%
% What follows is a plain TeX file. 
%



\magnification1200
\tolerance=10000
\hsize=17truecm\vsize=23truecm
\multiply\baselineskip by 15\divide \baselineskip by10
\parindent=40pt
\font\eightpoint=cmsl8



%\input mssymb 
\def\Bbb{\bf} %if \Bbb not available
 
% shorthand

\def\<{\langle}
\def\>{\rangle}



% misc math stuff

\def\slim{\mathop{\hbox{\rm s-lim}}}
\def\Ker{\mathop{\rm Ker}}
\def\Ran{\mathop{\rm Ran}}
\def\Re{\mathop{\rm Re}}
\def\Im{\mathop{\rm Im}}
\def\ess{\hbox{\it ess}}
\def\d{\hbox{\it d}}
\def\c{\hbox{\it c}}
\def\pp{\hbox{\it pp}}

\def\const{{\rm const}}
\def\parderivs#1#2{{\partial#1\over \partial#2}}
\def\parderiv#1{{\partial\over \partial#1}}
\def\secparderivs#1#2{{\partial^2#1\over \partial#2^2}}
\def\secparderiv#1{{\partial^2\over \partial#1^2}}
\def\derivs#1#2{{d #1\over d #2}}
\def\deriv#1{{d\over d #1}}
\def\secderivs#1#2{{d^2#1\over d #2^2}}
\def\secderiv#1{{d^2\over d #1^2}}
\def\nderivs#1#2#3{{d^{#1}#3\over d #2^{#1}}}
\def\nderiv#1#2{{d^{#1}\over d #2^{#1}}}
\def\inprod#1{\langle#1\rangle}
\def\max{{\rm max}}
\def\min{{\rm min}}


%symbols

\def\L{{\rm L}}

\def\R{{\bf R}}
\def\Clow{\lower1pt\hbox{\bf C}}
\def\C{{\bf C}}

\def\O{{\cal O}}
\def\H{{\cal H}}
\def\W{{\cal W}}
\def\J{{\cal J}}
\def\F{{\cal F}}
\def\P{{\cal P}}
\def\V{{\cal V}}
\def\U{{\cal U}}
\def\N{{\cal N}}
\def\Q{{\cal Q}}
\def\B{{\cal B}}
\def\D{{\cal D}}

% misc

%
% 	the stuff after \ifdraftmode only appears if \draftmode has previously 
%	appeared
%
\def\draftmode{\let\dfmd 1}
\def\ifdraftmode#1{\ifx\dfmd 1{#1}\fi}

\def\today{1\slash 7\slash 1996}



\def\quote#1{{\narrower\smallskip\noindent\it#1\smallskip}}

\def\Box{\vbox{\hrule
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%
%       the following macros read in the files \jobname.eqr 
%       and \jobname .thm which 
%       contain the names of equations and how they are nested 
%       within sections in a format like 
%       \\{name1}\bgnscn\\{name2}\bgnscn\\{name3}\ndscn\ndscn
%       In this example \eqlabelname1 will expand to 1
%                       \eqlabelname2 will expand to 1.1
%                       \eqlabelname3 will expand to 1.1.1
%       This is the maximum nesting supported
%       In the theorem list each \\{name} is preceded by \typetheorem,
%       \typelemma or \typeproposition to indicate what it is
%



\def\bgnsctn{
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                \edef\prenum{\number\seclevelone .\number\secleveltwo .}
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        }
\def\ndsctn{\advance\secnest by-1}

\def\thmtype{} %initialize
\def\typetheorem{\def\thmtype{Theorem}}
\def\typelemma{\def\thmtype{Lemma}}
\def\typeproposition{\def\thmtype{Proposition}}
\def\typecorollary{\def\thmtype{Corollary}}
\def\typehypothesis{\def\thmtype{Hypothesis}}




%
%       \beginsection begins a new section
%       writes \bgnsctn to token list in \eqrtoks
%       \endsection ends a section
%       writes \ndsctn to token list in \eqrtoks
%
\def\beginsection{
        \advance\secnumber by1
        \probctr=0
	\toksdef\ta=0
	\ta={\bgnsctn}
        \immediate\write\eqroutfile{\the\ta}
        \immediate\write\thmoutfile{\the\ta}
        }
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        \toksdef\ta=0
	\ta={\ndsctn}
        \immediate\write\eqroutfile{\the\ta}
        \immediate\write\thmoutfile{\the\ta}
        }

