A representation formula related to Schr\"odinger operators
Shijun Zheng
Let $H=-d^2/dx^2 +V$ be a Schr\"odinger operator on the real line, where $V\in L^1 \cap L^2$. We define the perturbed Fourier transform $\mathcal{F}$ for $H$ and show that $\mathcal{F}$ is an isometry from the absolute continuous subspace onto $L^2(\mathbb{R})$. % consider the Schr\"odinger operator $H$ % on the real line, with short-range potentials. % construct a kernel formula for spectral operators. This property allows us to construct a kernel formula for the spectral operator $\vphi(H)$. %Schr\"odinger operator is a central subject in the mathematical study %of quantum mechanics.
math0412314.tex
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\begin{document}

\title[1D Schr\"odinger Operators]%
      {A representation formula related to
   Schr\"odinger operators} 
\author[S.~Zheng]%
       {Shijun Zheng}
\address{Department of Mathematics \\
        Louisiana State University \\
        Baton Rouge, LA 70803}
\email{szheng@math.lsu.edu}
 \urladdr{http://www.math.lsu.edu/\textasciitilde{szheng}}
 \thanks{The author is supported in part by DARPA 
(Defense Advanced Research Projects Agency)}         
\keywords{spectral theory, Schr\"odinger operator}
\subjclass[2000]{Primary: 42C15; Secondary: 35P25}
\date{\today}

%\input{abs0.tex}
\begin{abstract}

Let $H=-d^2/dx^2 +V$ be a Schr\"odinger operator on the real line, 
where $V\in L^1 \cap  L^2$.  We define the perturbed Fourier transform 
$\mathcal{F}$ for $H$ and show
that $\mathcal{F}$ is an isometry from the absolute continuous
subspace  onto $L^2(\mathbb{R})$. % consider the Schr\"odinger operator $H$ 
%  on the real line, with short-range potentials. 
% construct a kernel formula for spectral operators.
This property allows
us to construct a kernel formula for the spectral operator $\vphi(H)$.  

%Schr\"odinger operator is a central subject in the mathematical study
%of quantum mechanics. 

\end{abstract}

\maketitle 

%\makeindex

%\section{Introduction}
%\input{sec1.tex}
Schr\"odinger operator is a central subject in the mathematical study
of quantum mechanics. 
% consider the Schr\"odinger operator $H$ 
%  on the real line, with short-range potentials.
Consider the Schr\"odinger operator $H=-\triangle +V$ on $\mathbb{R}$,
where $\triangle=d^2/dx^2$ and the potential function $V$ is real valued.
In Fourier analysis, it is well-known that a square integrable function admits 
an expansion with exponentials as eigenfunctions of $-\triangle$. A
natural conjecture is that an $L^2$ function admits a similar 
expansion in terms of
``eigenfunctions'' of $H$, a perturbation of the Laplacian (see \cite{RS}.  Ch.XI and the notes), under certain condition on $V$.

The three dimension analogue was proven true by T.Ikebe \cite{Ik}, a
member of Kato's school, in 1960.  Later his result was extended by
Thor to
the higher dimension case \cite{Th}. %In an earlier time Povzner
%studied the eigenfunction expansion for $H$, where $V$ is assumed to
%be continuous \cite{Pov}.   
In one dimension, recent related results
can be found in e.g., Guerin-Holschneider \cite{GH}, Christ-Kiselev \cite{ChK}
and Benedetto-Zheng \cite{BZhen}.
% Ikebe-Povzner expansion has connections with scattering theory,
%assymptotic completeness and the absence of singular continuous
%spectrum. %I did not make the change because in IP expansion
          %\sigma_{sc} is not alway empty

Throughout this paper we assume $V: \R\rightarrow \R $ is in $L^1\cap 
L^2$. We shall prove a one-dimensional version of Ikebe's
theorem for $L^2$ functions (Theorem 1).  
% main idea of Ikebe is to seek a relation
%\int_{\alpha\le k^2\le \beta} \vert \cal{F}f(k)\vert^2dk= \Vert
%P_{[\alpha,\beta]}(H)f  \Vert^2,
Theorem 2 presents an integral formula for the kernel 
of the spectral operator 
$\vphi(H)$ for a continuous function $\vphi$ with compact 
support. In a sequel to this
paper we shall use this explicit formula to study function spaces
associated with $H$ (see  \cite{BZhen}).

