^{o}$ and hence extend the 
results of \S2 to scattering off impurities (defects) in 
one-dimensional solids.

Again, most of the results in this section are valid under minimal 
smoothness assumptions on $V^o$ and $W$.  However, since our main 
result in Theorem 4.3 requires a certain regularity of $V^o$ and $W$, 
we shall avoid technicalities and suppose these regularity assumptions 
throughout this section.

We start by briefly reviewing the necessary Floquet theory associated 
with the periodic background potential $V^o$ satisfying
$$
V^{o}\in H^{1, 2}([0, a]), \quad V^{o} \ \text{real-valued}, \quad
V^{o}(x+a)=V^{o}(x), \quad x\in\Bbb R \tag 4.1
$$
for some $a>0$.  The corresponding Hill operator $H^o$ in $L^{2}(\Bbb 
R)$ is then defined by
$$
H^{o}=-\frac{d^2}{dx^2}+V^{o}, \quad \Cal D(H^{o})=H^{2, 2}(\Bbb R). \tag 4.2
$$
The spectrum of $H^o$ is purely absolutely continuous of the type
$$
\sigma(H^{o})=\bigcup_{n\in\Bbb N} 
[E^{o}_{2(n-1)}, E^{o}_{2n-1}], E^{o}_{0}<E^{o}_{1}\leq E^{o}_{2}<E^{o}_{3}
\leq E^{o}_{4}<E^{o}_{5}\leq \cdots \tag 4.3
$$
with uniform multiplicity two on $\sigma (H^{o})^{o}$.  (We recall 
that $A^o$ denotes the interior of $A\subset\Bbb R$.)  An entire 
fundamental system of distributional solutions of
$$
H^{o}\psi(z, x)=z\psi(z, x) \tag 4.4
$$
with respect to $z$ is then provided by $s^{o}(z, x)$ and $c^{o}(z, 
x)$ defined as
$$\gathered
s^{o}(z, x)=z^{-1/2}\sin(z^{1/2}x)+\int\limits^{x}_{0}dx'z^{-1/2}\sin
[z^{1/2}(x-x')]V^{o}(x')s^{o}(z, x'), \\
c^{o}(z, x)=\cos(z^{1/2}x)+\int\limits^{x}_{0}dx'z^{-1/2}\sin[z^{1/2}
(x-x')]V^{o}(x')c^{o}(z, x'), \quad z\in\Bbb C, 
\endgathered \tag 4.5
$$
$$
W(s^{o}(z), c^{o}(z))=-1, \quad z\in\Bbb C. \tag 4.6
$$
The discriminant $\Delta(z)$ and Floquet parameter (Bloch momentum) 
$\theta(z)$ are then defined by
$$
\Delta(z)=[c^{o}(z, a)+s^{o\prime}(z, a)]/2, \tag 4.7
$$
$$
e^{\pm i\theta(z)a}=\Delta(z)\mp\sqrt{\Delta(z)^{2}-1}. \tag 4.8
$$
Thus
$$
\cos[\theta(z)a]=\Delta(z), \quad 
\sin[\theta(z)a]=i\sqrt{\Delta(z)^{2}-1} \tag 4.9
$$
and the branch of $\sqrt{\cdot}$ is chosen such that 
$\sqrt{\Delta(\lambda)^{2}-1} >0$ for $\lambda < E^{o}_0$ and hence
$$\gathered
-i\theta(\lambda)>0, \lambda <E^{o}_{0}, \quad \theta(\lambda)\in\Bbb R\iff
\lambda\in\sigma(H^{o}), \\ -i\theta(\lambda)\in (0, 
\infty)\iff\lambda\in\Bbb R\backslash\sigma(H^{o}). 
\endgathered \tag 4.10
$$
The Floquet solutions of $H^o$ are defined by
$$\align
f^{o}_{\pm}(z, x)&=c^{o}(z, x)+s^{o}(z, x)\left[\Delta(z)\mp 
\sqrt{\Delta(z)^{2}-1} -c^{o}(z, a)\right]/ s^{o}(z, a) \\
&=e^{\pm i\theta(z)x} p_{\pm}(z, x), \quad p_{\pm}(z, x+a)=p_{\pm}(z, 
x). \tag 4.11
\endalign
$$
They satisfy
$$
f^{o}_{\pm}(z, .)\in L^{2}((R, \pm\infty)), \quad R\in\Bbb R, \quad
z\in\Bbb C\backslash \sigma(H^{o}), \tag 4.12
$$
$$
f^{o}_{-}(\lambda, x)=\overline{f^{o}_{+}(\lambda, x)}, \quad 
\lambda\in\sigma(H^{o}), \tag 4.13
$$
$$
f^{o}_{\pm}(\lambda, x) \ \text{are real-valued for} \ \lambda\in\Bbb 
R\backslash\sigma(H^{o})^{o}, \tag 4.14
$$
$$
W(f^{o}_{-}(z), f^{o}_{+}(z))=-2\sqrt{\Delta(z)^{2}-1}/s^{o}(z, a), 
\tag 4.15
$$
$$
W(f^{o}_{\mp}(\lambda), \overline{f^{o}_{\mp}(\lambda)}) = \cases
\pm 2i\sin[\theta(\lambda)a]/s^{o}(\lambda, a), 
\lambda\in\sigma(H^{o})^{o} \\ 0, \lambda\in\Bbb 
R\backslash\sigma(H^{o})^{o}. \endcases \tag 4.16
$$
For details in connection with (4.3)--(4.16), see, for example, [9], 
[10], [28], Sect.~7.4, [29], Sect.~3.4, [36], [37], and the references 
therein.

