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absolutely continuous spectrum, Schrodinger operator
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\begin{thebibliography}{10}
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one-dimensional Schr\"o\-dinger operators with slowly decaying potentials}, J. Funct. Anal. 179 (2001) 426--447
\bibitem{CHRIST-KIS-SV} M.~Christ, A.~Kiselev, {\it WKB and Spectral Analysis of One-Dimensional Schr\"odinger Operators with slowly varying potentials}, to appear in Comm. Math. Phys.
\bibitem{CHRIST-KIS-NOTES} M.~Christ, A.~Kiselev, {\it One-Dimensional Schr\"odinger Operators with
Slowly Decaying Potentials: Spectra and Asymptotics}, Notes for IPAM tutorial, March 19--23, 2001.
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New York-Toronto-London, 1955.
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operators with square summable potentials}, Comm. Math. Phys. {\bf 203} (1999) 341--347.
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completeness in two body quantum systems}, J. Funct. Anal. {\bf 23} (1976) 218--238.
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Schr\"odinger operators}, J. Statist. Phys. {\bf 40} (1985) 621--630.
\bibitem{DELYON-S-S} F.~Delyon, B.~Simon, B.~Souillard, {\it From power pure point to continuous spectrum in
disordered systems}, Ann. Inst. H. Poincar\'e {\bf 42} (1985) 283--309.
\bibitem{DENISOV} S.~A.~Denisov {\it On the application of some M. G. Krein's results to the spectral
analysis of Sturm--Liouville operators} Preprint.
\bibitem{EASTHAM} M.~S.~P.~Eastham, {\it The Spectral Theory of Periodic Differential Equations,}
Scottish Academic Press, Edinburgh and London, 1973.
\bibitem{GILB-PEAR} D.~J.~Gilbert, D.~B.~Pearson, {\it On subordinacy and analysis of the spectrum of
one-dimensional Schr\"odinger operators}, J. Math. Anal. Appl. {\bf 128} (1987), 30--56
\bibitem{HORM} L.~H\"ormander, {\it The analysis of linear partial differential operators IV.}, Springer Verlag, New York,
1995.
\bibitem{JITO-LAST} S.~Jitomirskaya, Y.~Last, {\it Power law subordinacy and singular spectra. I.
Half line operators,} Acta Math., to appear
\bibitem{JOST-PAIS} R.~Jost, A.~Pais {\it On the Scattering of a Particle by a Static Potential},
Phys. Rev. {\bf 82} no. 6 (1951) 840--851.
\bibitem{KILLIP-PHD} R.~Killip {\it Perturbations of One-Dimensional Schr\"odinger Operators Preserving the
Absolutely Continuous Spectrum}, Ph.D. Thesis, California Institute of Technology, 8 August 2000.
Preprint number 00-326 at {\tt http://rene.ma.utexas.edu/mp\verb!_!arc/}
\bibitem{KISELEV-1} A.~Kiselev, {\it Absolutely continuous spectrum of one-dimensional Schr\"o\-dinger operators and
Jacobi matrices with slowly decreasing potentials}, Comm. Math. Phys. {\bf 179} (1996), 377--400.
\bibitem{KISELEV-PRE} A.~Kiselev, {\it Preservation of the absolutely continuous spectrum of Schr\"o\-dinger equation under
perturbations by slowly decreasing potentials and a.e. convergence of integral operators}, Preprint number 96-470 at
{\tt http://rene.ma.utexas.edu/mp\verb!_!arc/}
\bibitem{KISELEV-2} A.~Kiselev, {\it Stability of the absolutely continuous spectrum of the Schr\"o\-dinger equation
under slowly decaying perturbations and a.e. convergence of integral operators}, Duke Math. J. {\bf 94} (1998), 619--646.
\bibitem{KLS} A.~Kiselev, Y.~Last, B.~Simon, {\it Modified Pr\"ufer and EFGP transforms and the spectral
analysis of one-dimensional Schr\"o\-dinger operators,} Comm. Math. Phys. {\bf 194} (1998), 1--45.
%\bibitem{KOTANI} S.~Kotani, {\it Ljapunov indices determine absolutely continuous spectra of
%stationary random one-dimensional Schr\"odinger operators} in {\it Stochastic analysis
%(Katata/Kyoto, 1982)}, 225--247, North-Holland, Amsterdam, 1984
\bibitem{KOTANI-USHIROYA} S.~Kotani, N.~Ushiroya, {\it One-dimensional Schr\"o\-dinger operators with random
decaying potentials}, Comm. Math. Phys. {\bf 115} (1988), 247--266.
\bibitem{OLEG} O.~Kovrijkine, private communication.
%\bibitem{MAGNUS} W.~Magnus, S.~Winkler, {\it Hill's Equation} Dover, New York, 1979.
\bibitem{MNV} S.~Molchanov, M.~Novitskii, B.~Vainberg, {\it First KdV integrals and absolutely continuous
spectrum for 1-D Scr\"odinger operator}, Comm. Math. Phys. {\bf 216} (2001), 195--213.
\bibitem{MOLCHANOV} S.~Molchanov, unpublished.
\bibitem{NABOKO} S.~N.~Naboko, {\it On the dense point spectrum of Schr\"o\-dinger and Dirac operators}, Teoret. Mat. Fiz.
{\bf 68} (1986) 18--28.
\bibitem{NOVIKOV} S.~Novikov, S.~V.~Manakov, L.~P.~Pitaevskii V.~E.~Zakharov
{\it Theory of Solitons: the Inverse Scattering Method}, a translation of
{\it Teoriia solitonov}, Consultants Bureau, New York, 1984.
\bibitem{PEARSON} D.~B.~Pearson, {\it Singular Continuous measures in scattering theory}, Comm. Math. Phys. {\bf 60} (1978), 13--36.
%\bibitem{REED-SIMON2} M.~Reed, B.~Simon, {\it Methods of Modern Mathematical Physics. II.
%Fourier Analysis, Self-Adjointness} Academic Press, New York-London, 1975.
%\bibitem{REED-SIMON4} M.~Reed, B.~Simon, {\it Methods of Modern Mathematical Physics. IV. Analysis of Operators},
%Academic Press, New York-London, 1978.
%\bibitem{REED-SIMON3} M.~Reed, B.~Simon, {\it Methods of Modern Mathematical Physics. III. Scattering Theory},
%Academic Press, New York-London, 1979.
\bibitem{REMLING-1} C.~Remling, {\it A probabilistic approach to one-dimensional Schr\"o\-dinger operators with sparse
potentials}, Comm. Math. Phys. {\bf 185} (1997) 313--323.
\bibitem{REMLING-2} C.~Remling, {\it The absolutely continuous spectrum of one-dimensional Schr\"o\-dinger operators with
decaying potentials}, Comm. Math. Phys. {\bf 193} (1998) 151--170.
\bibitem{REMLING-3} C.~Remling, {\it Bounds on embedded singular spectrum for one-dimensional Schr\"o\-dinger operators},
Proc. Amer. Math. Soc. {\bf 128} (2000) 161--171.
\bibitem{SIMON-TI} B.~Simon, {\it Trace Ideals and Their Applications}, London Mathematical Society Lecture Note Series
35, Cambridge University Press, 1979.
\bibitem{SEMIGROUPS} B.~Simon, {\it Schr\"o\-dinger semigroups}, Bull. Amer. Math. Soc. {\bf 7} (1982) 442--526.
\bibitem{SIMON-SJM} B.~Simon, {\it Some Jacobi matrices with decaying potential and dense point spectrum},
Comm. Math. Phys. {\bf 87} (1982) 253--258.
%\bibitem{SIMON-KT} B.~Simon, {\it Kotani theory for one dimensional stochastic Jacobi matrices},
%Comm. Math. Phys. {\bf 89} (1983), 227-234
\bibitem{SIMON-SCSI} B.~Simon, {\it Operators with singular continuous spectrum: I. General operators},
Ann. of Math. {\bf 141} (1995) 131--145.
\bibitem{SIMON-BNDD} B.~Simon, {\it Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schr\"odinger
operators}, Proc. Amer. Math. Soc. {\bf 124} (1996) 3361--3369.
\bibitem{SIMON-DPS} B.~Simon, {\it Some Schr\"o\-dinger operators with dense point spectrum},
Proc. Amer. Math. Soc. {\bf 125} (1997) 203--208.
\bibitem{STOLZ} G.~Stolz, {\it On the absolutely continuous spectrum of perturbed Sturm-Liouville
operators,} J. Reine Angew. Math. {\bf 416} (1991) 1--23.
\bibitem{TITCHMARSH} E.~Titchmarsh, {\it Eigenfunction expansions associated with second-order differential
equations}, Oxford, Clarendon Press, 1958.
\bibitem{WEIDMANN} J.~Weidmann, {\it Spectral Theory of Ordinary Differential Operators}, Lecture Notes in Mathematics 1258,
Springer-Verlag, Berlin, 1987.
\end{thebibliography}
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\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
%\usepackage{showkeys}
\hyphenation{di-men-sion-al}
% ac subsripts
\newcommand{\ac}{\mathrm{ac}}
\newcommand{\sigac}{{\sigma_\ac}}
\newcommand{\Sigac}{{\Sigma_\ac}}
% Sets
\newcommand{\Reals}{{\bf R}}
\newcommand{\Cmplx}{{\bf C}}
\newcommand{\Ints}{{\bf Z}}
\newcommand{\Nats}{{\bf N}}
\newcommand{\psipm}{\psi^{ _\pm}}
\newcommand{\psip}{\psi^{ _+}}
\newcommand{\psim}{\psi^{ _-}}
\newcommand{\tri}{\triangle}
\newcommand{\barr}{\overline}
\newcommand{\dde}{\frac{d\phantom{\epsilon}}{d\epsilon}}
\newcommand{\Ltwoloc}{L^2_{\mathrm{loc}}}
\renewcommand{\Im}{{\mathrm{Im}}}
\renewcommand{\Re}{{\mathrm{Re}}}
\renewcommand{\det}{\mathrm{det}}
% Math Operators
\DeclareMathOperator*{\wlim}{weak-lim}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\dist}{dist}
% Simple list
\newcounter{smalllist}
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\begin{list}{{\rm\roman{smalllist})}}%
{\setlength{\topsep}{0mm}\setlength{\parsep}{0mm}\setlength{\itemsep}{0mm}%
\setlength{\labelwidth}{2em}\setlength{\leftmargin}{2em}\usecounter{smalllist}}%
}{\end{list}}
% Theorem environments
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{prop}[lemma]{Proposition}
\newtheorem{coro}[lemma]{Corollary}
\newtheorem*{coroFL3}{Corollary \ref{FreeHs}}
\newtheorem*{coroFL4}{Corollary \ref{FreeL4}}
\newtheorem*{coroPL3}{Corollary \ref{PeriL3}}
\newtheorem*{coroSV}{Corollary \ref{SVary}}
\newtheorem*{coroST}{Corollary \ref{Stark}}
\newtheorem{definition}{Definition}
\title[Perturbations of 1-D Schr\smash{\"o}dinger operators]{Perturbations of One-Dimensional Schr\"odinger
Operators Preserving the Absolutely Continuous Spectrum}
\author{Rowan Killip}
\address{Institute for Advanced Study, Princeton NJ 08540, USA}
\date{\today}
\begin{document}
\begin{abstract}
The stability of the absolutely continuous spectrum of one-di\-men\-sion\-al Schr\"o\-dinger operators,
$$
[Hu](x) = -u''(x) + q(x)u(x),
$$
under perturbations of the potential is discussed. The focus is on demonstrating this stability
under minimal assumptions on how fast the perturbation decays at infinity.
A general technique is presented together with sample applications. These include the following:
for an operator with a periodic potential, any perturbation $V\in L^2$ preserves the a.c. spectrum.
For the Stark operator, the same is true for pertubations with $\int |V(t^2)|^2\, dt <\infty$.
Both of these results are known to be optimal, in the sense that the integrability index cannot
be increased.
\noindent
2000 MSC: 34L40, 47E05, 34B20, 81Q10 and 81Q15.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{S:In}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main object we shall study, is the Schr\"odinger operator on $\Reals^+=[0,\infty)$
with a Dirichlet boundary condition at the origin:
\begin{equation}\label{Defn:H}
[Hu](x) = -u''(x) + q(x)u(x) \ \text{ with }\ u(0)=0.
\end{equation}
Except in Section 6, where $q(x)=-x$, we consider only potentials $q\in\Ltwoloc$ for which
\begin{equation}\label{Kay}
\sup_n \int_n^{n+1} |q_-(x)|^2 \,dx < \infty.