%For general potential $V=V_+ - V_-$ on
%$\mathbb{R}^d$, where 
%$V_+\in \cal{K}_{loc}$ and $V_-\in \cal{K} $ in the Kato class 
%$\cal{K}$ (for $d=1$, $\int_{|x-y|\le 1} |V(y)|dy < C$ with $C$ independent of $x$),
%there is a similar result as stated in Theorem C.5.4 in
%\cite{Si2}; especially $(vi)$ and $(v)$ of this theorem show the
%possibility of realizing a spectral operator $\phi(H)$, with
% $\phi$ being bounded.
% However the spectral measure involved was not explicitly constructed.

%(The assumptions imply that $V$ is in $L^1$ and $\iint |V(x)|\cdot |V(%y)|\cdot |x-y|^2 \ud x \ud y < \infty$.)

%conditions on $V$ could be relaxed, Titchmarsh

% $H_{ac} \sigma_{ac}(H)  V= O(e^{-a|x|}) $ 
%\mathcal{E}$ is discrete. $ \sigma_{sc}= \emptyset$

The {\em generalized eigenfunctions} $e(x,\xi),\; \xi\in \R$ of $H$ satisfy
\begin{equation}
(-d^{\,2}/dx^2 + V(x)) e(x,\xi) = \xi^2 e(x,\xi) 
\label{S_eq} 
\end{equation} 
in the sense of distributions. 

%\section{ $L^2$ theory for the perturbed Fourier transform }
%\input{sec2.tex} 
\noindent
Definition.
  The {\em perturbed Fourier 
transform} $\mathcal{F}$ on $L^2$ is given by  
\begin{align}
\mathcal{F}f(\xi) = \textrm{l.i.m}.(2\pi)^{-1/2}\int f(x) 
\overline{e(x,\xi)} \, dx\\
=\lim_{N\rightarrow \infty} (2\pi)^{-1/2}\int_{-N}^N f(x) 
\overline{e(x,\xi)} \, dx,  \notag %:= \tilde{f} (\xi)  
\end{align}
where the convergence is in $L^2$ norm as $N \rightarrow \infty$.  
By Theorem 1, $\cal{F}$ is a well-defined isometry from $\cal{H}_{ac}$ onto 
$L^2$.

\begin{theorem} Suppose
$V\in L^1\cap L^2$.  Then there
exists a family of solutions $e(x,\xi)$, %$\textrm{a.e}.\;
$|\xi| \in [0,\infty)\setminus\cal{E}_0$, $\cal{E}_0 $ being a bounded closed set of measure zero,
to equation   
(\ref{S_eq}) with the following properties.  \n\n
(i) If $f \in L^2$, then %for every $N > 0$, $\xi \in \R$, the integral
%(2\pi)^{-1/2} \int_{-N}^N f(x) \overline{e(x,\xi)} \, dx
%converges absolutely. As a function of $\xi$ this integral defines an element
%$of $L^2$. Furthermore,
 there exists an element $\til{f} \in L^2$ such that
%(2\pi)^{-1/2} \int_{-N}^N f(x) 
%\overline{e(x,\xi)} \, dx  \rightarrow \tilde{f} (\xi) :=
$$ \cal{F}f(\xi)= \tilde{f} (\xi)
\qquad \textrm{in}\; L^2.  $$  
(ii) The adjoint operator $\cal{F}^*$ is given by 
$$
\cal{F}^*g=\textrm{l.i.m.}_{N\rightarrow\infty}\; \sum_{i=1}^N (2\pi)^{-1/2} \int_{\alpha_i \leq \xi^2 \leq \beta_i}
g(\xi)e(x,\xi) \, d\xi %\rightarrow P_{ac}f(x)
$$
in $L^2$,
 where  $[\alpha_i,\beta_i) \subset (0,\infty)$ are
a countable collection of disjoint intervals 
with $[0,\infty)\setminus\cal{E}_0^2$ equal to $\cup_i [\alpha_i,\beta_i)$.\n\n 
%\sum_{i=1}^N (2\pi)^{-1/2} \int_{\alpha_i \leq \xi^2 \leq \beta_i}
%\til{f}(\xi)e(x,\xi) \, d\xi \rightarrow P_{ac}f(x)
(iii) If $f \in L^2$, then $\Vert P_{ac}f \Vert_{L^2} = \Vert \tilde{f}
\Vert_{L^2}$, where $P_{ac}$ is the projection onto $\cal{H}_{ac}$, the  
absolute continuous subspace in $L^2$.   \n\n
(iv) $\cal{F}: L^2\rightarrow L^2$ is a surjection.
Moreover, $\cal{F}\cal{F}^*=Id$ and  $\cal{F}^*\cal{F}=P_{ac}$.\n\n  
(v) If $f \in \mathcal{ D}(H)$, then ${(Hf)}\sptilde(\xi) = \xi^2 \til{f}(\xi)$
in $L^2$. %for almost every $\xi$.
\end{theorem} 