%
%       \num{name} inserts (<number>) where <number> is the equation number
%       referred to by "name"
%
\def\num#1{(\csname eqlabel#1\endcsname )}

%
%       \thmnum is the analogous thing for theorems
%
\def\thmnum#1{\hbox{\csname typethmlabel#1\endcsname}}



%
% first do the references - define \\ to be a macro which defines \eqlabelname 
% to be a reference number and then evaluate the input string as in Appendix D
%
\def\\#1{
        \advance\cntnumber by1
        \expandafter\xdef\csname 
        eqlabel#1\endcsname{\prenum\number\cntnumber}
        }
\newcount\seclevelone \seclevelone=0
\newcount\secleveltwo \secleveltwo=0
\newcount\cntnumber \cntnumber=0
\newcount\secnest \secnest=0
\def\prenum{}
\newread\eqrinfile 
\openin\eqrinfile=\jobname .eqr 
\ifeof\eqrinfile
  \def\eqrinput{\relax}
\else\loop
  \read\eqrinfile to\eqrinput
  \eqrinput
  \ifeof\eqrinfile\let\endfile 1\else\let\endfile 0\fi
  \ifx\endfile 0 \repeat
\fi
\closein\eqrinfile

%
% now the same thing for theorems
%
\def\\#1{
        \advance\cntnumber by1
        \expandafter\xdef\csname 
        thmlabel#1\endcsname{\prenum\number\cntnumber}
        \expandafter\xdef\csname 
        typethmlabel#1\endcsname{\thmtype\ \prenum\number\cntnumber}
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\seclevelone=0
\secleveltwo=0
\cntnumber=0
\secnest=0
\def\prenum{}
\newread\thminfile 
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\ifeof\thminfile
  \def\thminput{\relax}
\else\loop
  \read\thminfile to\thminput
  \thminput
  \ifeof\thminfile\let\endfile 1\else\let\endfile 0\fi
  \ifx\endfile 0 \repeat
\fi
\closein\thminfile
        
%
% theorems etc
% These are numbered by section
%

\def\begintheoremlabel#1{
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	\ta={\typetheorem\\{#1}}
        \immediate\write\thmoutfile{\the\ta}
        \medbreak\par\noindent{\ifdraftmode{\mcomment{#1}}\bf Theorem \csname thmlabel#1\endcsname :}
        \begingroup \it
}
\def\endtheoremlabel{\par\medbreak\endgroup}

\def\begintheorem{
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        \edef\temp{{noname\the\nonamectr}}
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}
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        \begingroup \it
}
\def\endlemmalabel{\par\bigbreak\endgroup}

\def\beginlemma{
        \advance\nonamectr by1
        \edef\temp{{noname\the\nonamectr}}
        \expandafter\beginlemmalabel\temp
}
\def\endlemma{\endlemmalabel}



\def\beginpropositionlabel#1{
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}
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        \medbreak\par\noindent{\ifdraftmode{\mcomment{#1}}\bf Corollary \csname thmlabel#1\endcsname :}
        \begingroup \it
}
\def\endcorollarylabel{\par\medbreak\endgroup}

\def\begincorollary{
        \advance\nonamectr by1
        \edef\temp{{noname\the\nonamectr}}
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}
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\def\beginhypothesislabel#1{
        \toksdef\ta=0
	\ta={\typehypothesis\\{#1}}
        \immediate\write\thmoutfile{\the\ta}
        \medbreak\par\noindent{\ifdraftmode{\mcomment{#1}}\bf Hypothesis \csname thmlabel#1\endcsname :}
        \begingroup \it
}
\def\endhypothesislabel{\par\medbreak\endgroup}

\def\beginhypothesis{
        \advance\nonamectr by1
        \edef\temp{{noname\the\nonamectr}}
        \expandafter\beginhypothesislabel\temp
}
\def\endhypothesis{\endhypothesislabel}

%
% The following allows one to give a theorem or lemma name
% from a different part of the paper 
% to a result quoted here. 
%

\def\begintheoremref#1{
        \bigbreak\par\noindent{\bf \hbox{\csname typethmlabel#1\endcsname}:}
        \begingroup \it
                       }
\def\endtheoremref{\par\medbreak\endgroup}