\end{equation}
Here $q_-(x) = \min\{0,q(x)\}$.
For the questions
we address, operators with other choices of boundary condition or on the whole line can be reduced
to (\ref{Defn:H}). This is explained near the end of this section.
Much is known about the spectral analysis of Sturm-Liouville operators \cite{COD-LEV,TITCHMARSH}.
For example, there is a measure $d\mu$ on $\Reals$ so that the operator $H$ is unitarily equivalent to
$$
f(E) \mapsto Ef(E) \text{ acting in } L^2(\Reals;d\mu).
$$
This measure, known as the spectral measure, expresses all the spectral properties
of $H$. In particular, the the absolutely continuous spectrum of $H$, $\sigac(H)$, is equal to the
support of the absolutely continuous part of $d\mu$, $\supp(d\mu_\ac)$. The measure
$d\mu$ also permits a simple definition of the essential support of the absolutely continuous
spectrum of $H$, $\Sigac(H)$. It is the unique, up to sets of Lebesgue measure zero, set $\Sigma$
such that $\chi_\Sigma(E)dE$ and $d\mu_\ac$ are mutually absolutely continuous.
The problem we wish to discuss is the following: which perturbations of the potential,
$q(x)\mapsto q(x)+V(x)$, preserve $\Sigac(H)$? More generally, which perturbations preserve
some subset of $\Sigac(H)$? This question has generated considerable activity recently.
Attracting most attention, is the question of how slowly $V$ may decay at infinity,
if no smoothness assumptions are imposed.
The stimulus for these studies was a dichotomy that arose in the study of sparse and
random decaying potentials. Let $H_0$ denote the operator $H$ with $q\equiv0$ then, $H_0$
has purely absolutely continuous spectrum with $\sigma_\ac=\Sigac=[0,\infty)$.
A number of authors (e.g.~\cite{KLS,KOTANI-USHIROYA,PEARSON,REMLING-1}) have studied the
spectral properties of $H_0+V$ when $V$ decays at infinity and is either sparse or
random.
It was noticed that as the decay rate was reduced, the absolutely continuous spectrum disappeared
and was replaced by point or singular continuous spectrum. But in each case, the transition
point was the same. Namely, for $V\in L^2$, the a.c.~spectrum remained, $\Sigac=[0,\infty)$.
While for $V\notin L^2$, it dissolved completely, $\Sigac=\emptyset$.
Building on these results, Simon \cite{SIMON-SCSI} showed that if $p>2$, then
$\sigac(H_0+V)=\emptyset$ for a dense $G_\delta$ of $V\in L^p$.
A similar $\ell^2$-dichotomy was observed for the descretized operator
\begin{equation}\label{Defn:h}
[hu](n) = u(n+1) + u(n-1) + q(n)u(n) \ \text{ with }\ u(0)=0
\end{equation}
acting in $\ell^2(\Nats)$. The relevant publications are \cite{DELYON,DELYON-S-S,SIMON-SJM,SIMON-SCSI}.
Naturally, this phenomenon led to the following statement, originally conjectured by Kiselev,
Last and Simon \cite{KLS} and later proved by Deift and Killip:
\begin{theorem}\label{Deift-K}\cite{DEIFT-K}
If $V\in L^2$ then $\Sigac(H_0+V)=\Sigac(H_0)=[0,\infty)$.
\end{theorem}
\noindent
From the results described earlier, this is optimal in the sense that $L^2$ cannot be replaced by $L^p$
for any $p>2$. One should also note that the spectrum may be far from purely
absolutely continuous. For any function $f(x)$ with $\lim_{x\to\infty}f(x)=\infty$, there are examples of
potentials $V(x)\leq f(x)/x$ (for large $x$) which have point
spectrum dense in the interval $[0,\infty)$ \cite{NABOKO,SIMON-DPS}. For potentials with power-law decay,
Remling \cite{REMLING-3} has given a bound on the possible Hausdorff dimension of any embedded singular continuous
spectrum. However, the question of whether such singular continuous spectrum may exist is still open.
The initial work in the direction of Theorem \ref{Deift-K}, \cite{CHRIST-KIS-JAMS,CHRIST-KISELEV-REMLING,KISELEV-1,
KISELEV-PRE,KISELEV-2,MOLCHANOV,REMLING-2}, employed the following potent theorem from subordinacy theory
\cite{GILB-PEAR,JITO-LAST, SIMON-BNDD}: suppose that for almost every energy $E$, in a Borel set $S$,
all solutions of
\begin{equation}\label{ODE:1}
-u''(x) + V(x)u(x) = E u(x)
\end{equation}
are uniformly bounded, then $S\subseteq \Sigac(H_0+V)$. Showing that solutions of (\ref{ODE:1}) are bounded
is a difficult problem. By employing Pr\"ufer variables (Remling), or the Harris--Lutz transformation
(Christ--Kiselev), and a careful analysis of the resulting (multi-dimensional) Volterra operators
a weakener version of Theorem \ref{Deift-K} was proved. Specifically, if there exists $\epsilon>0$
such that $x^\epsilon V(x)\in L^2$ then $\Sigac(H_0+V)=[0,\infty)$. Quite recently, Christ and Kiselev
extended their method to treat $V\in L^p$, $p<2$ \cite{CHRIST-KIS-PRE} and announced a proof of
the existence of wave operators in a related setting \cite{CHRIST-KIS-NOTES}.
While it may seem natural to prove the existence of absolutely continuous spectrum directly
by showing the existence of modified wave operators, no one has succeeded in doing so under
such weak assumptions. The strongest known
results on the existence of wave operators require that $V$ or one of it's derivatives is at least
integrable \cite{HORM}. In a similar vein, is the following result of Weidmann \cite{WEIDMANN}:
If $V$ is the sum of two terms, one integrable and the other tending to zero
at infinity and of bounded variation, then $\Sigac(H_0+V)=[0,\infty)$. Extending this result,
Stolz showed that if $q$ is periodic then $\Sigac(H+V)=\Sigac(H)$ for such $V$ \cite{STOLZ}.
The method of Christ and Kiselev is quite robust and can be used to prove results for perturbations
of $H$ when $q\not\equiv0$. For example:
\begin{theorem}\label{Christ-K}
\cite{CHRIST-KIS-JAMS} Let $S$ be a Borel subset of $\Reals$ and let $V$ be a potential
which obeys $x^\epsilon V(x)\in L^2$ for some $\epsilon>0$.
Suppose that for each $E\in S$, there is a solution, $\theta(x,E)$, to
\begin{equation}\label{ODE:1q}
-u''(x) + q(x)u(x) = E u(x)
\end{equation}
for which
\begin{SmallList}
\item $\theta(x;E)$ is a uniformly bounded function of $x$;
\item $\theta(x;E)$ and its complex conjugate, $\bar\theta(x;E)$, are linearly independent; and
\item the integral operator
\begin{equation}\label{CKIntOp}
[Tf](E) = \int_0^\infty \exp\Big\{
\tfrac{i}{\Im\,\theta\rule{0mm}{1.7ex}{\bar\theta}^\prime}\int_0^x V(t)\,dt
\Big\} \theta(x,E)^2 f(x)\,dx
\end{equation}
is a bounded mapping $L^2(\Reals^+;dx)\to L^2(S;dE)$.
\end{SmallList}
Then for Lebesgue almost every $E\in S$, all solutions of
\begin{equation}\label{CK-ode}
-u''+qu+Vu=Eu
\end{equation}
are bounded. By subordinacy theory, this implies $S\subseteq \Sigac(H+V)$.
\end{theorem}
\noindent
{\it Remarks.} 1. Notice that conditions i) and ii) imply $S\subseteq \Sigac(H)$, by the subordinacy theorem.\\
2. The methods from \cite{CHRIST-KIS-PRE}, suggest a similar result for $V\in L^p$,
when $p<2$.\\
3. The hypotheses of this theorem have been verified for background potentials, $q$, which are periodic
\cite{CHRIST-KIS-JAMS}.
\smallskip
Our first result is an analogue of this theorem for $V\in L^2$. Or, to say it differently,
an analogue of Theorem \ref{Deift-K} in the presence of a background potential $q$. Before we
state this theorem, we wish to recall a definition.
For each $z\in \Cmplx\setminus\Reals$, the resolvent $R(z)=(H-z)^{-1}$ can
be represented as an integral operator
$$
[R(z)f](x) = \int G(x,y;z) f(y)\,dy.
$$
The kernel of this operator, $G$, is known as the Green function and for each $x,y$ it is an analytic function
of $z\in\Reals\setminus\Cmplx$.
\begin{theorem}\label{Th:det2}
Suppose $q\in\Ltwoloc$ obeys (\ref{Kay}), $V\in L^2(\Reals^+)$ and let $I$ be a closed interval contained in $\Sigac(H)$. If
\begin{SmallList}
\item for all $x,y\in\Reals^+$, $G(x,y;z)$ extends continuously from $z\in\Cmplx^+$ to $z\in I$; and
\item for some $r\in\Nats$ and all functions $w\in C^r(\Reals)$, supported by $I$, the integral
\begin{equation}\label{fHilb}
\int_I w(E)\; \Re\big[G(x,x;E)\big] \, dE
\end{equation}
defines a function in $L^2(\Reals^+;dx)$;
\end{SmallList}
then $I\subseteq\Sigac(H+V)$.
\end{theorem}
\noindent
{\it Remarks.} 1. Assumption i) is equivalent to the condition that $G(x,y;z)$ extends continuously
for {\it some}\/ pair $x,y\in\Reals^+$.
\noindent
2. For assumption ii), it is sufficient that (\ref{fHilb}) defines a function in $L^2(\Reals^+;dx)$
for a very specific family of functions $w$. The functions we will use originate from the harmonic
measure of tents sitting over $I$.
\noindent
3. The absence of the exponentiated integral, makes (\ref{fHilb}) easier to control than
(\ref{CKIntOp}).
\noindent
4. The assumptions on $q$ are not intended to be optimal. However a trace ideal estimate we use
(\ref{ti-est}) is only known in slightly more generality ($q\in K^1_{\mathrm{loc}}$ and $q_-\in K^1$ see
\cite{SEMIGROUPS}).
In particular, it is not known for general classes when $H$ is unbounded below.
We will treat one such example, $q(x)=-x$, by ad hoc methods.
\smallskip
Hypothesis ii) above, is quite close to the third hypothesis of Theorem \ref{Christ-K}.
Compare $\int [Tf](E) w(E)dE$ with (\ref{fHilb}) noting that $\Re\,G(x,x;E)$ is the product of
two solutions of (\ref{ODE:1q}). Note also that there is no real freedom in the choice for the
solution $\theta(x,E)$ in Theorem \ref{Christ-K}. For example, in the free case ($q\equiv0$),
any choice for $\theta$, other that multiples of $\exp(\pm ix\sqrt{E})$, makes $T$ unbounded.
Similarly, while (\ref{fHilb}) holds in all the examples examined later, it does not if one
considers the imaginary part rather than the real part. This is because $\Im\,G(x,y;z)$
is the kernel of a positive operator. We would hazard to conjecture that for energies $E$
in the interior of $\Sigac(H)$, $\Re\,G(x,x;E)$ is an oscillatory function of $x$. If
true, this would suggest that hypothesis ii) above may be redundant.
The technique we will employ to prove Theorem \ref{Th:det2} is completely different from that of
Christ and Kiselev. By their own admission \cite{CHRIST-KIS-PRE}, extending their methods to $V\in
L^2(\Reals^+)$ would be extremely difficult.
As discussed above, Theorem \ref{Christ-K} was proved by showing that for almost every energy $E\in
S$, all solutions of (\ref{CK-ode}) are uniformly bounded (in $x$). The secret to \cite{DEIFT-K},
expressed more directly by Molchanov, Novitskii and Vainberg \cite{MNV}, is the following:
it suffices to control the average (in energy) behaviour of solutions at each $x$. Moreover,
this is equivalent to the control of a scattering theoretic quantity.
In both \cite{DEIFT-K} and \cite{MNV} this is done using the the Buslaev--Faddeev--Zakharov trace formulae
\cite{NOVIKOV}.
These formulae express the first integrals of the KdV
equation in terms of scattering data, specifically, the transmission co-efficient and the
bound-state eigenvalues. In this way, the first integrals for KdV can be used to bound the
transmission co-efficient. This in turn, can be used to control the behaviour of solutions \cite{MNV},
or may be used directly to obtain the necessary spectral information \cite{DEIFT-K}.