\nd
Remark 1. %$V\in L^1 . \int \int |V(x)| |V(y)| |x-y|^2\ud x \ud y < \infty  
The proof is based on the ideas of \cite{Ik} for $3D$.  We also use some
simplifications as found in Reed and Simon(\cite{RS}) and 
Simon \cite{Si2}. 
 
\nd
Remark 2. If $ \vert e(x,\xi)\vert \le C $ a.e. $(x,\xi)\in \R^2$, then
we have a ``better-looking'' form in ($ii$) of the theorem 
$$ \cal{F}^*g = \textrm{l.i.m}.(2\pi)^{-1/2} \int g(\xi)e(x,\xi) \, d\xi. 
%\rightarrow P_{ac}
$$

%e(x,\xi) = e^{ix\xi} + 
%(2i \vert \xi \vert)^{-1} \int_{-\infty}^\infty e^{i\vert \xi \vert  
%\vert x-y \vert} V(y) e(y,\xi) \, dy.

% $V(\cdot)e(\cdot,\xi) \in 
%L^1(\R)$, then this solution is a {\em weak solution} to the differential 
%equation (\ref{S_eq}). It follows from standard theory and the assumption 
%$V \in L^1\cap L^{\infty}$ that such a weak solution $e(x,\xi)$ is 
%in fact $C^\infty$ in the $x$-variable (see Theorem IX.26 in \cite{RS}

%(Here $V^{1/2}(x)$ is notation for $\vert V(x) \vert^{1/2} {\rm sgn}\ V(x)$.) 

%There exists a bounded set $\mathcal{ E}_0 \subset (0,\infty)$ 
%such that: (a) the only possible limit point of $\mathcal{ E}_0$ is zero; 
%and (b) equation  
%(\ref{LS_mod_eq}) has a unique solution $\psi(\cdot,\xi) \in L^2$ for all
%$\xi \in \R $ with $\vert \xi \vert \in (0,\infty) \setminus \mathcal{ E}_0$.  
%If $\vert \xi \vert \in (0,\infty) \setminus \mathcal{ E}_0$, then the function  

%($C^\infty$ function of $x$ when $V$ is $C^{\infty}$ ) and 
%obeys equation (\ref{LS_eq}) (uniquely).

% e(x,\xi) jointly continuous in $x,\xi$ for $x \in \R$, 

%Let $f \in C^\infty_0$. Suppose $0 < \alpha < \beta$ are such that 
%$[ \sqrt{\alpha},\sqrt{\beta} \, ] \cap {\cal E}_0 = \emptyset$. Then 
%$$
%\Vert P_{[\alpha,\beta)} f \Vert_{L^2}^2 = 
%\int_{\alpha \leq \xi^2 \leq \beta} \vert \til{f}(\xi) \vert^2 \, d\xi.   