If $H^{o, D}_{y}$ denotes the associated Dirichlet operator in 
$L^{2}(\Bbb R)$ defined analogously to (2.7) with spectrum
$$\gathered
\sigma(H^{o, D}_{y})=\{\mu^{o}_{n}(y)\}_{n\in\Bbb N} 
\cup\sigma(H^{o}), \quad \sigma_{\text{ess}}(H^{o, 
D}_{y})=\sigma(H^{o}), \\
E^{o}_{2n-1}\leq \mu^{o}_{n}(y)\leq E^{o}_{2n}, \quad n\in\Bbb N, 
\endgathered \tag 4.17
$$
the trace formula (1.6) applied to $V^{o}(x)$ yields (see [16], [19])
$$\align
V^{o}(x)&=E^{o}_{0}+\lim_{\epsilon\downarrow 0} 
\int\limits^{\infty}_{E^{o}_{0}} d\lambda e^{-\epsilon\lambda}[1-
2\xi^{o}(\lambda, x)] \tag 4.18 \\
&=E^{o}_{0}+\sum^{\infty}_{n=1}[E^{o}_{2n-1}+E^{o}_{2n}-
2\mu^{o}_{n}(x)]. \tag 4.19
\endalign
$$
Here $\pi\xi^{o}(\lambda, x)$ denotes the argument of the Green's 
function $G^{o}(z, x, x)$ of $H^o$ (see (2.24) and one explicitly 
obtains
$$
\xi^{o}(\lambda, x)=\cases 1, E^{o}_{2n-1}<\lambda<\mu^{o}_{n}(x) \\
0, \mu^{o}_{n}(x)<\lambda <E^{o}_{2n} \\
\frac12 , E^{o}_{2(n-1)} <\lambda <E^{o}_{2n-1} . \endcases
\tag 4.20
$$
Moreover, the assumption $V^{o}\in H^{1, 2}([0, a])$ implies the 
finiteness of the total gap length (see, e.g., [23] or Theorem 1.5.2 in 
[29])
$$
\sum^{\infty}_{n=1} \left|E^{o}_{2n}-E^{o}_{2n-1}\right| <\infty 
\tag 4.21
$$
and this has been used to infer (4.19) from (4.18) and (4.20).