Examples of first integrals are $\int V$, $\int V^2$, $\int [V']^2+2V^3$. The second of these was
used to prove Theorem~\ref{Deift-K}. Molchanov, Novitskii and Vainberg used the higher order
formulae to prove that the condition $V\in L^2$ in this theorem may be replaced by $V\in L^{n+1}$
and $V^{(n-1)}\in L^2$ for some integer $n\geq1$.
Recent work by Denisov \cite{DENISOV} has shown that if $q\equiv 0$ and the integral of
$V$, $\int_x^\infty V(t) dt$, is square integrable then the absolutely continuous spectrum
is preserved. This result rests on work of Krein with intimate connections to the forward
and inverse spectral theory of the Dirac system. As this suggests, the result may also
be derived by applying the methods of \cite{DEIFT-K} to the non-linear Schr\"odinger equation.
Unfortunately, the trace formula method is restricted to the case when there is no background
potential, $q\equiv0$, and so does not provide a method for proving Theorem \ref{Th:det2}.
As we shall also see, the trace formula method does not always give the best results:
While the $L^2$ is optimal in the $L^p$ setting, for $V\in L^3$ (or $L^4$) the condition
$V'\in L^2$ (respectively $V''\in L^2$) of \cite{MNV} can be improved. See Corollaries
\ref{FreeHs} and \ref{FreeL4}.
A third, though perhaps purely aesthetic, fault the trace formula method is the necessity of
using whole-line operators to study a half-line problem. (For results on whole-line operators,
one must then glue together results on each half-line.)
In order to state the main theorem, let us introduce the following notation:
\begin{equation}\label{Kdefn}
K(x_1,\ldots,x_n;z) = \Re \prod_{j=1}^n G(x_j,x_{j+1};z)
\end{equation}
with the identification $x_{n+1}=x_1$.
\begin{theorem}\label{th4}
Suppose $q\in\Ltwoloc$ obeys (\ref{Kay}) and $I\subseteq\Sigac(H)$ is a compact interval.
If there is an integer $p\geq2$ so that
\begin{SmallList}
\item $V\in L^p(\Reals^+);$
\item for any $x,y\in\Reals^+$, $G(x,y;z)$ extends continuously from $z\in\Cmplx^+$ to $z\in I$; and
\item there is a positive integer, $r$, and a sequence, $V_n$, of compactly supported functions,
converging to $V$ in $L^p$, such that
\begin{equation*}
\sup_{n}\left| \int_{[0,\infty)^\ell}\!K(x_1,\ldots,x_\ell;E+i0) V_n(x_1)\cdots V_n(x_\ell)\,
dx_1\ldots dx_\ell\,w(E)\,dE\right| < \infty
\end{equation*}
for all integers $1\leq \ell\leq (p-1)$ and all functions $w\in C^r(\Reals)$ which are supported by $I$;
\end{SmallList}
then $I\subseteq\Sigac(H+V)$.
\end{theorem}
Theorem \ref{Th:det2} is a direct consequence of the above with $p=2$. This
is because $K(x;E+i0)=\Re\,G(x,x;E)$.
Here are some sample applications. The details and further applications are given in
Sections \ref{S:Ex} and \ref{S:St}.
\begin{coroFL3}
Suppose $q\equiv 0$ and $V\in L^3(\Reals^+)$. If $V$ can be written as a finite sum of functions
each of which belongs to a homogeneous Sobolev space $\dot {\sf H}{}^s$ $(s\in\Reals)$ then
$\Sigac(H+V)=[0,\infty)$.
\end{coroFL3}
As mentioned earlier, this improves upon the results obtained by the trace formula method.
A still stronger statement is given as Corollary \ref{FreeL3}. Also improving on \cite{MNV}
is
\begin{coroFL4}
Suppose $q\equiv0$. If for some integer, $N\geq0$, $V$ can be written as
$V(x)=\sum_{\nu=-1}^N U_\nu(x)$ such that
$U_\nu \in L^4$ and $U_\nu^{(\nu)}\in L^2$ then $\Sigac(H+V)=[0,\infty)$.
For $\nu\geq0$, $U^{(\nu)}$ denotes the $\nu^{\mathrm{th}}$ derivative while
$U^{(-1)}(x)=\int_x^\infty U(t)\,dt$.
\end{coroFL4}
We consider three examples of non-zero background potentials. For $q$ periodic, we not only
extend the result of \cite{CHRIST-KIS-JAMS} to $V\in L^2$ but also obtain a result for $V\in L^3$:
\begin{coroPL3}
Suppose $q\in L^2_{\mathrm{loc}}$ is periodic with period one.
If $V\in L^2$, or both $V\in L^3$ and $\hat V\in \ell^\infty(L^2)$, then
$\Sigac(H+V)=\sigac(H)$.
\end{coroPL3}
We also treat a case where $q$ is slowly varying:
\begin{coroSV}
If $q(x)\to 0$ as $x\to\infty$, $\int^\infty_x |q'(t)|^2 dt \in L^2(dx)$ and
$\int^\infty_x |q''(t)| dt \in L^2(dx)$ then for each $V \in L^2$, $\Sigac(H+V)=[0,\infty)$.
\end{coroSV}
Corollaries \ref{FreeHs} and \ref{SVary} should be compared to a recent result of Christ and
Kiselev \cite{CHRIST-KIS-SV}: Suppose $q\equiv0$ and $V=V_0 + V_1$ where for some $p<2$,
$V_0\in L^p$, $V_1^{(n)}\in L^p$ and $V_1(x)\to0$ at infinity, then $\Sigac(H+V)=[0,\infty)$.
Our third example is the Stark operator, $q(x)=-x$. This
is not within the scope of Theorem \ref{th4}. Unable to treat any suitably broad class of
potentials, we develop an analogue of Theorem \ref{th4} for
this specific example: Theorem \ref{St:th} in Section~\ref{S:St}. In a recent pre-print,
Christ and Kiselev showed that the absolutely continuous spectrum is preserved for perturbations
which obey $|V(x)| \lesssim (1+|x|)^{-\epsilon-1/4}$ for some $\epsilon>0$. They further
prove that there are perturbations $\lesssim (1+|x|)^{-1/4}$ under which there is no
absolutely continuous spectrum. The following therefore, is optimal in the sense that
$\int |V(t^2)|^2 \,dt$ cannot be replaced by $\int |V(t^2)|^p \,dt$ with any $p>2$.
\begin{coroST}
If $\int |V(t^2)|^2\, dt < \infty$ then $\Sigac(-\frac{d^2\phantom{x}}{dx^2}-x+V(x))=\Reals$.
\end{coroST}
As mentioned at the beginning of this Introduction, the results given above can easily
be extended to whole-line operators, or to operators with other boundary conditions at
the origin. The reasoning is as follows: A change of boundary condition is a relatively
trace-class perturbation. The Birman--Kuroda Theorem thus implies that the absolutely continuous
parts of the two operators are unitarily equivalent. That is, the essential support of the
absolutely continuous spectrum is unchanged.
In the case of a whole-line operator, we can enforce a Dirichlet boundary condition at the
origin. This is also a relatively trace class perturbation and it decomposes the whole-line
operator into the direct sum of two half-line operators. Not only does the Birman--Kuroda Theorem
then show that the essential support of the absolutely continuous spectrum is the union of
those of the direct summands; it also shows that on their intersection, the absolutely continuous
spectrum of whole-line operator has multiplicity two. For a further discussion of the use
of Dirichlet decoupling, particularly in higher dimensions, see \cite{DEIFT-SIMON}.
Much of the material presented here appeared in a some-what different form in the author's
PhD thesis \cite{KILLIP-PHD}.
The remainder of this article is arranged as follows: in the next section we discuss the
Weyl theory of one-dimensional Schr\"odinger operators and relate it to the perturbation
determinant or Jost function. Section 3 ties this together with the theory of regularized
determinants to prove Theorem 4. Sections 4 and 5 give sample applications.
\medskip
\noindent{\it Acknowledgement}: The author received support from NSF grant DMS 9729992.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weyl theory and the Perturbation Determinant}\label{S:We}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As in the Introduction, we define $H$ by
\begin{equation}\label{Defn:H:2}
[Hu](x) = -u''(x) + q(x)u(x) \ \text{ with }\ u(0)=0.
\end{equation}
The material presented in this section requires very little of $q$. We will assume that
$q\in\Ltwoloc$ and that (\ref{Defn:H:2}) is limit point at infinity. This ensures that
$H$ is essentially self-adjoint. For (\ref{Defn:H:2}) to be limit point at infinity,
it is sufficient that $q\in\Ltwoloc$ obey (\ref{Kay}) or $q(x)\geq -C - Cx^2$ for some $C>0$.
For each $z\in \Cmplx$ we let $\theta(x;z),\phi(x;z)$ denote the solutions of
\begin{equation}\label{ODE:2}
-u'' + q u = z u
\end{equation}
which obey $\theta(0;z)=\phi'(0;z)=1$, $\theta'(0;z)=\phi(0;z)=0$. Here $u'$ denotes the derivative
with respect to $x$. For each $z\in \Cmplx\setminus\Reals$ there is a unique solution $\psi(x;z)$
of (\ref{ODE:2}) which obeys
\begin{gather}
\int_0^\infty |\psi(x;z)|^2\,dx < \infty \ \text{ and }\ \psi(0;z)=1.
\end{gather}
This solution is known as the Weyl solution. The Weyl $m$-function is defined by
$\psi(x;z)=\theta(x;z)+m(z)\phi(x;z)$ and is an analytic function of $z\in\Cmplx\setminus\Reals$.
In fact, $m(z)$ admits a unique representation
\begin{equation}\label{Hergoltz}
m(z) = B + \int \Big[\frac{1}{\omega-z} - \frac{\omega}{1+\omega^2}\Big] d\mu(\omega)
\qquad \forall z\in\Cmplx\setminus\Reals
\end{equation}
where $B\in\Reals$ and $d\mu$ is a positive measure with $\int (1+\omega^2)^{-1} d\mu$ finite.
Given $m(z)$ one may recover $d\mu$ as weak limit:
$$
\int\! f(E) d\mu(E) = \lim_{\epsilon\downarrow0}\tfrac{1}{\pi}\!\!\int\! f(E) \; \Im\,m(E+i\epsilon)\,dE
$$
for any continuous function $f$ of compact support. Moreover, if we write $m(E+i0)$ for
$\lim_{\epsilon\downarrow0} m(E+i\epsilon)$ then $m(E+i0)$
exists for Lebesgue-a.e.~$E\in\Reals$ and $\Im\,m(E+i0)=\pi\frac{d\mu}{dE}$.
The measure, $d\mu$, is called the spectral measure for $H$ and, as mentioned in the Introduction,
$H$ is unitarily equivalent to $f(E)\mapsto Ef(E)$ acting in $L^2(d\mu)$.
This implies $\Sigac(H)=\{E:\Im\,m(E+i0)>0\}$.
Given a perturbation $V$, one may similarly write down a Weyl theory for the operator $H+V$. Actually,
in this section, we will not deal with the true perturbation, but rather, obtain some {\it a priori}\/ information.
To this end, let $V_n$ be a sequence of square integrable functions of compact support and let $m_n$
denote the Weyl $m$-function for the operator $H+V_n$.
The resolvent of $H$, $R(z)=(H-z)^{-1}$, can be represented as an integral operator whose kernel is the
Green function:
\begin{equation}\label{Defn:G}
G(x,x';z) = \phi(x^{<};z)\psi(x^{>};z)
\end{equation}
where $x^{>}=\max\{x,x'\}$ and $x^{<}=\min\{x,x'\}$.
As $V_n$ has compact support, the operator $R(z)V_n$ is trace class for any
$z\in\Cmplx\setminus\Reals$. In this way,
\begin{equation}\label{Defn:a}
a_n(z) = \det\big[1+R(z)V_n\big]
\end{equation}
defines an analytic function on $\Cmplx\setminus\Reals$, known as the perturbation determinant.
For such $z$, $a_n(z)$ is non-zero: if $a_n(z)=0$ then $R(z)(H+V_n-z)=1+R(z)V_n$ is not invertible
which means $z\in\sigma(H+V_n)$. This is impossible because $\sigma(H+V_n)\subseteq\Reals$.
\begin{lemma}\label{2:lemma}
Suppose the Green function, $G$, extends continuously from $\Cmplx^+$ to a compact set
$I\subseteq\Reals$ then $a_n(z)$ also extends continuously to $I$.