%
%
%	open files for writing
%
%
\newwrite\eqroutfile 
\immediate\openout\eqroutfile=\jobname .eqr 
\newwrite\thmoutfile 
\immediate\openout\thmoutfile=\jobname .thm




%
%       \be{name} and \ee begin and end a numbered displayed equation 
%       the tokens in the name are added to the token list in \eqrtoks
%       if \eqlabelname has been defined uses this as a number
%       \lastdisplayline{name}{...} gives a referenced number  to the last 
%       line in a displaylines series
%
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	\toksdef\ta=0
	\ta={\\{#1}}
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	}
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        }
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        \eqno{(\csname eqlabel\temp\endcsname #1)
	\ifdraftmode{\rlap{\ {\smallfont\temp}}}}$$
        }
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	\ifdraftmode{\rlap{\ {\smallfont#1}}}}\cr
        \toksdef\ta=0
	\ta={\\{#1}}
        \immediate\write\eqroutfile{\the\ta}
        }



\def\beginproof{\bigbreak\par\noindent{\bf proof:\ }}
%\def\endproof{\thinspace\Box\par\medbreak}
\def\endproof{\hfill\vrule height .75em width .75em depth 0pt\bigbreak}

\def\beginproofof#1{\bigbreak\par\noindent{\bf proof of #1:\ }}
%\def\endproofof{\thinspace\Box\par\medbreak}



\def\begindefn{\medbreak\par\noindent{\it Definition:\/\ }}
\def\enddefn{\par\medbreak}

\def\beginexample{\medbreak\par\noindent{\it Example:\/\ }}
\def\endexample{\par\medbreak}




\def\chead#1{\medbreak\centerline{#1}\medskip}

\def\bea#1{$$\eqalign\begingroup}
\def\eea{\endgroup $$}
\def\beann{$$\eqalign\begingroup}
\def\eeann{\endgroup $$}


%
%       These go at the beginning  of input
%
\global\newtoks\eqrtoks
\global\newtoks\thmtoks
\global\newcount\secnumber
\global\newcount\nonamectr
\global\newcount\probctr

\message{reference macros need to run twice to get references right}








\ \vskip1cm

\centerline{\bf Generalized Eigenfunction Expansion Near Resonances}

\vskip1cm
\centerline{Roger Waxler}
\medskip
\centerline{\it Department of Mathematics, S.U.N.Y. at Buffalo}
\centerline{\it Buffalo, N.Y., U.S.A. 14214-3093}
\centerline{\it rwax@newton.math.buffalo.edu}


\bigskip\bigskip
{\narrower\bigskip\noindent{\bf Abstract:\/} The non-relativistic quantum mechanical description of meta-stable states which arise by perturbation of embedded eigenvalues is considered. The model given by the Hamiltonian
$$
H(\lambda)=\pmatrix{-{d^2\over dx^2} & \lambda u \cr
           \lambda u & -{d^2\over dx^2}+x^2 \cr}
$$
is studied for small $\lambda$. This model is a simple resonant model. At $\lambda=0$ there is a sequence of embedded eigenvalues, $\{1,3,5,\dots\}$, which become resonances for $\lambda\ne0$. The eigenfunction expansion for $H(\lambda)$ is obtained for small $\lambda$. Systematic approximations for the continuum eigenfunctions are obtained which are uniform in the spectral parameter. The scattering theory for this model is considered and it is shown that scattering cross sections have the expected Breit-Wigner peaks near the resonances.
\bigskip}


\vfill\eject




\beginsection

\def\header#1#2{
              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint #2\hfill \today}}
              }


\header{\S 1: Introduction and Statement of Results}{Introduction}



One of the mechanisms which produces meta-stable states in Quantum Mechanics is the Feshbach resonance. Let $H(\lambda)$ be the Hamiltonian describing some physical system; $\lambda$ is some real parameter. Feshbach resonances arise when $H(0)$ has eigenvalues embedded in its continuous spectrum while $H(\lambda)$, for $\lambda\ne0$, does not. Thus there are bound states at $\lambda=0$ which become continuum states for any $\lambda\ne0$; these are the so called meta-stable states. In particular, the $\lambda\to0$ limit of the time evolution operator $e^{-itH(\lambda)}$ is singular. The physics associated with this singular behavior is that of meta-stability and resonant scattering.