Next we briefly turn to the impurity potential $W$ assuming
$$
W\in H^{2, 1}(\Bbb R), \quad W \ \text{real-valued}, \quad W\in 
L^{1}(\Bbb R; (1+|x|)dx). \tag 4.22
$$
The total Hamiltonian $H$ in $L^{2}(\Bbb R)$ is then defined by
$$
H=-\frac{d^2}{dx^2} + V, \quad \Cal D(H)=H^{2, 2}(\Bbb R), \quad
V(x)=V^{o}(x)+W(x). \tag 4.23
$$
The spectrum $\sigma(H)$ of $H$ is now of the type
$$
\sigma(H)=\sigma_{p}(H)\cup\sigma(H^{o}), \quad 
\sigma_{\text{ess}}(H)=\sigma(H^{o}), \tag 4.24
$$
where the point spectrum $\sigma_{p}(H)$ (the set of eigenvalues) of 
$H$ may be denoted by
$$
\sigma_{p}(H)=\bigcup^{\infty}_{n=0} \sigma_{p, n},
$$
$$
\sigma_{p, 0}=\{e_{0, j}\}_{j\in J_{0}}\subset(-\infty, E^{o}_{0}), 
\quad \sigma_{p, n}=\{e_{n, j}\}_{j\in J_{n}}\subset(E^{o}_{2n-1},
E^{o}_{2n}), \quad e_{n, j} < e_{n, j+1}, \tag 4.25
$$
with
$$
J_{n}=\cases \emptyset \\ \{0, 1, 2, \dots, N_{n}\} \endcases \ , \quad
n\in\Bbb N_{0} \tag 4.26
$$
an appropriate index set.  Similarly to \S 2, we shall also need the 
notation
$$\gathered
J_{0, +}=\cases \emptyset \\ \{1, 2, \dots, N_{0}, N_{0}+1\} \endcases
\ , \quad J_{n, +} = \cases \emptyset \\ \{0, 1, 2, \dots, N_{n}, 
N_{n}+1\} \endcases \ , \quad n\in\Bbb N \\ 
e_{0, N_{0}+1}=E^{o}_{0}, \quad e_{n, -1}=E^{o}_{2n-1}, \quad
e_{n, N_{n}+1}=E^{o}_{2n} 
\endgathered \tag 4.27
$$
depending on whether $J_n$ is empty or finite.  Each eigenvalue of $H$ 
is simple, $\sigma_{p}(H)\cap\sigma(H^{o})=\emptyset$, and the 
spectrum of $H$ in $\sigma(H^{o})^{o}$ is purely absolutely continuous 
and of uniform multiplicity two under hypotheses (4.1) and (4.22).

\remark{Remark \rom{4.1}} We have chosen to add the hypothesis $W\in 
L^{1}(\Bbb R; (1+|x|)dx)$ from the outset in (4.22) since it guarantees 
finiteness of the discrete spectrum $\sigma_{p, n}$ of $H$ in any of 
its essential spectral gaps $(-\infty, E^{o}_{0}), (E^{o}_{2n-1}, 
E^{o}_{2n})$ as proven in [31].  Moreover, it can be shown that $H$ has 
at most two eigenvalues in $(E^{o}_{2n-1}, E^{o}_{2n})$ for $n$ 
sufficiently large (i.e., $N_{n}\leq 2$ for $n$ large enough), and if 
$\int\limits_{\Bbb R} dx W(x)\neq 0$ precisely one eigenvalue in 
$(E^{o}_{2n-1}, E^{o}_{2n})$ for $n$ sufficiently large 
(i.e., $N_{n}=1$ if $\int\limits_{\Bbb R} dxW(x)\neq 0$ for $n$ large 
enough), see [9], [15], [32], [36], [37]. The following material can be 
developed without the assumption $W\in L^{1}(\Bbb R; (1+|x|)dx)$ but 
only at the expense of introducing a considerably more involved bulk 
of notations since elements of $\sigma_{p, n}$ may then accumulate at 
$E^{o}_{2n-1}$ and/or $E^{o}_{2n}$.
\endremark

Next we briefly introduce the associated Dirichlet operator 
$H^{D}_{y}$ in $L^{2}(\Bbb R)$ defined as in (2.7).  Its spectrum is 
then given by
$$
\sigma(H^{D}_{y})=\sigma_{p}(H^{D}_{y})\cup\sigma(H^{o}), \quad
\sigma_{\text{ess}}(H^{D}_{y})=\sigma(H^{o}), \tag 4.28
$$
where
$$
\sigma_{p}(H^{D}_{y})=\bigcup\limits^{\infty}_{n=0} \sigma^{D}_{p, 
n}(y), \tag 4.29
$$
$$\gathered
\sigma^{D}_{p, 0}(y)=\{\mu_{0, j}(y)\}_{j\in J_{0, +}}\cap (e_{0, 0}, 
E^{o}_{0}), \\
e_{0, 0}<\mu_{0, 1}(y)\leq e_{0, 1}, \quad e_{0, j-1}\leq \mu_{0, 
j}(y)\leq e_{0, j}, \quad j\in J_{0, +}\backslash \{1\},
\endgathered \tag 4.30
$$
$$\gathered
\sigma^{D}_{p, n}(y)=\{\mu_{n, j}(y)\}_{j\in J_{n, +}}\cap 
(E^{o}_{2n-1}, E^{o}_{2n}), \quad n\in\Bbb N, \\
e_{n, j-1}\leq \mu_{n, j}(y)\leq e_{n, j}, \quad j\in J_{n, +}, \quad 
n\in\Bbb N. 
\endgathered \tag 4.31
$$
(Our notation in (4.30), (4.31) indicates that the limiting 
cases $\mu_{n, 0}(y)=e_{0, -1}=E^{o}_{2n-1},\mathbreak n\in\Bbb N, \mu_{n, 
N_{n}+1}(y)=e_{n, N_{n}+1}=E^{o}_{2n}, n\in\Bbb N$ are not Dirichlet 
eigenvalues since $H$ has no $L^{2}((y, \pm\infty))$ eigenfunctions
at $E^{o}_{n}, n\in\Bbb N_{0}$.)  The spectrum of $H^{D}_{y}$ in 
$\sigma(H^{o})^{o}$ is purely absolutely continuous and of uniform 
multiplicity two.  Similarly to \S2, $\mu_{n, j}(y)$ is a simple 
eigenvalue of $H^{D}_{y}$ if and only if $e_{n, j-1}<\mu_{n, 
j}(y)<e_{n,j}$, whereas if $\mu_{n, j}(y)\in\{e_{n, j-1}, e_{n, j}\}$, 
then $\mu_{n, j}(y)$ has multiplicity two (excluding the cases 
$\mu_{n, j}(y)=E^{o}_{2n-1}, E^{o}_{2n}$).  For details in the context 
of (4.24)--(4.31) see, for example, [9], [10], [15], [31], [32], [36], [37], 
and the references therein.