Moreover, if\/ $\Im\,m(E+i0)$ is non-zero on $I$ then $\Im\,m_n(E+i0)$ is also non-zero
and
\begin{equation}\label{amm}
\big|a_n(E+i0)\big|^{-2} = \frac{\Im[m_n(E+i0)]}{\Im[m(E+i0)]} \qquad \forall E\in I.
\end{equation}
\end{lemma}
\begin{proof}
Because the Green function $G(x,y;z)$ extends continuously from $\Cmplx^+$ to $I$, so does
the Weyl solution $\psi(x;z)$. As $V_n$ is square integrable and of compact support,
one may solve the Volterra equation
\begin{equation*}
u_n(x;z) = \psi(x;z) + \int_x^\infty \big[\theta(x;z)\phi(y;z) - \phi(x;z)\theta(y;z)\big] V_n(y) u_n(y;z)\,dy
\end{equation*}
for any $z\in\Cmplx^+\cup I$. Moreover, $u_n(x;z)$ depends continuously on $z$,
solves $-u_n''+qu_n+Vu_n=zu_n$ and $u_n(x;z)=\psi(x;z)$ for all $x>L$, $L=\sup[\supp V_n]$.
The function $u_n$ is known as the Jost solution and $u(0;z)$ as the Jost function. The latter is
equal to the perturbation determinant: $u_n(0;z)=a_n(z)$ \cite{JOST-PAIS}.
The stated properties of $u_n$ prove that $a_n$ extends
continuously from $\Cmplx^+$ to $I$. In addition, the limiting value $u(0;E+i0)$ cannot be zero
because $\Im\,m(E+i0)\neq 0$. Therefore $a_n$ is continuous and non-zero on $\Cmplx^+\cup I$.
Notice that $u_n(x;z)/a_n(z)$ is the Weyl solution for $H+V_n$---it is square integrable for $z\in\Cmplx^+$
and $u_n(0;z)=a_n(z)$. To prove (\ref{amm}), recall that
$$
\Im\,m(E+i\epsilon) = \epsilon\int_0^\infty \big|\psi(x;E+i\epsilon)\big|^2 dx
$$
and similarly for $m$-function and Weyl solution for $H+V_n$. Namely $m_n$ and $u_n(x;z)/a_n(z)$.
Thus for $E\in I$,
\begin{align}
\notag
\Im\,m(E+i0) &= \lim_{\epsilon\to 0} \epsilon\int_0^\infty \big|\psi(x;E+i\epsilon)\big|^2 dx \\
\notag
&= \lim_{\epsilon\to 0} \epsilon\int_0^\infty \big|u_n(x;E+i\epsilon)\big|^2 dx \\
\label{asdf}
&= \lim_{\epsilon\to 0} |a_n(E+i\epsilon)|^2\Im\,m_n(E+i\epsilon).
\end{align}
While $\psi(x;z)=u_n(x;z)$ only for $x\geq L$, the continuity of $\psi,u$ (as functions of $\epsilon$)
shows that $\int |u_n|^2-|\psi|^2 dx$ is uniformly bounded as $\epsilon\to0$.
As $a_n(z)$ is known to be continuous, (\ref{asdf}) implies that $\Im\,m_n$ extends continuously to $I$ and
that (\ref{amm}) holds there.
\end{proof}
\noindent
{\it Remark.}\/ The equation (\ref{amm}) holds under much less stringent conditions---it is
just an expression of the fact that wave operators are isometries.
\smallskip
Now we come to the main result of this section.
\begin{prop}\label{mainprop} Suppose $V\in L^p(\Reals^+)$ {\rm(}for any $1\leq p\leq\infty${\rm)} and
$V_n$ is a sequence of square integrable functions of compact support which converge to $V$ in $L^p$.
Let $I$ be a compact
subset of $\Reals$ and let $w\!:\!I\to\Reals$ be continuous and almost everywhere positive. If the Green
function extends continuously from $\Cmplx^+$ to $I$, \/$\Im\,m(E+i0)>0$ for all $E\in I$ and
there is a constant $C$ such that
\begin{equation}\label{2:abound}
\int_I \log\big|a_n(E+i0)\big| w(E) dE \leq C
\end{equation}
for all $n$, then $I\subseteq\Sigac(H+V)$.
\end{prop}
\begin{proof} Let $n(z)$ denote the Weyl function for $H+V$ and let $d\nu$ denote the corresponding
spectral measure. As $V_n\to V$ in $L^p$, $m_n(z)\to n(z)$ uniformly on compact subsets of $\Cmplx^+$.
This implies that $d\mu_n$, the spectral measures for $H+V_n$, converge weakly to $d\nu$. In
particular, $\mu_n(I)$ is uniformly bounded.
Let us write $\log^\pm|z| = \max\{0,\pm\log|z|\}$ so that $\log|z|=\log^+|z|-\log^-|z|$. As $I$ is
compact, the continuous function, $\log[\Im\,m(E+i0)]$, is uniformly bounded from both below and above.
Employing the lower bound,
\begin{align*}
\int_I\log^-\Big|\frac{\Im\,m(E+i0)}{\Im\,m_n(E+i0)}\Big| w(E)dE
&\leq c + \int_I\log^+\big|\Im\,m_n(E+i0)\big| w(E)dE\\
&\leq c + \int_I \Im\,m_n(E+i0)\,w(E)dE\\
&\leq c + \int_I w(E)d\mu_n(E)\\
&\leq C'
\end{align*}
independently of $n$. Here we have used $\log^+|z|\leq |z|$, $\Im\,m_n=\frac{d\mu_n}{dE}$, $w\in L^\infty$
and the fact that $\mu_n(I)$ is uniformly bounded.
Using the above estimate, (\ref{2:abound}) implies
\begin{equation*}
\int_I\log^+\Big|\frac{\Im\,m(E+i0)}{\Im\,m_n(E+i0)}\Big| w(E)dE \leq C + C'
\end{equation*}
and so, as $\log[\Im\,m(E+i0)]$ is bounded above on $I$, there is a constant $C''$ such that
\begin{equation*}
\int_I\log^- \big|\Im\,m_n(E+i0)\big| w(E)dE \leq C''.
\end{equation*}
Employing Jensen's inequality and $\Im\,m_n=\frac{d\mu_n}{dE}$, we have
\begin{equation}\label{Jensen}
\int_K w(E)d\mu_n(E) \geq w(K) \exp\big\{-C''/w(K)\big\}
\end{equation}
for any compact $K\subset I$ of positive Lebesgue measure. Here $w(K)=\int_K w(E)dE$.
The measure $w(E)d\nu(E)$ is the weak limit of $w(E)d\mu_n(E)$ and so
$$
\int_K w(E)d\nu(E) \geq \limsup_{n\to\infty} \int_K w(E) d\mu(E)
$$
follows from the inner regularity of Borel measures (c.f. Lemma 2 of \cite{DEIFT-K}).
This combined with (\ref{Jensen}), shows that $\nu(K)>0$ for all compact $K\subseteq I$ of positive Lebesgue
measure. Hence $I\subseteq \mbox{ess-supp}(d\nu)=\Sigac(H+V)$.
\end{proof}
In Section \ref{S:Ex} it will be helpful to have an expression for the Green function other
than that given in (\ref{Defn:G}).
\begin{lemma}\label{Alt:G}
Suppose the Weyl solution, $\psi(x;z)$, extends continuously from $z\in \Cmplx^+$ to $E\in \Reals$.
If $\psi(\,\cdot\;\!;E)$ and $\bar\psi(\,\cdot\;\!;E)$ are linearly independent then the Green
function also has a continuous extension to $E$. Moreover, it is given by
\begin{align*}
G(x,y) &= \frac{\big[\psi(x)-\bar\psi(x)\big]\psi(y)}{W[\bar\psi,\psi]}
\qquad \forall\ x\leq y,\\
2\,\Re\, G(x,x) &= \frac{\psi(x)^2-\bar\psi(x)^2}{W[\bar\psi,\psi]}
\end{align*}
Here $W[\bar\psi,\psi](x) =\bar\psi(x)\psi'(x)-\psi(x)\bar\psi'(x)$ is the Wronskian of
$\bar\psi(x;E)$ and $\psi(x;E)$ and so is independent of $x$.
By homogeneity, the same formula holds if $\psi$ is merely a multiple of the Weyl solution.
\end{lemma}
\begin{proof} This is a purely computational consequence of $\phi(x;z)\,\Im\,m(z) = \Im\,\psi(x;z)$
and $2i\,\Im\,m(z) = \bar\psi(0)\psi'(0)-\psi(0)\bar\psi'(0)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Regularized determinants}\label{S:Re}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Fredholm determinants are only defined for trace class operators, however it is possible to
extend them to higher trace ideals, $\mathfrak{I}_p=\{A:\tr(|A|^p)<\infty\}$, by adding
convergence factors. Specifically, for each $p\in\Nats$ we write
\begin{align}
\det_p(1+A) &= \det(1+A)\,\exp\bigg\{\sum_{k=1}^{p-1} \tfrac{(-1)^k}{k} \tr\big(A^k\big)\bigg\}.
\end{align}
Whilst this definition only works for elements of trace class, $A\in\mathfrak{I}_1$, the resulting
function does extend continuously to $\mathfrak{I}_p$. Indeed one has
\begin{align}
\label{detest}
\big|\det_p(1+A)\big| &\leq \exp\{C_p\|A\|_p^p\}\qquad\forall p\in\Nats, A\in \mathfrak{I}_p
\end{align}
where the $C_p$ are universal constants and $\|A\|_p^p=\tr(|A|^p)$ denotes the $p$th power of the norm
on $\mathfrak{I}_p$. For more on regularized determinants, see Section 9 of \cite{SIMON-TI}.
\begin{lemma}\label{TIest}
Suppose $q\in\Ltwoloc$ obeys (\ref{Kay}) and $V$ is of compact support. For each integer $p\geq 2$ there is
a constant $C$ so that
\begin{equation}\label{ti-est}
\big\| R(E+i\epsilon)V\big\|_p^p
\leq C \epsilon^{-p} (1+\epsilon^2)^{1/2}(1+E^2)^{1/2} \|V\|^p_{L^p}
\end{equation}
for all $E\in\Reals$ and all $\epsilon>0$. Consequently, for a new constant $C$,
\begin{equation}\label{detpb}
\Big| \log\big| \det_p\big[1+R(E+i\epsilon)V\big] \big| \Big|
\leq C \epsilon^{-p}(1+\epsilon)(1+E^2)^{1/2} \|V\|_{L^p}^p.
\end{equation}
\end{lemma}
\begin{proof} The first estimate follows directly from Theorem B.9.3 of \cite{SEMIGROUPS}
using $f(x)=(1+|x|)^{-1/2}$.
For the remainder of this proof, we will write $R$ for $(H-E-i\epsilon)^{-1}$ and $\tilde{R}$
for $(H+V-E-i\epsilon)^{-1}$. To prove the second inequality we use
\begin{equation}\label{dlog}
\dde \log\big[ \det_p(1+RV)\big] = (-1)^p \tr\big( i\tilde{R} (RV)^p \big)
\end{equation}
which may be checked by induction. By applying the
trace ideal version of H\"older's inequality and (\ref{ti-est}), this implies
\begin{align*}
\left| \dde \log\big[ \det_p(1+RV)\big] \right| & \leq \|RV\|_p^p\, \|\tilde{R}\|
\leq C \epsilon^{-p-1}(1+\epsilon)(1+E^2)^{1/2} \|V\|_{L^p}^p.
\end{align*}
Here $\|\tilde{R}\|$ is the operator norm of $\tilde{R}$ and so bounded by $\epsilon^{-1}$.
As $V$ is of compact support, it is easy to show that
$$
\lim_{\epsilon\to\infty} \det_p\big[1+R(E+i\epsilon)V\big] = 1 \quad \forall E\in\Reals.
$$
Equation (\ref{detpb}) now follows from our derivative bound and the Fundamental Theorem of Calculus.
\end{proof}
This lemma shows how to control the regularized determinant, $\det_p[1+R(z)V]$ for $\Im\,z>0$. But Proposition
\ref{mainprop} suggests we need an {\it a priori}\/ estimate of the normal determinant for real energies.
The key to studying real energies is the fact that $f_n(z)=\log\big|\det_p[1+R(z)V_n]\big|$ is harmonic.
The regularization is controlled by fiat: assumptions ii) and iii) of
Theorems 3 and 4 respectively.
In the next section we will see that that these assumptions do in fact hold in applications.