Formal schemes for estimating some of the physical quantities relevant to meta-stability and resonant scattering were developed early in the history of Quantum Mechanics ([GW], [WW]). Some important features of meta-stability are exponential decay and natural line width ([WW]). Important to resonant scattering are the Breit-Wigner peaks ([LL]). Let $\phi$ be a meta-stable state for $H(\lambda)$ corresponding to an unperturbed embedded eigenvalue of $H(0)$, $\epsilon_0$. Exponential decay refers to the expectation that there is some number $\Gamma>0$, depending on $\lambda$, for which
$$
|\inprod{\phi,e^{-itH(\lambda)}\phi}|\sim e^{-t\Gamma}+\O(\Gamma).
$$
Natural line width refers to the expectation that in the $t\to\infty$ limit $e^{-itH(\lambda)}\phi$ is contained in the absolutely continuous spectral subspace of $H(0)$ and that, in a representation in which $H(0)$ is diagonal,
$$
\lim_{t\to\infty}|\big(e^{-itH(\lambda)}\phi\big)(\epsilon)|^2\sim \const\ {1\over(\epsilon-\epsilon_0)^2+\Gamma^2};
$$
here $H(0)$ is diagonal in the sense that $\big(H(0)\psi\big)(\epsilon)=\epsilon\psi(\epsilon)$ for all $\psi$ in the absolutely continuous spectral subspace of $H(0)$; $\epsilon$ is the spectral parameter. The Breit-Wigner peaks refer to peaks in the scattering cross sections again of the form ${\const\over(\epsilon-\epsilon_0)^2+\Gamma^2}$. These peaks presumably reflect the excitation and subsequent decay of a meta-stable state. In all of these expressions $\Gamma$ is the same positive real number which goes to zero as $\lambda\to0$ (if $H(\lambda)$ is a polynomial in $\lambda$ then typically $\Gamma\sim\lambda^2$).


There has been only partial success in mathematically justifying these formal expectations. An early review of the problems is found in [S]. The strongest results on exponential decay are found in [H]. There are very few rigorous results on either line width ([W]) or resonant scattering for Feshbach resonances. In this paper a model is studied which is exactly solvable in the sense that a closed form expression for the continuum eigenfunctions of $H(\lambda)$ can be obtained. These continuum eigenfunctions are then used to control the scattering theory in this model and to establish the existence of Breit-Wigner peaks. In a subsequent paper it is shown how to obtain small $\lambda$ asymptotics for $e^{-itH(\lambda)}$ which are uniform in $t$.



The model considered is described formally by the Hamiltonian
$$
H(\lambda)=\pmatrix{-{d^2\over dx^2} & \lambda u \cr
           \lambda u & -{d^2\over dx^2}+x^2 \cr}
$$
acting in $\L^2(\R)\oplus\L^2(\R)$. Here $u$ is the multiplication operator corresponding to the function $u(x)$. We take $u$ to be continuous and compactly supported in $\R$. This model is a simple resonant model and is very similar to the model studied in [W]. For $\lambda=0$ the spectrum of $H(0)$ is $[0,\infty)$ and consists of absolutely continuous spectrum and the embedded eigenvalues $\{2j+1\, \big|\, j\in{\bf N}\}$. For $\lambda\neq 0$ the essential spectrum of $H(\lambda)$ is still $[0,\infty)$ since $H(\lambda)-H(0)$ is a relatively compact perturbation. There are, however, no longer any embedded eigenvalues since the equation $\big(H(\lambda)-\epsilon\big)\Psi=0$ has no non-zero $L^2$ solutions if $\epsilon\ge0$.




In this paper the spectral measure for $H(\lambda)$, $\derivs{\mu}{\epsilon}(\lambda,\epsilon)$, is constructed for $\lambda$ sufficiently small. We will see that the spectrum of $H(\lambda)$ consists of one negative eigenvalue, $E(\lambda^2)$, and absolutely continuous spectrum $[0,\infty)$. Our chief concern is with the absolutely continuous spectrum and the resonant behavior caused by the unperturbed embedded eigenvalues $2j+1$. Using $\derivs{\mu}{\epsilon}(\lambda,\epsilon)$ a generalized eigenfunction expansion associated to $H(\lambda)$ is obtained. The continuum eigenfunctions, $\Psi(\lambda,k)$, are found to be meromorphic functions of $k$ in an open neighborhood of the real line excluding $k=0$. The $\Psi(\lambda,k)$ have simple poles in $k$ when $k^2=E(\lambda^2)$ as well as for $k^2=\epsilon_j(\lambda^2)$, for some numbers $\epsilon_j(\lambda^2)$. Here the number $\epsilon_j(\lambda^2)$ is generally referred to as the resonance associated with the unstable eigenv!
 alue $2j+1$. It is shown that the resonances are analytic functions of $\lambda^2$, have negative imaginary part and are given, for each $j\in\{0,1,2,\dots\}$, by $\epsilon_j(\lambda^2)=2j+1+\O(\lambda^2)$. 