Impurity (defect) scattering associated with the pair $(H, H^{o})$ can 
then be summarized as follows (see, e.g., [9], [10], [13], and the 
references therein).  The Jost solutions $f_{\pm}(z, x)$ of $H$ are 
defined by
$$
f_{\pm}(z, x)=f^{o}_{\pm}(z, x)-\int\limits^{\pm\infty}_{x} dx'g^{o}(z, 
x, x')W(x')f_{\pm}(z, x'), \quad z\in\Bbb 
C\backslash\{E^{o}_{n}\}_{n\in\Bbb N_{0}}, \tag 4.32
$$
$$
g^{o}(z, x, x')=[f^{o}_{+}(z, x)f^{o}_{-}(z, x')-f^{o}_{+}(z, x')
f^{o}_{-}(z, x)]/W(f^{o}_{-}(z), f^{o}_{+}(z)) \tag 4.33
$$
such that
$$
Hf_{\pm}(z, x)=zf_{\pm}(z, x), \quad z\in\Bbb C\backslash 
\{E^{o}_{n}\}_{n\in\Bbb N_{0}} \tag 4.34
$$
in the distributional sense. The unitary scattering matrix 
$S(\lambda), \lambda\in\sigma(H^{o})^{o}$ in $\Bbb C^2$ associated 
with the pair $(H, H^{o})$ then reads as follows
$$
S(\lambda) = \pmatrix T(\lambda) & R^{r}(\lambda) \\
R^{\ell}(\lambda) & T(\lambda) \endpmatrix \ , \quad 
\lambda\in\sigma(H^{o})^{o}, \tag 4.35
$$
$$
T(\lambda)=\frac{2i\sin[\theta(\lambda)a]/s^{o}(\lambda, a)}
{W(f_{-}(\lambda), f_{+}(\lambda))}=\left\{1-\frac{s^{o}(\lambda, a)}
{2i\sin[\theta(\lambda)a]}\int\limits_{\Bbb R} dx W(x)f^{o}_{\pm}
(\lambda, x) f_{\mp}(\lambda, x)\right\}^{-1}, \tag 4.36
$$
$$
R^{\ell}(\lambda)=-\frac{W(\overline{f_{-}(\lambda)}, f_{+}(\lambda))}
{W(f_{-}(\lambda), f_{+}(\lambda))} = \frac{T(\lambda)s^{o}(\lambda, 
a)}{2i\sin[\theta(\lambda)a]} \int\limits_{\Bbb R} dxW(x)f^{o}_{+}
(\lambda, x), f_{+}(\lambda, x), \tag 4.37
$$
$$
R^{r}(\lambda)=-\frac{W(f_{-}(\lambda), \overline{f_{+}(\lambda)})}
{W(f_{-}(\lambda), f_{+}(\lambda))}=\frac{T(\lambda)s^{o}(\lambda, a)}
{2i\sin[\theta(\lambda)a]} \int\limits_{\Bbb R} dxW(x)f^{o}_{-}
(\lambda, x)f_{-}(\lambda, x), \tag 4.38
$$
where $f_{\pm}(\lambda, x)=\lim\limits_{\epsilon\downarrow 0} 
f_{\pm}(\lambda +i0, x)$.  The Green's function of $H$ satisfies
$$\gather
G(\lambda +i0, x, x)=\frac{f_{+}(\lambda, x)f_{-}(\lambda, x)}
{W(f_{+}(\lambda), f_{-}(\lambda))} \\ =\frac{is^{o}(\lambda, a)}
{2\sin[\theta(\lambda)a]} |f_{\pm}(\lambda, x)|^{2} 
\left[1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}
{|f_{\pm}(\lambda, x)|^{2}}\right], \quad \lambda\in\sigma(H^{o})^{o},
x\in\Bbb R. \tag 4.39
\endgather
$$