If $I=[a,b]$, we define the tent $\tri_\alpha$ over $I$ to be the solid triangle with
vertices $a$, $b$ and $\frac{a+b}{2} + i\frac{b-a}{2}\tan(\alpha)$. We also define $z_\alpha$ to be
the centroid (centre of mass) of this triangle and partition $\partial\tri_\alpha$ into
$I\cup\Lambda_\alpha$. This construction is depicted more clearly in Figure 1.
\begin{figure}[ht]
\begin{center}
%\fbox{
\setlength{\unitlength}{1mm}
\begin{picture}(80,40)(-40,-5)
\put(-23.5,23.5){$\Lambda_\alpha$}
\put(-19,25){\vector(2,-1){8}}
\put(-19,25){\vector(4,-1){30}}
\put(-30, 0){\circle*{2}}\put(-31,-3.9){$a$}
\put( 30, 0){\circle*{2}}\put( 29,-3.9){$b$}
%\put( 0,30){\circle*{2}}\put( 1,31){$c$}
\put( 0,10){\circle*{1}}\put( 2,9.5){$z_\alpha$}
\put(-1,-3.9){$I$}
\put(-40, 0){\line(1, 0){80}}
\put(-30, 0){\line(1, 1){30}}
\put( 0,30){\line(1,-1){30}}
%
%\qbezier(-25,0)(-25,2.0711)(-26.4644,3.5355)
\qbezier(-23,0)(-23,2.8995)(-25.0503,4.9497)
\put(-22, 3){$\alpha$}
\linethickness{2pt}
\put(-30, 0){\line(1,0){60}}
\end{picture}\\
%}
Figure 1: The definition of $\tri_\alpha,\Lambda_\alpha$ and $z_\alpha$.
\end{center}
\end{figure}
Let $\mathcal{P}_\alpha(z;z_\alpha)$ denote the Poisson kernel for the triangle $\tri_\alpha$
evaluated at the point $z_\alpha$.
That is to say, for any function $F$, continuous and harmonic throughout the solid triangle $\tri_\alpha$,
\begin{equation}\label{poisson}
F(z_\alpha) = \int_{I\cup\Lambda_\alpha} F(z) \mathcal{P}_\alpha(z;z_\alpha)\, |dz|.
\end{equation}
We need the following information about $\mathcal{P}_\alpha(z;z_\alpha)$.
\begin{lemma}\label{PoisLemma} Given $I,\alpha$, there is a constant $C$ such that
\begin{equation}
\label{PoisEst}
\mathcal{P}_\alpha(z;z_\alpha) \leq C \big[\Im\,z\big]^{(\pi-\alpha)/\alpha} \quad \forall z\in\Lambda.
\end{equation}
While $E$ is in the interior of $I$, $\mathcal{P}_\alpha(E;z_\alpha)$ is $C^\infty$ and positive. At
the ends of $I$, $\mathcal{P}_\alpha(E;z_\alpha)$ vanishes to order $(\pi-\alpha)/\alpha$.
\end{lemma}
We will not prove this here. It is a fairly standard result which can be proved by conformal
transplantation to the disk.
\begin{proof}[Proof of Theorem 4]
As $I\subseteq\Sigac(H)$, we know that $\Im\,m(E+i0)>0$ for a.e. $E\in I$. Also, by assumption i), it is a
continuous function on $I$. Therefore, in order to prove that $I\subseteq\Sigac(H+V)$, it will suffice to show
that $J\subseteq\Sigac(H+V)$ for all closed intervals $J\subseteq I$ on which $\Im\,m(E+i0)$ is positive. By this
argument, we may freely assume that $\Im\,m(E+i0)>0$ for all $E\in I$. We will do so for the remainder of this proof.
The statement of the theorem provides a sequence of compactly supported functions, $V_n$, which converges to
$V$ in $L^p$. We define $a_n(z)=\det[1+R(z)V_n]$, as in Section 2, and
\begin{align}
\notag
f_n(z) &= \log\big|\det_p[1+R(z)V_n]\big| \\
\label{fatr}
&= \log\big|a_n(z)\big| - \Re \sum_{\ell=1}^{p-1} \tr\Big\{ {\big[R(z)V_n\big]}^\ell \Big\}.
\end{align}
Our goal is to apply Proposition \ref{mainprop} for which we need to obtain the bound
(\ref{2:abound}) on $\log|a_n|$.
We will do so by estimating the contribution of $f_n$ and each of the traces separately.
For the function $w$ of (\ref{2:abound}), we choose
$$
w(E) = \begin{cases} \hfill0\hfill & E\notin I \\
\mathcal{P}_\alpha(E;z_\alpha) & E\in I \end{cases}
$$
with $\alpha=\pi/\max\{p+2,r+2\}$ ($r$ is the integer from the third assumption). Notice
that Lemma~\ref{PoisLemma} implies $w\in C^r(\Reals)$.
Because $f_n$ is harmonic on $\tri_\alpha$,
\begin{equation}\label{3:fest}
\int_I f_n(E) w(E) dE = f_n(z_\alpha) -
\int_{\Lambda_\alpha} f_n(E) \mathcal{P}_\alpha(z;z_\alpha) |dz|.
\end{equation}
Lemma \ref{TIest} immediately gives a ($n$-independent) bound on $f_n(z_\alpha)$.
Combining this Lemma with Lemma~\ref{PoisLemma} shows that the integral over $\Lambda_\alpha$
is also bounded (the $\epsilon^{-p}$ growth of $f_n(E+i\epsilon)$ is cancelled by
$\epsilon^{(\pi-\alpha)/\alpha}$ decay of $\mathcal{P}_\alpha$). Thus, we see that
$\int f_n(E) w(E) dE$ is uniformly bounded.
Consider now $\Re\tr\{[R(z)V_n]^\ell\}$. The integral kernel for the resolvent, $R(z)$, is
just the Green function, $G(x,y;z)$ which extends continuously from $z\in\Cmplx^+$ to $I$
(by assumption). Therefore, for each $\ell$, $\tr\{[R(z)V_n]^\ell\}$ also has a continuous
extension to $z\in I$ and
$$
\Re \int \tr\Big\{ {\big[R(E+i0)V_n\big]} \Big\} w(E)\,dE =
\iint \big[ \Re\,G(x,x;E+i0)\big] V_n(x) w(E)\,dx\,dE
$$
or, more generally,
\begin{gather*}
\Re \int \tr\Big\{ {\big[R(E+i0)V_n\big]}^\ell \Big\} w(E)\,dE\\
=\iint K(x_1,\ldots,x_\ell;E+i0)\, V_n(x_1)\cdots V_n(x_\ell)\,
dx_1\ldots dx_\ell\,w(E)\,dE
\end{gather*}
The kernel $K$ is exactly the same one defined in (\ref{Kdefn}). In this way, we see that
assumption iii) has been contrived precisely to bound the integrals of the traces.
To recap, we have shown that $\int f_n(E) w(E)\,dE$ and each
$$
\int \Re\tr\{[R(E+i0)V_n]^\ell\} w(E)\,dE \qquad 1\leq \ell \leq p-1
$$
is bounded uniformly in $n$. Combining these with (\ref{fatr}) demonstrates that
$$
\sup_n \int \log|a_n(E+i0)| w(E)\,dE < \infty
$$
With this, we apply Proposition~\ref{mainprop} to conclude that $I\subseteq\Sigac(H+V)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples}\label{S:Ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We begin with the simplest case. Namely, when there is no background potential, $q\equiv0$.
In this case, the Green function is given by
\begin{equation}\label{FreeG}
G(x,x';k^2) = \tfrac{1}{k}\sin(kx^<)e^{ikx^>} \qquad k\in \Cmplx^+.
\end{equation}
(The notation $x^{\gtrless}$ was introduced for (\ref{Defn:G}).)
This extends continuously to positive energies. We now begin the more
arduous task of checking hypothesis iii) of Theorem \ref{th4}. By simple computation,
\begin{align}\label{FreeK1}
K(x;k^2+i0) &= \tfrac{1}{2k} \sin(2kx),\\
\label{FreeK2}
K(x,x';k^2+i0) &= \tfrac{1}{2k^2} [1-\cos(2kx^<)]\cos(2kx^>).
\end{align}
As these formulae suggest, the Fourier transform will be important. We adopt
the convention $\hat f(k) = \int f(x)e^{ikx} dx$.
\begin{lemma}\label{lem41}
Given $a>0$, $1\leq p \leq \infty$ and any $w\in C^3(\Reals)$, which is supported on
the interval $[a^2,b^2]$,
\begin{align}\label{Free1}
\int_{a^2}^{b^2}\!\!\int K(x;E+i0) V(x)\,dx\,w(E)dE
&\lesssim \|\hat V\|_{L^2([2a,2b])} \\
\label{Free2}
\int_{a^2}^{b^2}\!\!\int\!\!\!\int K(x,y;E+i0) V(x)V(y)\,dx\,dy\,w(E)dE
&\lesssim \|\hat V\|^2_{L^2([2a,2b])} + \|V\|^2_{L^p(\Reals)}
\end{align}
for all real-valued Schwartz functions $V$.
\end{lemma}
\begin{proof}
It is important that $w$ is supported away from zero. For (\ref{Free2}), we break $K(x,y;E)$ into
two parts using (\ref{FreeK2}). The first part is bounded by employing
$$
\tfrac{1}{4k^2}\int\!\!\!\int \cos(2kx)\cos(2ky)V(x)V(y)\,dx\,dy \lesssim |\hat V(2k)|^2
$$
and
$$
\int_{a^2}^{\smash{b^2}}\! |\hat V(2\sqrt{E})|^2 w(E) dE \lesssim \|\hat V\|^2_{L^2([2a,2b])}.
$$
The second with
$$
\int_{a^2}^{b^2} \tfrac{1}{2E} \cos(2y\sqrt{E}) w(E)dE \lesssim (1+y)^{-3}
$$
and
$$
\int_0^\infty\int_0^y |V(x)|\, |V(y)|\, (1+y)^{-3}\,dx\,dy \lesssim \|V\|^2_{L^p(\Reals)}.
$$
Using (\ref{FreeK1}), the proof of (\ref{Free1}) is straight-forward.
\end{proof}
\begin{coro}\label{FreeL3} Suppose $q\equiv 0$ and $V\in L^3(\Reals^+)$.
If the distribution $\hat V$ restricts to an $L^2$ function on an interval
$[2a,2b]\subset(0,\infty)$ then $[a^2,b^2]\subset\Sigac(H+V)$.
\end{coro}
\begin{proof} There is a sequence of compactly supported functions $V_n$ which converge to
$V$ in $L^3$ and so that for each $J=[2c,2d]$, properly contained in $[2a,2b]$,
$\int_J |\hat V_n(k)|^2 dk$ is uniformly bounded. The proof of this fact is not very edifying
and will be postponed (Lemma \ref{YUKKY}). The preceding lemma then shows that the hypotheses
of Theorem \ref{th4} hold with $I=[c^2,d^2]$, $p=3$ and $r=3$.
Thus $[c^2,d^2]\subset\Sigac(H+V)$.
We have just shown that any proper sub-interval of $[a^2,b^2]$ is contained in $\Sigac(H+V)$.
It follows that $[a^2,b^2]\subset \Sigac(H+V)$.
\end{proof}
We say that a distribution $V$ belongs to the homogeneous Sobolev space $\dot {\sf H}{}^s$
if $\int |\hat V^2(k)|^2 |k|^{2s} dk < \infty$.
\begin{coro}\label{FreeHs}
Suppose $q\equiv 0$ and $V\in L^3(\Reals^+)$. If $V$ can be written as a finite sum of functions
each of which belongs to a homogeneous Sobolev space $\dot {\sf H}{}^s$ $(s\in\Reals)$ then
$\Sigac(H+V)=[0,\infty)$.
\end{coro}
\begin{proof}
The previous corollary shows that each compact sub-interval of $(0,\infty)$ is contained in
$\Sigac(H+V)$. This implies that $\Sigac(H+V)\supset [0,\infty)$.
Conversely, $\Sigac(H+V)$ is a subset of the essential spectrum. By Weyl's
Theorem on relatively compact perturbations, we know that $\sigma_{\mathrm{ess}}(H)=
\sigma_{\mathrm{ess}}(H+V)=[0,\infty)$.