The techniques used are constructive and provide systematic approximations to $\Psi(\lambda,k)$ for small $\lambda$. These approximations are uniform in $k$ and have a particularly simple form in leading order when $k^2$ is sufficiently close to $2j+1$. It is clear that $\Psi(\lambda,k)$ cannot be analytic in $\lambda$ since the nature of the spectrum of $H(\lambda)$, for $\lambda\ne0$, is drastically different from that of $H(0)$. Rather, we will see that $\Psi(\lambda,k)$ is given by a term which is singular as $\lambda\to0$ plus a term which is analytic in $\lambda$.


Finally the scattering theory for this model is considered. The comparison operator appropriate to scattering in this model is $-{d^2\over dx^2}$ acting in $\L^2(\R)$. The stationary wave operators ([RS] [Y]) for this system are easily extracted from $\Psi(\lambda,k)$. Standard methods ([RS]) then show that the stationary wave operators are in fact the wave operators
$$
\Omega_\pm=\slim_{t\to\mp\infty}e^{itH(\lambda)}\pmatrix{1\cr0\cr}e^{it{d^2\over dx^2}}.
$$
The $S$ matrix, $S=\Omega_-^*\Omega_+$, is then computed and is found to have the expected Breit-Wigner peaks.



The results obtained are stated below.


\noindent{\bf Results:}\begingroup\it\ There is a $\lambda_0>0$ such that if $0<|\lambda|<\lambda_0$ then the following is true:

\noindent a) The spectrum of $H(\lambda)$ consists of one negative eigenvalue and absolutely continuous spectrum $[0,\infty)$. Explicitly,
$$
\sigma\big(H(\lambda)\big)=\{E(\lambda^2)\}\cup [0,\infty),
$$
$\sigma_{pp}\big(H(\lambda)\big)=\{E(\lambda^2)\}$, $\sigma_{ac}\big(H(\lambda)\big)=[0,\infty)$ and $\sigma_{sing}\big(H(\lambda)\big)=\emptyset$. 