Since $V=V^{o}+W$ is continuous and bounded from below, relations 
(2.20)--(2.27), in particular, the definition (2.24) of $\xi(\lambda, 
x)$ and the trace formula (2.27), are valid.

In attempting to generalize Lemma 2.1, however, one is faced with the 
following problem.  Although
$$
s^{o}(\lambda, a)\operatornamewithlimits{=}_{\lambda\to\infty}
\lambda^{-1/2}\sin(\lambda^{1/2}a)-(2\lambda)^{-1}\cos(\lambda^{1/2}a)
\int\limits^{a}_{0} dxV^{o}(x) +O(\lambda^{-3/2}), \tag 4.40
$$
$$
\theta(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty}\lambda^{1/2}+O(1) 
\tag 4.41
$$
and hence
$$
T(\lambda)\operatornamewithlimits{\longrightarrow}_{\lambda\to\infty} 
1, \quad R^{r(\ell)}(\lambda)\operatornamewithlimits{\longrightarrow}_
{\lambda\to\infty}0 \ \text{pointwise for 
$\lambda\in\sigma(H^{o})^{o}$}, \tag 4.42
$$
that is, away from the essential spectral band edges 
$\{E^{o}_{n}\}_{n\in\Bbb N_{0}}$ of $H$, the factors 
$\sin[\theta(\lambda)a]$ in the denominators of (4.36)--(4.38) prevent 
the convergence in (4.42) at the band edges.  In fact, as we will 
briefly explore at the end of this section, one generally has 
$R^{\ell, r}(E^{o}_{n})=-1$ and hence a result such as $R^{\ell 
(r)}(\lambda)\operatornamewithlimits{=}\limits_{\lambda\to\infty}
o(\lambda^{-3/2})$ in (2.36) is usually false in the present impurity 
scattering situation.  Nevertheless, by separately considering sufficiently 
small compact intervals $\sigma_{n}\subset\sigma(H^{o})$ with 
$E^{o}_{n}\in\partial\sigma_{n}$ and the remaining spectral band 
$[E^{o}_{2(n-1)}, E^{o}_{2n-1}]\backslash\{\sigma_{2n-
1}\cup\sigma_{2n}\}$, a device studied in detail by Firsova [9], [10], 
we will be able to prove a suitable analog of Lemma 2.1.

\proclaim{Lemma 4.2}  Suppose $V=V^{o}+W$ where $V^{o}\in H^{1,2}([0, 
a])$ is real-valued, $V^{o}(x+a)=V^{o}(x)$ for some $a>0$, $W\in 
H^{2,1}(\Bbb R)$ is real-valued, and $W\in L^{1}(\Bbb R; (1+|x|)dx)$.  
Then for all $x\in\Bbb R$,
$$
\xi(\lambda, x)=0, \quad \lambda<E_{o}=\inf\,\sigma(H), \tag 4.43
$$
$$
\xi(\lambda, x)=\frac12 + \pi^{-1}\text{\rom{Im}}\left\{\ln\left[
1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, 
x)^{2}}{|f_{\pm}(\lambda, x)|^{2}}\right]\right\}, \quad 
\lambda\in\sigma(H^{o}), \tag 4.44
$$
and $\xi(\lambda, x)$ is continuous for $\lambda\in\sigma(H^{o})^{o}$.  
Moreover,
$$
|1-2\xi(\lambda, x)|\leq |R^{r(\ell)}(\lambda)|, \quad 
\lambda\in\sigma(H^{o})^{o}. \tag 4.45
$$
In addition, there exist compact intervals 
$\sigma_{n}\subset\sigma(H^{o}), E^{o}_{n}\in\partial\sigma_{n}, 
n\in\Bbb N_{0}$ with $\sum\limits_{n\in\Bbb 
N_{0}}|\sigma_{n}|\mathbreak
<\infty$ \rom($|\cdot |$ denoting Lebesgue measure\rom) such that
$$
[1-2\xi(\lambda, x)]
\operatornamewithlimits{=}_{\lambda\to\infty}o(\lambda^{-3/2}) \ 
\text{\rom{for}} \ \lambda\in\sigma(H^{o})\backslash\bigcup_{n\in\Bbb 
N_{0}}\sigma_{n} \tag 4.46
$$
uniformly with respect to $x\in\Bbb R$.  In addition, 
$$
R^{r(\ell)}\in L^{1}(\sigma(H^{o}); d\lambda). \tag 4.47
$$
Finally, we note that
$$\align
&\xi(\lambda, x)=\cases 0, \lambda < e_{0, 0}, \mu_{n, j}(x) <
\lambda < e_{n, j} \\ 1, e_{n, j-1} < \lambda < \mu_{n, j}(x) 
\endcases \quad \text{\rom{if}} \ \mu_{n, j}(x)\in (e_{n, j-1}, 
e_{n, j}), \tag 4.48 \\
&\xi(\lambda, x)=\cases 0, e_{n, j-1} < \lambda < e_{n, j} 
\ \text{\rom{if}} \ \mu_{n, j}(x)=e_{n, j-1} \\
1, e_{n, j-1} < \lambda < e_{n, j} \ \text{\rom{if}} \
\mu_{n, j}(x)=e_{n, j} \endcases \tag 4.49
\endalign
$$
whenever $\sigma_{p, n}(H)\neq\emptyset$.
\endproclaim