\vspace{.3in}

%\section{Kernel formula for spectral operator} 
%\input{sec3.tex}  
%\section{}
If $H=\int \lam dE_\lam$ is the spectral resolution of $H$, 
define the spectral operator $\vphi(H):=\int \vphi(\lam) dE_\lam $ 
by  functional calculus.
We prove a representation formula for  the integral
kernel of $\varphi(H)$. %=\cal{F}^{-1}\vphi(\xi^2)\cal{F}$.

%(i) If $f \in L^2$ and $\xi^2 \til{f}(\xi) \in L^2$, 
%then $f \in \cal{ D}(H)$. \n\n

%Then $ ( P_{[\alpha, \beta]} f)\sptilde{ (\xi )}= \one_{[\sqrt{\alpha}, \sqrt{\beta}] } \til{f} (\xi)$. 

%\Vert (P_{[\alpha,\beta]}f)\sptilde  \Vert^2_{L^2}  
%= \Vert P_{[\alpha,\beta]}f \Vert^2_{L^2}  = \int_{\alpha < \xi^2 < \beta}
%\vert (P_{[\alpha,\beta]}f)\sptilde{(\xi)} \vert^2 \, d\xi. \\
%(P^2_{[\alpha, \beta)} = P_{[\alpha, \beta)} \;\;  by \; Lem \;2.7)

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis in $\cal{H}_p$, the 
subspace of eigenfunctions in $L^2$ for $H$ and let $\lam_k$ be the eigenvalue corresponding to $e_k$.

\begin{theorem}    
Let the operator $H$ be as in Theorem 1. %$\sigma_{sc}(H)=\emptyset$.
Suppose $\varphi : \R \rightarrow \C$ is continuous  and has a compact 
 support disjoint from $\cal{E}_0^2:=\{\eta^2:\eta\in \cal{E}_0  \}$. 
%\subset [\alpha,\beta]$, where $0 < \alpha < \beta$, $[\sqrt{\alpha},\sqrt{\beta}] \cap \cal{ E}_0 = \emptyset$. 
Then for $f \in L^1 \cap L^2$
\begin{equation}
\varphi(H)f(x) = \int_{-\infty}^\infty K(x,y) f(y) \, dy 
\label{eq:kernel}
\end{equation}
where
$K=K_{ac}+K_p$,
$$
K_{ac}(x,y) = (2\pi)^{-1} \int_{-\infty}^\infty \varphi(\xi^2) e(x,\xi)
\overline{e(y,\xi)} \, d\xi.
$$
 and 
$$ 
K_p(x,y)= \sum_k \vphi(\lam_k)e_k(x)\bar{e}_k(y).
$$
%is in $\cal{S}\cap C^\infty$ 
\end{theorem}

\nd
Remark 1. If $|e(x, \xi) | \le C$, a.e. $(x,\xi) \in \R^2$, 
then, under the same condition the integral expression (\ref{eq:kernel}) is
valid for any $\vphi \in C(\R)$ with compact support. 
%when $H=H_0, where $P_{[\alpha, \beta)} f= \mathcal{F}^{-1}\chi_{[\alpha, \beta)} (\xi^2) \mathcal{F} f$.   

\nd
Remark 2. When $\vphi$ is smooth with rapid decay and $V$ is compactly
supported in $\R^3$, a formula
of this type appeared in \cite{Tao} by Tao.

\begin{thebibliography}{9}
%\input{ref.tex}
\bibitem{AS} P.~Alsholm, G.~Schmidt, Spectral and scattering theory
for Schr\"odinger operators, {\em Arch. Rational Mech. Anal.} {\bf 40}
(1971), 281--311.  

\bibitem{Ben} J.~J. Benedetto, {\em Harmonic Analysis and Applications},
CRC Press, Inc., Boca Raton. FL, 1997.