Further, for each $j\in\{0,1,2,\dots\}$ there is a resonance $\epsilon_j(\lambda^2)$ associated with the unperturbed embedded eigenvalue $2j+1$. The resonances $\epsilon_j(\lambda^2)$ are estimated by
$$
|\epsilon_j(\lambda^2)-\lambda^2\delta_j+i\lambda^2\gamma_j|\le\const\ \lambda^4
$$
where $\delta_j$ and $\gamma_j$ are real numbers and
$$
\gamma_j={|\inprod{e^{i\sqrt{2j+1}x}u,\phi_j}|^2\over2\sqrt{2j+1}}.
$$
Here $\phi_j$ is the eigenfunction of $-{d^2\over dx^2}+x^2$ corresponding to the eigenvalue $2j+1$.\hfill\break
b) The eigenvalue $E(\lambda^2)$, satisfies the estimate
$$
E(\lambda^2)=-{\lambda^4\over2}\Big(\inprod{u,\big(-{d^2\over dx^2}+x^2\big)^{-1}u}\Big)^2+\O(\lambda^6).
$$
Further, $E(\lambda^2)$ is simple, and its eigenfunction satisfies the $L^2$ norm estimate
$$
\Psi_0={1\over{\cal N}_0}\pmatrix{-\lambda \big(-{d^2\over dx^2}-E(\lambda^2)\big)^{-1}u\big(-{d^2\over dx^2}+x^2\big)^{-1}u\cr\big(-{d^2\over dx^2}+x^2\big)^{-1}u\cr}+\O(\lambda^2).
$$
Here
$$
{\cal N}_0=\sqrt{\lambda^2\|\big(-{d^2\over dx^2}-E(\lambda^2)\big)^{-1}u\big(-{d^2\over dx^2}+x^2\big)^{-1}u\|^2+\|\big(-{d^2\over dx^2}+x^2\big)^{-1}u\|^2}
$$
is a normalization constant.\hfill\break
c) The generalized eigenfunctions are given by
$$
\Psi(\lambda,k;x)={1\over\sqrt{2\pi}}\pmatrix{e^{-ik x}+i{\lambda^2\over2|k|}\int e^{i|k|\, |x-r|}u(r)\psi(k;r)\, dr\cr 
\lambda\psi(k;x)\cr}
$$
where $\psi(k)\in\L^2(\R)$ satisfies the integro-differential equation
$$
\big(-{d^2\over dx^2}+x^2-k^2\big)\psi(k;x)-i{\lambda^2\over2|k|}u(x)\int e^{i|k|\, |x-r|}u(r)\psi(k;r)\, dr=e^{-ikx}u(x)
$$
and, for $k$ such that $\min\{{1\over2},2j-{1\over2}\}<k^2<2j+{5\over2}$, satisfies the estimate
$$
\|\psi(k)-{\inprod{\phi_j,e^{- ik\cdot}u}\over k^2-\epsilon_j(\lambda^2)}\, \phi_j\|\le\const\ .
$$
As above, $\phi_j$ is the eigenfunction of $-{d^2\over dx^2}+x^2$ corresponding to the eigenvalue $2j+1$.\hfill\break
d) The scattering matrix is given in momentum space by
$$
S(k,q)=\alpha(k)\delta(k-q)+\beta(k)\delta(k+q);
$$
here $S(k,q)$ is defined by
$$
[S\phi](x)={1\over\sqrt{2\pi}}\int S(q,k)\hat\phi(k)e^{iqx}\, dk\, dq,
$$
$\hat\phi$ is the Fourier transform of $\phi$. For $k$ such that $\min\{{1\over2},2j-{1\over2}\}<k^2<2j+{5\over2}$ the coefficients $\alpha$ and $\beta$ have the expected Breit-Wigner form
$$
|\alpha(k)|^2={\lambda^4|\inprod{\phi_j,ue^{ik\cdot}}|^4\over (k^2-2j-1-\lambda^2\delta_j)^2+\lambda^4\gamma_j^2}+\O(\lambda^2)
$$
and
$$
|\beta(k)|^2={\lambda^4|\inprod{\phi_j,ue^{-ik\cdot}}|^4\over (k^2-2j-1-\lambda^2\delta_j)^2+\lambda^4\gamma_j^2}+\O(\lambda^2).
$$
\endgroup




\vfill\eject

\endsection








\beginsection
\def\header#1#2{
              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint #2\hfill \today}}
              }


\header{\S 2: Preliminary Results}{Preliminaries}



As stated in the introduction, the model being considered is described formally by the Hamiltonian
$$
H(\lambda)=\pmatrix{-{d^2\over dx^2} & \lambda u \cr
           \lambda u & -{d^2\over dx^2}+x^2 \cr}
$$
acting in $\L^2(\R)\oplus\L^2(\R)$. Here $u$ is the multiplication operator corresponding to the continuous, compactly supported function $u(x)$. The spectrum of $H(0)$ is very different from that of $H(\lambda)$ for $\lambda\ne0$. The point spectrum of $H(0)$ consists of the sequence of positive embedded eigenvalues
$$
\sigma_{pp}\big(H(0)\big)=\{1,3,5,\dots\}.
$$
The proposition below establishes that for any $\lambda\ne0$ the point spectrum of $H(\lambda)$ consists of one negative eigenvalue, $E(\lambda^2)\sim -\lambda^4$.



\beginpropositionlabel{eigenvalues} The operator $H(\lambda)$ has no non-negative eigenvalues. Further, if $\lambda$ is small enough, $H(\lambda)$ has a unique eigenvalue
$$
E(\lambda^2)=-{\lambda^4\over2}\Big(\inprod{u,\big(-{d^2\over dx^2}+x^2\big)^{-1}u}\Big)^2+\O(\lambda^6).
$$
This eigenvalue is simple and has eigenfunction
$$
\Psi_0={1\over{\cal N}_0}\pmatrix{-\lambda (-{d^2\over dx^2}-E(\lambda^2))^{-1}u\big(-{d^2\over dx^2}+x^2\big)^{-1}u\cr\big(-{d^2\over dx^2}+x^2\big)^{-1}u\cr}+\O(\lambda^2).
$$
Here
$$
{\cal N}_0=\sqrt{\lambda^2\|\big(-{d^2\over dx^2}-E(\lambda^2)\big)^{-1}u\big(-{d^2\over dx^2}+x^2\big)^{-1}u\|^2+\|\big(-{d^2\over dx^2}+x^2\big)^{-1}u\|^2}
$$
is a normalization constant.\endpropositionlabel