\demo{Proof}  (4.43), (4.48) and (4.49) reflect the general behavior 
of $\xi(\lambda, x)$ in spectral gaps of $H$ since $G(z, x, x)>0$ for 
$z<\inf\,\sigma(H)$ and $G(z, x, x)$ is real-valued for $z$ in any 
(non-empty) spectral gap of $H$. Equation (4.44) follows from (2.24) 
and (4.39) (we note that $s^{o}(\lambda, a)$ is real-valued for 
$\lambda\in\Bbb R$ and $\theta(\lambda)\in\Bbb R$ for 
$\lambda\in\sigma(H^{o})$ by (4.10)).  Since $G(\lambda +i0, x, x)$ is 
continuous and zero-free for $\lambda\in\sigma(H^{o})^{o}$, $\xi(\lambda, 
x)$ is continuous in $\lambda\in\sigma(H^{o})$.  Inequality (4.45) is 
then clear from (4.44).  For the explicit construction of the compact 
intervals $\sigma_n$ with $E^{o}_{n}\in\partial\sigma_{n}, 
\sum\limits_{n\in\Bbb N_{0}}|\sigma_{n}|<\infty$ such that 
$$
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty} 
o(\lambda^{-1/2}) \ \text{for} \ \lambda\in\sigma(H^{o})\backslash
\bigcup_{n\in\Bbb N_{0}}\sigma_{n}, \tag 4.50
$$
we refer to [9], [10].  Here we only mention that (4.50) is implied by 
the asymptotic relation (4.41) and, assuming $\lambda\in
\sigma(H^{o})\backslash
\operatornamewithlimits{\cup}\limits_{n\in\Bbb N_{0}}\sigma_{n}$, by
$$
W(f^{o}_{-}(\lambda), f^{o}_{+}(\lambda))=
\frac{2i\sin[\theta(\lambda)a]}{s^{o}(\lambda, a)}
\operatornamewithlimits{=}_{\lambda\to\infty}
2i\lambda^{1/2}[1+O(|\lambda |^{-1/8})], \tag 4.51
$$
$$
T(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty} 
1+(1/2i\lambda^{1/2})\left[\int\limits_{\Bbb R}dxW(x)+
O(|\lambda |^{-1/8})\right], \tag 4.52
$$
$$
R\Sp r\\ \ell\endSp(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty}
(1/2i\lambda^{1/2})\left[\int\limits_{\Bbb R} dxW(x)e^{\mp 
2i\theta(\lambda)x} +O(|\lambda |^{-1/8})\right] \tag 4.53
$$
as proven in [9] (see also [10]).  In particular, in order to arrive at 
(4.52), (4.53) one combines (4.41), (4.51), (4.36)--(4.38), and
$$
p_{\pm}(\lambda, x)\operatornamewithlimits{=}_{\lambda\to\infty}
1+O(|\lambda |^{-1/8}), \quad \lambda\in\sigma(H^{o})\backslash
\bigcup_{n\in\Bbb N_{0}}\sigma_{n}, \tag 4.54
$$
$$
|f_{\pm}(\lambda, x)|\leq C(1+|x|), \quad \lambda\in\sigma(H^{o}), \tag 
4.55
$$
$$
|f_{\pm}(\lambda, x)-f^{o}_{\pm}(\lambda, x)|\leq C(1+|x|)(1+|\lambda 
|)^{-1/2}, \quad \lambda\in\sigma(H^{o}). \tag 4.56
$$
Here $p_{\pm}(\lambda, x)$ has been introduced in (4.11), and (4.55) 
and (4.56) follow from (4.32), (4.33) (see [9], [10]).  Analogous 
relations for the first and second $x$-derivatives of $p_{\pm}(\lambda, 
x)$ and $f_{\pm}(\lambda, x)$ then yield
$$
R^{r(\ell)}(\lambda)\operatornamewithlimits{=}_{\lambda\to\infty}
o(\lambda^{-3/2}), \quad \lambda\in\sigma(H^{o})\backslash
\bigcup_{n\in\Bbb N_{0}}\sigma_{n} \tag 4.57
$$
after two integrations by parts in (4.37), (4.38).  Together with 
(4.45) this proves (4.46).  Using $|R^{r(\ell)}(\lambda)|\leq 1, 
\lambda\in\sigma(H^{o})$ (as a consequence of the unitarity of the 
scattering matrix (4.35)), one then infers (4.47) from
$$
\int\limits_{\sigma(H^{o})} d\lambda |R^{r(\ell)}(\lambda)| \leq 
\int\limits_{\operatornamewithlimits{\cup}\limits_{n\in\Bbb N_{0}}
\sigma_{n}}d\lambda + \int\limits_{\sigma(H^{o})\backslash
\operatornamewithlimits{\cup}\limits_{n\in\Bbb N_{0}}\sigma_{n}}
|o(\lambda^{-3/2})|<\infty. \qed \tag 4.58
$$
\enddemo