\bibitem{BZhen} J.~J. Benedetto and S.~Zheng, 
Besov spaces for the Schr\"odinger operator with barrier potential, 
{\em submitted}.

%\bibitem{BEKS} J.F.Brasche, P.Exner, Yu.A.Kuperin,
%P.\u Seba, Schr\"odinger operators with singular interactions,
%{\em J.Math.Anal.Appl.} {\bf 184}, 112-139(1994).

\bibitem{ChK} M.~Christ and A.~Kiselev, One-Dimensional Schr\"odinger
operators with slowly decaying potentials: spectra and asymptotics, or, 
{\em Baby Fourier Analysis Meets Toy Quantum Mechanics}, Notes for
IPAM tutorial, 2001 Workshop on Oscillatory Integrals and
Dispersive Equations.  

%\bibitem{ChK3}---, WKB asymptotic behavior of (almost all) generalized
%eigenfunctions for one-dimensional Schr\"odinger operators (with slowly
%decaying potentials), {\em J.Func.Anal.} {\bf 179}, (2001), 426-447. 

%\bibitem{DOS} X.T. Duong, E.M. Ouhabaz and A. Sikora, Plancherel type
%estimates and sharp spectral multipliers, to appear in {\em J.Func.Anal}.

%\bibitem{DZ} J.Dziuba\'nsk, Triebel-Lizorkin spaces associated with
%Laguerre and Hermite expansions, {\em Proc. Amer. Math. Soc.}{\bf
%125}(1997), 3547-3554.

%\bibitem{DZ2} J.Dziuba\'nski and J. Zienkiewicz, Hardy space $H^1$
%associated to Schr\"odinger operator with potential satisfying
%reverse H\"older inequality, {\em Rev. Mat. Iberoamericana} {\bf
%15}(2)( 1999), 279-296.

%\bibitem{FR} Fayyazuddin, Riazuddin, {\em Quantum Mechanics}, World Scientific
%Publishing Co.Pte.Ltd., 1990.

\bibitem{GH} C.-A.~Guerin, M.~Holschneider, Time-dependent scattering
on fractal measures,  {\em J. Math. Physics} {\bf 39}(8), 1998. 

%\bibitem{Her} J.~Herczynki, On Schr\"odinger operators with
%distributional potentials, {\em J. Operator Theory} {\bf 21}, 273-293(1989).

\bibitem{Ik} T.~Ikebe, Eigenfunction expansions associated with the 
Schr\"odinger operators and their applications to scattering theory, 
{\em Arch. Rational Mech. Anal.} {\bf 5} (1960), 1--34. 
 \quad (Erratum, Remarks on the orthogonality of eigenfunctions for the
Schr\"odinger operator on $\R^n$, J.~Fac.~Sci.~Univ. Tokyo Sect.I {\bf 17}, 1970)%,355-361)

%\bibitem{JN} A. Jensen, S. Nakamura, Mapping properties of functions
%of Schr\"odinger operators between $L^p$ spaces and Besov spaces, 
%in {\em Spectral and Scattering Theory and Applications}, Advanced 
%Studies in Pure Math. 23, Kinokuniya Co. Ltd., Tokyo, 1994. 

% -- $L^p$ -mapping properties of functions of
%Schr\"odinger operators and their applications to scattering theory


%\bibitem{Kuro} S.T.Kuroda, Perturbation of eigenfunction
%expansions. {\em Proc. Nat. Acad. Sci. U.S.A.} {\bf 57} (1967),
% 1213--1217.
  
%\bibitem{Pov} A.Povzner, On the expansion of arbitrary functions in
% terms of the eigenfunctions of the operator $-\triangle u+ cu$,
%{\em Mat.Sb. (N.S.)} {\bf 32} (1953 ), 109-156 (Russian).

%\bibitem{Pov2} ---, On eigenfunction expansions in terms of
%scattering solutions, {\em Dokl. Akad. Nauk. SSSR} {\bf 104} (1955). 