\end{proof}
Treating $V\in L^p$ becomes more involved as $p$ grows. We have just proved a result for
$p=3$ and will content ourselves with treating just one more case: $p=4$. For this we
use the tri-linear estimate below. The main idea is due to Kovrijkine \cite{OLEG}.
\begin{lemma}\label{trilin}
Let $\nu\geq0$ be an integer. Suppose $\rho\in C^{\nu+2}(\Reals)$ is compactly supported with
$0\not\in\supp(\rho)$. If for each pair of compactly supported functions $f\in C^\nu$ and
$g\in L^4$
$$
I(f,g) = \int\!\!\!\int\!\!\!\int_x^z\!\! f(y)\,dy\;g(x)g(z)\!
\int\! \sin[2k(z-x)] \rho(k)\, dk \,dx\,dz
$$
then
$
|I(f,g)| \lesssim \big\{ \|f^{(\nu)}\|_{L^2} + \|f\|_{L^4}+\|g\|_{L^4} +
\|\hat g\|_{L^2(\Omega)} \big\}^3
$ where $\Omega=\{k:\pm k/2\in\supp\rho\}$.
Further, if we write $F(x)=\int_x^\infty f(t)\,dt$ then
$|I(f,g)| \lesssim \|F\|_{L^2}^{} \|g\|_{L^4}^2$.
\end{lemma}
\noindent
{\it Remark.}\/ Both $\int_x^z f(y)dy$ and $\sin[2k(z-x)]$ have odd parity under the interchange
of $x$ and $z$. This makes the full integrand even.
\smallskip
\begin{proof}
If $r(\xi) = \int \sin[2k\xi] \rho(k)\, dk$ then $\rho\in C^{\nu+2}(\Reals)$ implies
$\xi^j r(\xi)\in L^1$ for all $0\leq j\leq\nu+1$. Moreover, for each such $j$
the Fourier transform of $\xi^j r(\xi)$ is uniformly bounded and is supported by $\Omega$.
This means that for $0\leq\ell\leq\nu$,
\begin{equation}\label{rest}
\Big\| \int (z-x)^{\ell+1} r(z-x) f(z)\,dz \Big\|_{L^2(dx)} \lesssim \|\hat g\|_{L^2(\Omega)}.
\end{equation}
Let $[Mh](x)=\sup_{z} \frac{1}{|z-x|} \int_x^z\!|h(y)|\,dy$ that is, the maximal function of
$|h|$. Then $\|Mh\|_{L^2} \lesssim \|h\|_{L^2}$ and
\begin{equation}\label{Maxml}
\sup_z \Big| \int_x^z f^{(\nu)}(t) \frac{(z-t)^\nu}{(z-x)^{\nu+1}} \,dt \Big|
\leq [M\,f^{(\nu)}](x).
\end{equation}
Consider the case $\nu=0$. Using the above and Young's inequality,
\begin{align*}
|I(f,g)| &= \Big| \int\!\!\!\int r(z-x)\int_x^z\!\! f(y)\,dy\; g(x)g(z)\,dx\,dz \Big| \\
&\leq \int\!\!\!\int\! \big|(z-x)r(z-x)\big|\,[Mf](x)\,|g(x)|\,|g(z)|\,dx\,dy \\
&\lesssim \|g[M f]\|_{L^{4/3}}^{} \|g\|_{L^4}^{} \lesssim \|f\|_{L^2}^{} \|g\|_{L^4}^2.
\end{align*}
For $\nu>0$, we employ the following consequence of Taylor's formula (with the
integral form of the remainder):
$$
\int_x^z f(y) \,dy = \sum_{\ell=0}^{\nu-1} \tfrac{1}{(\ell+1)!}(z-x)^{\ell+1} f^{(\ell)}(x) +
\tfrac{1}{\nu!} (z-x)^\nu \int_x^z f^{(\nu)}(t) \frac{(z-t)^\nu}{(z-x)^\nu} \,dt.
$$
Substituting it into the definition of $I(f)$,
\begin{equation*}
\begin{split}
I(f,g) &= \sum_{\ell=0}^{\nu-1} \tfrac{1}{(\ell+1)!} \int
g(x)\,f^{(\ell)}(x)\int(z-x)^{\ell+1}\,r(z-x)\,g(z)\,dz\,dx \\
&\qquad + \tfrac{1}{\nu!} \int\!\!\!\int\!\!\!\int_x^z f^{(\nu)}(t)
\frac{(z-t)^\nu}{(z-x)^{\nu+1}}\,dt \;(z-x)^{\nu+1}\,r(z-x)\,g(x)\,g(z)\,dx\,dz.
\end{split}
\end{equation*}
The summands from the first line are bounded using (\ref{rest}), H\"older's inequality and the
fact that for all $0\leq\ell<\nu$, $\|f^{(\ell)}\|_{L^4} \lesssim \|f\|_{L^4} + \|f^{(\nu)}\|_{L^2}$:
\begin{align*}
&\quad \int g(x)\,f^{(\ell)}(x)\int(z-x)^{\ell+1}\,r(z-x)\,g(z)\,dz\,dx \\
&\lesssim \|gf^{(\ell)}\|_{L^2} \|\hat g\|_{L^2(\Omega)} \lesssim
\big\{ \|f^{(\nu)}\|_{L^2} + \|f\|_{L^4} \big\} \|g\|_{L^4} \|\hat g\|_{L^2(\Omega)}.
\end{align*}
The term on the second line is estimated by using (\ref{Maxml}) much as in the $\nu=0$ case:
\begin{align*}
&\quad \Big| \int\!\!\!\int\!\!\!\int_x^z f^{(\nu)}(t)
\frac{(z-t)^\nu}{(z-x)^{\nu+1}}\,dt \;(z-x)^{\nu+1}\,r(z-x)\,g(x)\,g(z)\,dx\,dz \Big|\\
&\leq \int\!\!\!\int [M f^{(\nu)}](x)\,|g(x)|\,\big|(z-x)^{\nu+1}r(z-x)\big| \,|g(z)|\,dx\,dz \\
&\lesssim \|f^{(\nu)}\|_{L^2}^{} \|g\|_{L^4}^2
\end{align*}
Adding together these estimates does indeed show that
$
|I(f,g)| \lesssim \big\{ \|f^{(\nu)}\|_{L^2} + \|f\|_{L^4}+\|g\|_{L^4} +
\|\hat g\|_{L^2(\Omega)} \big\}^3
$.
The second estimate stated in the lemma, the one involving $F$, is easier to prove:
\begin{align*}
I(f,g) &= \int\!\!\!\int r(z-x)\int_x^z\!\! f(y)\,dy\; g(x)g(z)\,dx\,dz \\
&= \int\!\!\!\int r(z-x) [F(x)-F(z)]g(x)g(z)\,dx\,dz.
\end{align*}
Thus, by Young's and H\"older's inequalities, $|I(f,g)| \lesssim \|g\|_{L^4}^2 \|F\|_{L^2}^{}$.
\end{proof}
\begin{coro}\label{FreeL4}
Suppose $q\equiv0$. If for some integer, $N\geq0$, $V$ can be written as
$V(x)=\sum_{\nu=-1}^N U_\nu(x)$ such that
$U_\nu \in L^4$ and $U_\nu^{(\nu)}\in L^2$ then $\Sigac(H+V)=[0,\infty)$.
For $\nu\geq0$, $U^{(\nu)}$ denotes the $\nu^{\mathrm{th}}$ derivative while
$U^{(-1)}(x)=\int_x^\infty U(t)\,dt$.
\end{coro}
\begin{proof}
We will show that Theorem \ref{th4} (with $p=4$ and $r=N+100$) is applicable to any interval
$[a^2,b^2]$ where $00$ and smooth functions $A(E),\vartheta(E)$ so that writing
\begin{equation*}
\Re\,G(x,x;E+i0) = A(E) [E-q(x)]^{-1/2}
\sin\Big\{ \vartheta(E) + 2 \int^x_{x_0} \!\!\sqrt{E-q(t)}\,dt\Big\} + \zeta(x;E)
\end{equation*}
for all $x>x_0$ and all $E\in I$ implies that $\int_{x_0}^\infty \zeta(x;E)^2\,dx$ is a uniformly
bounded function of $E\in I$.
\end{lemma}
\begin{proof}
Let $\Delta(x;E)=\frac{5}{16}[q'(x)]^2[E-q(x)]^{-2} + \frac{1}{4} q''(x)[E-q(x)]^{-1}$
and choose $x_0$ so that for all $E\in I$, both $E>2q(x)$ for all $x>x_0$ and
$\int_{x_0}^\infty |\Delta(x;E)| dx < 10^{-6}\,\min_{x\geq x_0}\,\sqrt{E-q(x)}$.
Let us define operators $\tilde H^D,\tilde H^N$ as the restriction of $H$ to the interval
$[x_0,\infty)$ with a Dirichlet, respectively Neumann, boundary condition at $x_0$. We begin
by constructing the Green functions for these operators. For each $z\in \Cmplx^+$, define
$$
\tilde u_\pm(x)=\big[z-q(x)\big]^{-1/4} \exp\Big\{\pm i \int^x_{x_0} \!\!\sqrt{z-q(t)}\,dt\Big\}
$$
where the branch of the square root is chosen to make $\tilde u_+$ exponentially decaying. Next
write
$
\tilde G_0^D(x,x';z) = \tfrac{1}{2i}\big[\tilde u_+(x^<)-\tilde u_-(x^<)\big]\tilde u_+(x^>)
$
and
$
\tilde G_0^N(x,x';z) = \tfrac{1}{2i}\big[\alpha(z)\tilde u_+(x^<)-\tilde u_-(x^<)\big]\tilde u_+(x^>)
$
where $\alpha(z)=[q'(0)-4i(z-q(0))^{3/2}]/[q'(0)+4i(z-q(0))^{3/2}]$. These formulae are Green functions
for an operator with a `potential' other that $q-z$. The difference between this and $q-z$ is given by
$\Delta(x;z)$. These approximate Green functions, $\tilde G_0^{D/N}$, clearly extend continuously from
$z\in\Cmplx^+$ to $z\in I$. However, because $\Delta$ is so small, we can write the
true Green functions in a Neumann series
$$
\tilde G^{D/N} = \tilde G_0^{D/N} +
\sum_{k=1}^\infty \tilde G_0^{D/N} \Big\{ G_0^{D/N} \Delta G_0^{D/N} \Big\}^k
$$
which is convergent for $z$ in a neighbourhood of $I+i[0,1)\subset\overline{\Cmplx^+}$.
This shows that $\tilde G^{D/N}$ have continuous extensions from $z\in \Cmplx^+$ to
the interval $I$. It also permits us to calculate the large $x$ asymptotics of $\Re\,\tilde G^{D/N}(x,x;E)$.
We will do this for $\tilde G^D$, the calculation for $\tilde G^N$ is similar.
From the Neumann series,
\begin{align*}
\tilde G^D(x,x;E) &= \tilde G_0^D(x,x;E) + \int \tilde G_0^D(x,y;E) \Delta(y) \tilde G_0^D(y,x;E) \,dy \\
&\quad + \int\!\!\int \tilde G_0^D(x,y) \Delta(y) \tilde G^D(y,z;E) \Delta(z) \tilde G_0^D(z,x;E) \,dy\,dz.
\end{align*}
We will calculate the asymptotics of each term on the right hand side separately.
For the the second term we have,
\begin{align*}
-\tfrac{1}{4}\; \tilde u_+(x)^2 \int_{x_0}^\infty [\tilde u_+(y)-\tilde u_-(y)]^2\Delta(y)\,dy +
O\Big\{ \int_x^\infty \Delta(y)\,dy \Big\}
\end{align*}
and for the third,
\begin{gather*}
-\tfrac{1}{4}\; \tilde u_+(x)^2 \int_{x_0}^\infty\!\!\int_{x_0}^\infty
[\tilde u_+(y)-\tilde u_-(y)][\tilde u_+(z)-\tilde u_-(z)]\tilde G^D(y,z;E) \Delta(y)\Delta(z)\,dy\,dz \\
{} + O\Big\{ \int_x^\infty \Delta(y)\,dy \Big\}.
\end{gather*}
By assumption, $\int_x^\infty \Delta(y) dy$ is a square integrable function of $x$. Thus the real part of
each of the previous two terms may be written as
\begin{equation*}
\tilde A(E)\sin\Big\{\tilde\vartheta(E)+2\int^x_{x_0}\!\!\sqrt{E-q(t)}\,dt \Big\} + \text{square integrable error}
\end{equation*}
for some smooth amplitude $\tilde A$ and phase $\tilde \vartheta$. Moreover, the $L^2$ norm of the error is
a uniformly bounded function of $E\in I$. The first term has the same asymptotics. In fact,
in this case, $\tilde A=1/2$, $\tilde \vartheta=0$ and there is no error term. Because the sum of sinusoids is
a sinusoid, adding the three terms proves that
$$
\tilde G^D(x,x;E) = \tilde A(E)\sin\Big\{\tilde\vartheta(E)+2\int^x_{x_0}\!\!\sqrt{E-q(t)}\,dt \Big\} + \text{square integrable error}
$$
for some smooth functions $\tilde A(E),\tilde\vartheta(E)$. Similar arguments show that the same is true for
$\tilde G^N(x,x;E)$.