As in the previous sections, Lemma 4.2 will enable us to remove the 
Abelian limit in the trace formula (2.27) for $V(x)$ and state the 
principal result of this section. (We recall our notational 
conventions in (4.3), (4.17), (4.25)--(4.31).)

\proclaim{Theorem 4.3}  Suppose $V=V^{o}+W$ where $V^{o}\in H^{1, 
2}([0, a])$ is real-valued, $V^{o}(x+a)=V^{o}(x)$ for some $a>0$, 
$W\in H^{2, 1}(\Bbb R)$ is real-valued, and $W\in L^{1}(\Bbb R; 
(1+|x|)dx)$.  Let $E_{o}=\inf\,\sigma(H)$.  Then $[1-2\xi(., x)]\in 
L^{1}((E_{o}, \infty); d\lambda), x\in\Bbb R$ and
$$\align
V(x)&=V^{o}(x)+W(x)=E_{o}+\int\limits^{\infty}_{E_o}d\lambda
[1-2\xi(\lambda, x)] \tag 4.59 \\
&=\{2e_{0, 0}+2\sum_{j\in J_{0, +}}[e_{0, j}-\mu_{0, j}(x)]-
E^{o}_{0}\} \\
&+ \sum^{\infty}_{n=1} \{E^{o}_{2n-1}+
2\sum_{j\in J_{n, +}} [e_{n, j}-\mu_{n, j}(x)]-E^{o}_{2n}\}
+\int\limits_{\sigma(H^{o})} d\lambda[1-2\xi(\lambda, x)] \tag 4.60 \\
&=\{2e_{0, 0}+2\sum_{j\in J_{0, +}} [e_{0, j}-\mu_{0, j}(x)]-
E^{o}_{0}\} \\
&+ \sum^{\infty}_{n=1}\{E^{o}_{2n-1}+2\sum_{j\in J_{n, +}}
[e_{n, j}-\mu_{n, j}(x)]-E^{o}_{2n}\} \\
&- (2/\pi)\int\limits_{\sigma(H^{o})} d\lambda\text{\rom{ Im}}\left\{\ln\left[
1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}
{|f_{\pm}(\lambda, x)|^{2}}\right]\right\}, \quad x\in\Bbb R. \tag 4.61
\endalign
$$
Similarly,
$$\align
W(x)&=2\{e_{0, 0}+\sum_{j\in J_{0, +}}[e_{0, j}-\mu_{0, j}(x)]-
E^{o}_{0}\} \\
&+2\sum^{\infty}_{n=1}\{\mu^{o}_{n}(x) 
+\sum_{j\in J_{0, +}}[e_{n, j}-\mu_{n, j}(x)]-E^{o}_{2n}\} 
+\int\limits_{\sigma(H^{o})} d\lambda [1-2\xi(\lambda, x)] 
\tag 4.62 \\
&=2\{e_{0, 0}+\sum_{j\in J_{0, +}}[e_{0, j}-\mu_{0, j}(x)]-
E^{o}_{0}\} \\
&+2\sum^{\infty}_{n=1}\{\mu^{o}_{n}(x)+\sum_{j\in J_{n, +}}
[e_{n, j}-\mu_{n, j}(x)]-E^{o}_{2n}\} \\
&-(2/\pi)\int\limits_{\sigma(H^{o})} d\lambda\text{\rom{ Im}}\left\{\ln
\left[1+R\Sp r\\ \ell\endSp(\lambda)\frac{f_{\pm}(\lambda, x)^{2}}{|f_{\pm}
(\lambda, x)|^{2}}\right]\right\}, \quad x\in\Bbb R. \tag 4.63
\endalign
$$
If $\sigma_{p, n}(H)=\emptyset$, the corresponding expression 
$\{\dots\}$ in \rom{(4.60)}, \rom{(4.61)} is to be replaced by 
$\{E^{o}_{2n-1}-2\mu_{n, 0}(x)+E^{o}_{2n}\}$ if $n\in\Bbb N$ and 
deleted if $n=0$. Similarly, if $\sigma_{p, n}(H)=\emptyset$, the 
corresponding expression $2\{\dots\}$in \rom{(4.62)}, \rom{(4.63)} is 
to be replaced by $2\{\mu^{o}_{n}(x)-\mu_{n, 0}(x)\}$ if $n\in\Bbb N$ 
and deleted if $n=0$.
\endproclaim