\bibitem{RS} M.~Reed and B.~Simon, {\em Methods of Modern Mathematical
Physics III: Scattering Theory}, 
Academic Press, New York, 1979.
% revised ed., Academic Press, New York, 1980.  
%\bibitem{RS2} ---, {\em Methods of Modern Mathematical 
%Physics II: Fourier Analysis, Self-Adjointness}, 
%Academic Press, New York, 1975. 
%\bibitem{RS3} ---, {\em Methods of Modern Mathematical
%Physics III: Scattering Theory}, Academic Press, New York, 1979. 

\bibitem{Si2} B.~Simon, {\em Quantum Mechanics for Hamiltonians Defined 
as Quadratic Forms}, Princeton University Press, Princeton, New Jersey, 
1971. 

%\bibitem{Si2}---, Schr\"odinger semigroups, {\em Bull.Amer.Math.Soc.}, {\bf
%7}(3), 1982.
%\bibitem{Si3}---, Bounded eigenfunctions and absolutely continuous
%spectra for one-dimensional Schr\"odinger operators,
%{\em Proc. Amer. Math. Soc.} {\bf 134}(11), 1996.

%\bibitem{Sto}
%G.Stolz, Bounded solutions and absolute continuity of Sturm-Liouville
%operators, J.Math.Anal.Appl. {\bf 169} (1992), 210-228.

\bibitem{Tao} 
T.~Tao, Scattering for the $3D$ Schr\"odinger equation with
compactly supported potential, \emph{Preprint}.

%\bibitem{Tay2} M.~Taylor, Partial Differential Equations II, {\em Qualitative
%Studies of Linear Equations}, Applied Mathematical Sciences {\bf 116},
%Springer Verlag, 1996. 

\bibitem{Th} D.~Thoe, Eigenfunction expansions associated with
Schr\"odinger operators in $\R^n$, $n \geq 4$, {\em Arch. Rational Mech. Anal.}
{\bf 26} (1967), 335--356. 

\end{thebibliography}
\end{document}


math0412314.extracted_bib
\begin{thebibliography}{9}
%\input{ref.tex}
\bibitem{AS} P.~Alsholm, G.~Schmidt, Spectral and scattering theory
for Schr\"odinger operators, {\em Arch. Rational Mech. Anal.} {\bf 40}
(1971), 281--311.  

\bibitem{Ben} J.~J. Benedetto, {\em Harmonic Analysis and Applications},
CRC Press, Inc., Boca Raton. FL, 1997.

\bibitem{BZhen} J.~J. Benedetto and S.~Zheng, 
Besov spaces for the Schr\"odinger operator with barrier potential, 
{\em submitted}.

%\bibitem{BEKS} J.F.Brasche, P.Exner, Yu.A.Kuperin,
%P.\u Seba, Schr\"odinger operators with singular interactions,
%{\em J.Math.Anal.Appl.} {\bf 184}, 112-139(1994).

\bibitem{ChK} M.~Christ and A.~Kiselev, One-Dimensional Schr\"odinger
operators with slowly decaying potentials: spectra and asymptotics, or, 
{\em Baby Fourier Analysis Meets Toy Quantum Mechanics}, Notes for
IPAM tutorial, 2001 Workshop on Oscillatory Integrals and
Dispersive Equations.  

%\bibitem{ChK3}---, WKB asymptotic behavior of (almost all) generalized
%eigenfunctions for one-dimensional Schr\"odinger operators (with slowly
%decaying potentials), {\em J.Func.Anal.} {\bf 179}, (2001), 426-447. 

%\bibitem{DOS} X.T. Duong, E.M. Ouhabaz and A. Sikora, Plancherel type
%estimates and sharp spectral multipliers, to appear in {\em J.Func.Anal}.

%\bibitem{DZ} J.Dziuba\'nsk, Triebel-Lizorkin spaces associated with
%Laguerre and Hermite expansions, {\em Proc. Amer. Math. Soc.}{\bf
%125}(1997), 3547-3554.

%\bibitem{DZ2} J.Dziuba\'nski and J. Zienkiewicz, Hardy space $H^1$
%associated to Schr\"odinger operator with potential satisfying
%reverse H\"older inequality, {\em Rev. Mat. Iberoamericana} {\bf
%15}(2)( 1999), 279-296.