We have shown that $\tilde G^{D/N}$ have continuous extensions from $\Cmplx^+$ to $I$ and that the
values on $I$ have the right asymptotic form. The result for the original operator now follows because,
for $x>x_0$, the Green function $G(x,x;E)$ is a smoothly-$E$-dependent linear combination of $\tilde G^D(x,x;E)$
and $\tilde G^N(x,x;E)$.
\end{proof}
\begin{coro}\label{SVary}
If $q(x)\to 0$ as $x\to\infty$, $\int^\infty_x |q'(t)|^2 dt \in L^2(dx)$ and
$\int^\infty_x |q''(t)| dt \in L^2(dx)$ then for each $V \in L^2$, $\Sigac(H+V)=[0,\infty)$.
\end{coro}
\begin{proof}
That $\Sigac(H+V)$ is a subset of $[0,\infty)$, follows from Weyl's theorem, as in Corollary~\ref{FreeHs}.
To prove the converse it suffices to show that Theorem \ref{Th:det2} is applicable to each
compact interval $I\subset(0,\infty)$. To this end, let $w(E)$ be a $C^1(\Reals)$ function
supported on such an interval. We wish to show that
$$
\int_I \Re\,G(x,x;E+i0) w(E) dE \in L^2(\Reals^+;dx)
$$
By referring to the preceding lemma, we see that the following
calculation suffices: Let $\varphi(x,E)=2\int^x_{x_0}\!\!\sqrt{E-q(t)}\,dt + \vartheta(E)$,
$\varphi'(x,E)=\frac{\partial\phantom{E}}{\partial E}\,\varphi(x,E)$ and $\tilde w(E)=w(E)A(E)$.
For $x_0$ sufficiently large and each $x>x_0$,
\begin{align*}
\Big| \int e^{i\varphi(x,E)} w(E)\,dE \Big| &=
\Big| \int e^{i\varphi(x,E)} \Big[ \tfrac{w'(E)}{\varphi'(x,E)}
- \tfrac{w(E)\varphi''(x,E)}{\varphi'(x,E)^2} \Big]\,dE \Big|
\lesssim x^{-1}.
\end{align*}
The first step is integration by parts after first multiplying and dividing by $\varphi'$. The
second step is the combination of that facts $x\lesssim\varphi'$
and $\varphi''\lesssim x$ for $x$ sufficiently large and $E\in I$.
\end{proof}
The final example for this section is the case of a periodic background potential. We assume that $q$ is
locally square integrable and, without loss of generality, has period one.
Much is known about such operators, but we shall
only describe those parts of Floquet Theory, as it is called, that we will require. For more information
see \cite{EASTHAM}.
There is a continuous function $\gamma:\Reals\to\overline{\Cmplx^+}$, so that, with the exception
of the discrete set where $e^{i\gamma(E)}=\pm 1$, there is a basis of solutions $\psi_\pm(x;E)$ of
$-\psi''+q\psi=E\psi$ with $\psi_\pm(x+1)=e^{\pm\gamma(E)}\psi_\pm(x)$. (The function $\gamma(E)$
is typically called the quasi-momentum and $\psi_\pm$ the Bloch (or Floquet) solutions.) The
absolutely continuous spectrum of $H$ is equal to the set where $|e^{i\gamma(E)}|=1$. Because of
the Dirichlet boundary condition at the origin, eigenvalues appear in the spectral gaps. That is
in the intervals where $|e^{i\gamma(E)}|<1$.
Let $\Sigma$ denote the set $\{E\in\sigma_{\ac}:e^{i\gamma(E)}\neq\pm 1\}$. While the closure of
$\Sigma$ is equal to $\sigma_{\ac}(H)$, it need not be the case that $\Sigma$ is equal to the
interior of the spectrum. This is because of the possibility of two bands touching.
The function $\gamma(E)$ is analytic and strictly monotone on $\Sigma$. For $E\in\Sigma$,
it is possible to define the Bloch solutions
uniquely by adding the constraint $\psi_\pm(0)=1$. This normalization makes $\psi_+$ equal to the
Weyl solution. It also follows that $\psi_+(x;E)=\overline{\psi_-(x;E)}$.
It is well known that the Green function extends continuously (indeed analytically) from
$\Cmplx^+$ to $\Sigma$. Indeed, we have the following formula
\begin{equation}\label{Gperiod}
G(x,x';E+i0)= \tfrac{1}{W(E)} \psi_-(x^<;E)\psi_+(x^>;E) - \tfrac{1}{W(E)} \psi_+(x;E)\psi_+(x';E)
\end{equation}
where $W(E)$ is the Wronskian of $\psi_+$ and $\psi_-$.
For part of the following Corollary, we will require that $\hat V$ be uniformly locally square integrable.
To this end, we introduce the Banach space $\ell^\infty(L^2)$ with norm
\begin{equation}
\label{ellL}
\big\|f\big\|_{\ell^\infty(L^2)}^2 = \sup_n \int^{2\pi}_{-2\pi}
\big|f\big(x+2\pi n\big)\big|^2 dx.
\end{equation}
\begin{coro}\label{PeriL3}
Suppose $q\in L^2_{\mathrm{loc}}$ is periodic with period one.
If $V\in L^2$, or both $V\in L^3$ and $\hat V\in \ell^\infty(L^2)$, then
$\Sigac(H+V)=\sigac(H)$.
\end{coro}
\begin{proof}
As in the previous results, $\Sigac(H+V)\subset \sigac(H)$ follows from Weyl's Theorem. We will
prove the converse inclusion by showing that each compact interval $I\subset\Sigma$ is contained
in $\Sigac(H+V)$. This will be a consequence of Theorem \ref{Th:det2}, for $V\in L^2$, and of
Theorem \ref{th4} in the other case. As mentioned above, the Green function extends continuously
to $I$. It remains, however, to estimate the behaviour of the Green function.
We use an idea from ``The Proof of Theorem 1.8'' in the pre-print
\cite{KISELEV-PRE} (the published version \cite{KISELEV-2} uses a different method). It is the
following: The function $\sigma(x;E)=\exp[-i\gamma(E)x]\psi_+(x;E)$ is periodic, with period 1,
a $C^\infty(I)$ function of $E$ and a $C^1(\Reals)$ function of $x$. Moreover, $\sup_{E\in I}
\|\sigma(\cdot;E)\|_{C^1}<\infty$ and $\sup_{x} \|\sigma(x;\cdot)\|_{C^2}<\infty$.
Let $I$ be as described above, and let $w\in C^2(\Reals)$ be supported by $I$. We begin by showing
that
\begin{equation}\label{pest1}
\Big| \Re \int\!\!\!\int_I G(x,x;E+i0) V(x)\,w(E)dE\,dx \Big| \lesssim
\big\|V\big\|_{L^p}
\end{equation}
for any $p>1$. Because $\psi_\pm$ are complex conjugates, the first term on right of (\ref{Gperiod})
is purely imaginary. As a result,
\begin{align*}
\Big|\int \Re\, G(x,x;E+i0)\,w(E)dE\Big| &=
\Big| \int e^{2i\gamma(E)x}\,\tfrac{\sigma(x;E)^2w(E)}{W(E)}\,dE \Big| \\
&\leq x^{-1} \int \big| \tfrac{\partial\phantom{E}}{\partial E}
\tfrac{\sigma(x;E)w(E)}{2\gamma'(E)W(E)}\big| \,dE
\end{align*}
by multiplying and dividing by $\gamma'(E)$ and then using integration by parts.
The last integral is finite because both $W(E)$ and $\gamma'(E)$ are bounded
away from zero on $I$ and because all terms are $C^1$ functions of $E$. Furthermore, since $\sigma$ and
its derivative are bounded independently of $x$, this proves (\ref{pest1}).
With $p=2$, (\ref{pest1}) shows that Theorem \ref{Th:det2} is applicable when $V\in L^2$.
This proves the first case of the theorem. The second, $V\in L^3$ and $\hat V\in \ell^\infty(L^2)$,
follows from Theorem \ref{th4} with $p=3$ and $r=2$. The third hypothesis of this theorem is
satisfied because the calculation performed above and because
\begin{equation}\label{pest2}
\int\!\!\!\int\!\!\!\int \Re\,[G(x,y;E+i0)^2] V(x)V(y)\,dx\,dy\,w(E)dE
\lesssim \big\|\hat V\big\|_{\ell^\infty(L^2)}^2 + \big\| V\big\|_{L^3}^2
\end{equation}
for every $w\in C^{2}(\Reals)$ supported by $I$. The remainder of this proof is devoted to the justification
of this inequality.
Using (\ref{Gperiod}), we can expand $G(x,x';E+i0)^2$ as three terms: $\tfrac{1}{W^2}\psi_+(x^<)^2\psi_+(x^>)^2$,
$\tfrac{2}{W^2}\psi_-(x^<)\psi_+(x^<)\psi_+(x^>)^2$ and $\tfrac{1}{W^2}\psi_-(x^<)^2\psi_+(x^>)^2$.
We will deal with them in this order. For the first, we have
\begin{align*}
&\phantom{{} = {}\!}\Big|\int \tfrac{1}{W(E)^2}\psi_+(x;E)^2\psi_+(y;E)^2 \,w(E)dE\Big| \\
&=\Big|\int \tfrac{1}{W(E)^2}\sigma(x;E)^2\sigma(y;E)^2 e^{2i\gamma(E)[x+y]} \,w(E)dE\Big| \\
&\lesssim (x+y)^{-2}.
\end{align*}
The last line follows by integrating by parts twice, each time multiplying and dividing by $\gamma'(E)$.
(Compare the proof of (\ref{pest1}) given above.) This calculation, together with an easy $L^\infty(dx\,dy)$
estimate, shows that
$$
\Big| \int\!\!\!\int\!\!\!\int\!\! \tfrac{1}{W(E)^2}\psi_+(x;E)^2\psi_+(y;E)^2 V(x)V(y)\,dx\,dy\,w(E)dE \Big|
\lesssim \big\| V \big\|_{L^3}^2.
$$
Consequently, we have controlled the first of the three terms. By the same arguments,
\begin{align*}
&\phantom{{} = {}\!}\Big|\int \tfrac{2}{W(E)^2}\psi_+(x;E)\psi_-(x;E)\psi_+(y;E)^2 \,w(E)dE\Big| \\
&=\Big|\int \tfrac{2}{W(E)^2}|\sigma(x;E)|^2\sigma(y;E)^2 e^{2i\gamma(E)y} \,w(E)dE\Big| \\
&\lesssim y^{-2}
\end{align*}
shows that the contribution the second term makes to (\ref{pest2}) is $\lesssim\|V\|_{L^3}^2$. This
leaves the third and most interesting term. This time, we will perform the spatial integrals first
(rather than the energy integral, as above). Because $\psi_+$ and $\psi_-$ are complex conjugates,
the Wronskian is purely imaginary and taking the complex conjugate of $\psi_+(x)^2\psi_-(y)^2$
is equivalent to interchanging $x,y$. Combining these facts,
\begin{equation}\label{pest3}
2\Re\,\underset{\!\!\!\!x2$, then the sequence $V_n(x)=\phi(x/n)V(x)$
converges to $V$ in $L^3$. Moreover, for any interval $J$ properly contained in $I$,
the integral $\int_J |\widehat{V_n}(k)|^2 dk$ is uniformly bounded.
\end{lemma}
\begin{proof}
We will write $\varphi_n(x)=\varphi(x/n)$. That $V_n\to V$ in $L^3$ is well known, the
difficulty lies in the second part. Let $J$ be a proper sub-interval of $I$.
It suffices to show that for any function $g$, whose Fourier
transform is $C^\infty$ and supported by $J$,
\begin{equation}\label{YUK1}
\int_J \hat V_n(k) \hat g(k) dk \lesssim \|\hat g\|_{L^2}.
\end{equation}
The implicit constant depends on the distance from $J$ to the complement of $I$.
To prove (\ref{YUK1}), choose a $\rho\in C^\infty(\Reals)$ which is supported by $I$ and
equal to one in a neighbourhood of $J$.