\demo{Proof} The trace formula (4.53) follows from (1.6), (4.45) 
(4.47), and the Lebesgue dominated convergence theorem.  Equalities 
(4.54) and (4.55) are then clear from (4.59), (4.44), (4.48), and 
(4.49).  Equations (4.62) and (4.63) are obtained by combining (4.60), 
(4.61) and (4.19) observing the finite total gap length (4.21). \qed
\enddemo

We note that the analog of Remark 2.2 clearly holds in the present 
context.  Moreover, the threshold behavior of $\xi(\lambda, x)$ in 
(2.43)--(2.51) near $\lambda=0$ extends to the essential spectral band 
edges $\{E^{o}_{n}\}_{n\in\Bbb N_{0}}$ of $H$ in the current impurity 
scattering situation. In particular, assuming
$$
W\in L^{1}(\Bbb R; (1+x^{2})dx) \tag 4.64
$$
in addition to (4.22), one again distinguishes two cases depending on 
whether or not $H$ has a threshold resonance at $E^{o}_{n}$.

\example{Case I} $W(f_{-}(E^{o}_{n}), f_{+}(E^{o}_{n}))\neq 0$ and 
$f_{-}(E^{o}_{n}, x)f_{+}(E^{o}_{n}, x)\neq 0$.
\endexample
\flushpar Then
$$
R^{r(\ell)}(E^{o}_{n})=-1 \tag 4.65
$$
and
$$
\xi(\lambda, x)\operatornamewithlimits{=}\Sb \lambda\to E^{o}_{n}\\ 
\lambda\in\sigma (H^{o}) \endSb 
O(|\lambda -E^{o}_{n}|^{1/2}) \ \text{in case I}. \tag 4.66
$$

\example{Case II} $W(f_{-}(E^{o}_{n}), f_{+}(E^{o}_{n}))=0$ and 
$f_{-}(E^{o}_{n},x)f_{+}(E^{o}_{n}, x)\neq 0$.
\endexample
\flushpar Then one can show that
$$
\xi(\lambda, x)\operatornamewithlimits{=}\Sb \lambda\to E^{o}_{n}\\
\lambda\in\sigma (H^{o}) \endSb 
\frac12 + O(|\lambda - E^{o}_{n}|^{1/2}) \ \text{in case II}. 
\tag 4.67
$$ 

\vskip 0.4in
\example{Acknowledgments}  We would like to thank Z.~Zhao for numerous 
helpful discussions. F.G.~is indebted to the Department of 
Mathematical Sciences of the University of Trondheim, Norway, and the 
Department of Mathematics at Caltech, Pasadena, CA for the hospitality 
extended to him in the summer of 1993.  Support by the Norwegian 
Research Council for Science and Humanities (NAVF) and by Caltech is 
gratefully acknowledged.  H.H.~is grateful to the Norwegian Research 
Council for Science and the Humanities (NAVF) for support.
\endexample

\vskip 0.4in
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\enddocument