%\bibitem{FR} Fayyazuddin, Riazuddin, {\em Quantum Mechanics}, World Scientific
%Publishing Co.Pte.Ltd., 1990.

\bibitem{GH} C.-A.~Guerin, M.~Holschneider, Time-dependent scattering
on fractal measures,  {\em J. Math. Physics} {\bf 39}(8), 1998. 

%\bibitem{Her} J.~Herczynki, On Schr\"odinger operators with
%distributional potentials, {\em J. Operator Theory} {\bf 21}, 273-293(1989).

\bibitem{Ik} T.~Ikebe, Eigenfunction expansions associated with the 
Schr\"odinger operators and their applications to scattering theory, 
{\em Arch. Rational Mech. Anal.} {\bf 5} (1960), 1--34. 
 \quad (Erratum, Remarks on the orthogonality of eigenfunctions for the
Schr\"odinger operator on $\R^n$, J.~Fac.~Sci.~Univ. Tokyo Sect.I {\bf 17}, 1970)%,355-361)

%\bibitem{JN} A. Jensen, S. Nakamura, Mapping properties of functions
%of Schr\"odinger operators between $L^p$ spaces and Besov spaces, 
%in {\em Spectral and Scattering Theory and Applications}, Advanced 
%Studies in Pure Math. 23, Kinokuniya Co. Ltd., Tokyo, 1994. 

% -- $L^p$ -mapping properties of functions of
%Schr\"odinger operators and their applications to scattering theory


%\bibitem{Kuro} S.T.Kuroda, Perturbation of eigenfunction
%expansions. {\em Proc. Nat. Acad. Sci. U.S.A.} {\bf 57} (1967),
% 1213--1217.
  
%\bibitem{Pov} A.Povzner, On the expansion of arbitrary functions in
% terms of the eigenfunctions of the operator $-\triangle u+ cu$,
%{\em Mat.Sb. (N.S.)} {\bf 32} (1953 ), 109-156 (Russian).

%\bibitem{Pov2} ---, On eigenfunction expansions in terms of
%scattering solutions, {\em Dokl. Akad. Nauk. SSSR} {\bf 104} (1955). 

\bibitem{RS} M.~Reed and B.~Simon, {\em Methods of Modern Mathematical
Physics III: Scattering Theory}, 
Academic Press, New York, 1979.
% revised ed., Academic Press, New York, 1980.  
%\bibitem{RS2} ---, {\em Methods of Modern Mathematical 
%Physics II: Fourier Analysis, Self-Adjointness}, 
%Academic Press, New York, 1975. 
%\bibitem{RS3} ---, {\em Methods of Modern Mathematical
%Physics III: Scattering Theory}, Academic Press, New York, 1979. 

\bibitem{Si2} B.~Simon, {\em Quantum Mechanics for Hamiltonians Defined 
as Quadratic Forms}, Princeton University Press, Princeton, New Jersey, 
1971. 

%\bibitem{Si2}---, Schr\"odinger semigroups, {\em Bull.Amer.Math.Soc.}, {\bf
%7}(3), 1982.
%\bibitem{Si3}---, Bounded eigenfunctions and absolutely continuous
%spectra for one-dimensional Schr\"odinger operators,
%{\em Proc. Amer. Math. Soc.} {\bf 134}(11), 1996.

%\bibitem{Sto}
%G.Stolz, Bounded solutions and absolute continuity of Sturm-Liouville
%operators, J.Math.Anal.Appl. {\bf 169} (1992), 210-228.

\bibitem{Tao} 
T.~Tao, Scattering for the $3D$ Schr\"odinger equation with
compactly supported potential, \emph{Preprint}.

%\bibitem{Tay2} M.~Taylor, Partial Differential Equations II, {\em Qualitative
%Studies of Linear Equations}, Applied Mathematical Sciences {\bf 116},
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\end{thebibliography}