Then write $\varphi_n(x)g(x)=G_1(x)+G_2(x)$ with
$\hat G_1(k) = \rho(k) \widehat{\varphi_n g}(k)$ and
$\hat G_2(k) = [1-\rho(k)]\widehat{\varphi_n g}(k)$. Let $p'$ denote the conjugate exponent
of $p$: $1/p + 1/p' = 1$. Using Parseval and H\"older,
\begin{align*}
\int_J \hat V_n(k) \hat g(k) dk &\lesssim \int V(x) \phi_n(x) g(x) dx
= \int V(x) [G_1(x) + G_2(x)] dx\\
&\lesssim \|\hat V\|_{L^2(I)}\|\hat G_1\|_{L^2} + \|V\|_{L^p}\|G_2\|_{L^{\smash{p'}}}
\end{align*}
so it suffices to bound $\|\hat G_1\|_{L^2}$ and $\|G_2\|_{L^{\smash{p'}}}$ by an $n$-independent
multiple of $\|\hat g\|_{L^2}$.
From $\|\varphi_n g\|_{L^2}\leq \|g\|_{L^2}$ it follows that both
$\|\hat G_1\|_{L^2} \leq \|\hat g\|_{L^2}$ and $\|\hat G_2\|_{L^2} \leq \|\hat g\|_{L^2}$.
This completes the study of $G_1$. For $G_2$ we continue with
\begin{equation*}
\|G_2\|_{L^{\smash{p'}}} \lesssim \|G_2\|_{L^2} + \|xG_2\|_{L^2}
\lesssim \|\hat G_2\|_{L^2} + \|\hat G_2^{\,\prime}\|_{L^2}
\lesssim \|\hat g\|_{L^2} + \|\hat G_2^{\,\prime}\|_{L^2}.
\end{equation*}
Showing that $\|\hat G_2^{\,\prime}\|_{L^2} \lesssim \|\hat g\|_{L^2}$ is all that remains
to be done. However, this is rather involved.
From its definition, $\hat G_2^{\,\prime}(k) = (1-\rho(k))
[\widehat{\varphi_n}'\ast\hat g](k) - \rho'(k)\widehat{\varphi_n g}(k)$. The $L^2$ norm
of the second term is $\lesssim \|\rho'\|_{L^\infty}\|\varphi_n g\|_{L^2}
\leq \|\rho'\|_{L^\infty}\|g\|_{L^2}$ as we need. If we denote the first term by $f(k)$ then
$$
f(k)=[1-\rho(k)]\,[\widehat{\varphi_n}'\ast\hat g](k)
= n^2 [1-\rho(k)] \int_J g(k') \hat\varphi'(nk-nk')\,dk.
$$
By employing Cauchy--Schwarz, this implies
\begin{align*}
| f(k) |^2 &\leq
n^4 [1-\rho(k)]^2 \, \|g\|^2_{L^2} \int_J \big| \hat\varphi'(nk-nk') \big|^2 \,dk'.
\end{align*}
%
As $1-\rho(k)$ is zero for $k\in J$, we may continue with
%
\begin{align*}
|f(k)|^2 &\leq [1-\rho(k)]^2 \, \|g\|^2_{L^2} \int_J \frac{|nk-nk'|^3}{\dist(k,J)^3}
\big| \hat\varphi'(nk-nk') \big|^2 \,d(nk')\\
&\leq \frac{(1-\rho(k))^2}{\dist(k,J)^3} \, \|g\|^2_{L^2} \int |y|^3 |\hat\varphi'(y)|^2 dy.
\end{align*}
Thus the assumptions on $\rho$ and $\varphi$ are sufficient to conclude that
$\|f\|_{L^2} \lesssim \|g\|_{L^2}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Stark Operator}\label{S:St}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we discuss perturbations of the Stark operator
\begin{equation}\label{HStark}
[Hu](x) = -u''(x) - xu(x)
\end{equation}
acting on $L^2(\Reals^+)$ with a Dirichlet boundary condition at the origin.
This operator has purely absolutely continuous spectrum with $\Sigac(H)=\Reals$.
Lemma \ref{TIest} is not applicable because the potential $q(x)=-x$
is unbounded below. However, as solutions to the equation $-u'' - xu=zu$ are well
understood---they are Airy functions---we are able to obtain the necessary estimates
by direct computation. Throughout this section, the notation for special functions
is that of Abramowitz and Stegun \cite{AB-S}.
\begin{lemma}
Suppose $V$ is continuous and compactly supported. For each $2\leq p<\infty$, we have
the following trace ideal estimate:
\begin{equation}\label{Stark:HS}
\big\| R(E+i\epsilon) V \|_p^p \leq C \epsilon^{-1}\!\!\int |V(t^2)|^p\, dt \qquad
\forall 0<\epsilon<1.
\end{equation}
The constant $C$ depends on $p$ and continuously on $E$. This implies that for a similar
constant,
\begin{equation}\label{Stark:logdet}
\Big| \log\big| \det_p\big[1+R(E+i\epsilon)V\big] \big| \Big|
\leq C \epsilon^{-p}\!\!\int |V(x^2)|^2\, dx \qquad
\forall 0<\epsilon<1.
\end{equation}
\end{lemma}
\begin{proof} The Green function is given by
$$
G(x,y;z) = \frac{\sqrt{x+z} H^{(1)}_{1/3} (\frac{2}{3}(x+z)^{3/2}) }%
{\pi\sqrt{z} H^{(1)}_{1/3} (\frac{2}{3}z^{3/2})}
\big[Ai(-z-y)Bi(-z)-Ai(-z)Bi(-z-y)\big]
$$
whenever $x>y$ and $\Im\,z>0$. It is important to know that the branch of $(x+z)^{3/2}$
is chosen such that as $x\to\infty$, $\Im\,(x+E+i\epsilon)^{3/2}=\frac{3}{2}\epsilon\sqrt{x}+O(1)$.
It follows from the asymptotics of the Airy and Hankel functions that if $x>y$ then
$$
|G(x,y;E+i\epsilon)|^2 \lesssim x^{-1/2} y^{-1/2} \exp\{ 2\epsilon(\sqrt{\smash[b]{y}} - \sqrt x)\}
\qquad \forall 0<\epsilon<1.
$$
The implicit constant depends on $E$. Employing this bound gives
\begin{align*}
\big\| R(E+i\epsilon) V \|_2^2 &= \iint |G(x,y;E+i\epsilon|^2 |V(x)|^2 \,dx\,dy
\lesssim \epsilon^{-1}\!\!\int |V(t^2)|^p\, dt.
\end{align*}
(Note that $\int y^{-1/2}\exp\{ \pm 2\epsilon \sqrt y\}\,dy =
\epsilon^{-1}\exp\{\pm2\epsilon\sqrt x\}$.)
This proves the $p=2$ case of (\ref{Stark:HS}). For the remaining values of $p$
we employ interpolation as in Theorem 4.1 of \cite{SIMON-TI}.
To do so, one need only note that for $V$ of compact support, $R(E+i\epsilon) V$ is compact
and moreover, $\|R(E+i\epsilon) V\| \lesssim \epsilon^{-1} \|V\|_{L^\infty}$.
The argument given in Lemma \ref{TIest} shows that (\ref{Stark:logdet}) follows
directly from (\ref{Stark:HS}).
\end{proof}
With Lemma 5.1 replacing Lemma 3.1, the original proof of Theorem \ref{th4} extends
to the Stark operator.
\begin{theorem}\label{St:th}
Suppose $\int |V(t^2)|^p\, dt < \infty$ and $I$ is a compact interval.
If there are integers $p\geq2$, $r>0$ and a sequence of compactly supported
functions $V_n$ so that
\begin{SmallList}
\item as $n\to \infty$, $\int |V(t^2)-V_n(t^2)|^p\, dt \to 0$; and
\item for each $1\leq \ell\leq (p-1)$ and every $w\in C^r(\Reals)$ which is supported on $I$,
\begin{equation*}
\sup_{n} \left| \int_{[0,\infty)^\ell}\!K(x_1,\ldots,x_\ell;E+i0) V_n(x_1)\cdots V_n(x_\ell)\,
dx_1\ldots dx_\ell\,w(E)\,dE\right| < \infty
\end{equation*}
\end{SmallList}
then $I\subseteq\Sigac(H+V)$.
\end{theorem}
\begin{coro}\label{Stark}
If $\int |V(t^2)|^2\, dt < \infty$ then $\Sigac(H+V)=\Reals$.
\end{coro}
\begin{proof}
Let $V_n=\chi_{[0,n]}(x)V(x)$ and let $I$ be a compact interval in $\Reals$.
To show that $I\subset\Sigac(H+V)$, we need only check hypothesis ii) above with $\ell=1$.
For each $z\in\Cmplx^+$, the Weyl solution is proportional to
$(x+z)^{-1/2} H^{(1)}_{1/3} \big(\frac{2}{3}(x+z)^{3/2}\big)$ so Lemma \ref{Alt:G} gives
\begin{align*}
\Re\, G(x,x;E+i0) &= \tfrac{\pi}{3} (x+E)\,\, \Im\;
\Big[H^{(1)}_{1/3} \big(\tfrac{2}{3}(x+E)^{3/2}\big)\Big]^2.
\end{align*}
Because $I$ is compact, there is an $x_0>0$ such that Hankel function asymptotics permit us to write
\begin{equation}\label{St:Gd}
\Re\, G(x,x;E+i0) = (x+E)^{-1/2} \sin\big\{\tfrac{4}{3}(x+E)^{3/2} -\tfrac{5\pi}{6}\big\}
+ \zeta(x,E)
\end{equation}
with $\sup_{E\in I} |\zeta(x,E)| \leq C x^{-3/2}$ for $x\geq x_0$. We also require
that $x_0>-2E$.
Let us suppose that $w\in C^1$ then
we may estimate
\begin{align}\label{St:thing}
\int\!\!\int V_n(x) \Re\,G(x,x;E+i0)\,dx\,w(E)dE
\end{align}
as follows. First, $\sup_{E\in I,x\in\Reals} |G(x,x;E+i0)|<\infty$ so
\begin{align*}
\int\!\!\int_0^{x_0} V_n(x) \Re\,G(x,x;E+i0)\,dx\,w(E)dE &\lesssim
\|w\|_{L^1} \int_0^{x_0} |V_n(x)|\,dx \\
&\lesssim \|w\|_{L^1} \left\{\int |V(t^2)|^2\,dt\right\}^{1/2}.
\end{align*}
By (\ref{St:Gd}), we can write the remainder of (\ref{St:thing}) as
\begin{align*}
& \int\!\!\int_{x_0}^\infty V_n(x)
(x+E)^{-1/2} \sin\big\{\tfrac{4}{3}(x+E)^{3/2} -\tfrac{5\pi}{6}\big\}\,dx\,w(E)dE \\
+&\int\!\!\int_{x_0}^\infty V_n(x)\zeta(x,E)\,dx\,w(E)dE.
\end{align*}
The absolute value of the second summand is
$\lesssim \|w\|_{L^1} \{ \int |V_n(t^2)|^2\,dt\}^{1/2}$.
For the first summand, we integrate by parts with respect to $E$ using
$\frac{d}{dE} \cos\big\{\tfrac{4}{3}(x+E)^{3/2} -\tfrac{5\pi}{6}\big\} =
-(x+E)^{1/2} \sin\big\{\tfrac{4}{3}(x+E)^{3/2} -\tfrac{5\pi}{6}\big\}$ to obtain
$$
\int\!\!\int_{x_0}^\infty \cos\big\{\tfrac{4}{3}(x+E)^{3/2}-\tfrac{5\pi}{6}\big\}
\frac{(x+E)w'(E)-w(E)}{(x+E)^2} V_n(x) \,dx\,dE.
$$
which is $\lesssim \|w\|_{C^1} \int_{x_0}^\infty (x+E)^{-1} V_n(x)\,dx
\lesssim \|w\|_{C^1} \{ \int |V_n(t^2)|^2\,dt\}^{1/2}$ in absolute value.
The second inequality used the fact that $x_0>-2E$.
To recap, we have shown that the absolute value of (\ref{St:thing}) is bounded by
a multiple of $\|w\|_{C^1} \{ \int |V_n(t^2)|^2\,dt\}^{1/2}$ and thus verified
hypothesis ii) of Theorem \ref{St:th}. This shows $I\subset \Sigac(H+V)$. Since $I$ was
arbitrary, $\Sigac(H+V)=\Reals$.
\end{proof}
\input{Biblio.tex}
\end{document}
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