\beginproof Consider the equation
$$\eqalign{
\pmatrix{0\cr0\cr}
&=\pmatrix{-{d^2\over dx^2}-\epsilon & \lambda u \cr
           \lambda u & -{d^2\over dx^2}+x^2-\epsilon \cr}\pmatrix{\psi\cr\phi\cr}\cr
&=\pmatrix{(-{d^2\over dx^2}-\epsilon)\psi+\lambda u\phi\cr (-{d^2\over dx^2}+x^2-\epsilon)\phi+\lambda u\psi\cr}.
}$$
For $\epsilon\ge 0$ the equation $(-{d^2\over dx^2}-\epsilon)\psi+\lambda u\phi=0$ has no solutions with $\psi\in\L^2(\R)$, regardless of $\phi$, except for the trivial solution $\psi=\phi=0$. It follows that $H(\lambda)$ has no non-negative eigenvalues.

For $\epsilon<0$ we have
\be{scam eig funct eq}
\psi=-\lambda (-{d^2\over dx^2}-\epsilon)^{-1}u\phi
\ee
so that
\be{eig equation}
\Big(-{d^2\over dx^2}+x^2-\epsilon-\lambda^2u(-{d^2\over dx^2}-\epsilon)^{-1}u\Big)\phi=0.
\ee
Writing $\epsilon=-\kappa^2$, for $\kappa>0$, we have the following integral kernel for $V(\epsilon)=u(-{d^2\over dx^2}-\epsilon)^{-1}u$.
$$\eqalign{
V(\kappa^2;x,y)&=u(x){1\over2\kappa}e^{-\kappa|x-y|}u(y)\cr
&={1\over2\kappa}u(x)u(y)+u(x){e^{-\kappa|x-y|}-1\over2\kappa}u(y)\cr
&={1\over2\kappa}u(x)u(y)+V^{(1)}(\kappa;x,y).
}$$
For all $\kappa>0$ we have $\|V^{(1)}(\kappa)\|\le\const\ $. Thus, if $\lambda$ is small enough, $-{d^2\over dx^2}+x^2+\kappa^2-\lambda^2V^{(1)}(\kappa)$ is invertible. It follows that $\num{eig equation}$ is equivalent to
\be{eig funct}
\phi=\inprod{u,\phi}\Big(-{d^2\over dx^2}+x^2+\kappa^2-\lambda^2V^{(1)}(\kappa)\Big)^{-1}u
\ee
Thus, $\num{eig equation}$ has a non-trivial solution in $\L^2(\R)$ if and only if $\kappa$ satisfies
\be{eig cond}
1={\lambda^2\over2\kappa}\inprod{u,\Big(-{d^2\over dx^2}+x^2+\kappa^2-\lambda^2V^{(1)}(\kappa)\Big)^{-1}u}.
\ee





Note that, as a function of $\kappa$, the right side of $\num{eig cond}$ is continuous, non-zero and, from considering large $\kappa$, positive. Further, since
$$
\|\Big(-{d^2\over dx^2}+x^2+\kappa^2-\lambda^2V^{(1)}(\kappa)\Big)^{-1}\|\le\const\ {1\over1+\kappa^2}
$$
if $\lambda$ is small enough, $\num{eig cond}$ has at least one solution, $\kappa\sim\O(\lambda^2)$. For such $\kappa$ it is clear that
$$
\inprod{u,\Big(-{d^2\over dx^2}+x^2+\kappa^2-\lambda^2V^{(1)}(\kappa)\Big)^{-1}u}=\inprod{u,\big(-{d^2\over dx^2}+x^2\big)^{-1}u}+\O(\lambda^2).
$$


Finally, for $\kappa\sim\O(\lambda^2)$, the derivative with respect to $\kappa$ of the right side of $\num{eig cond}$ is equal to $-{1\over\lambda^2}+\O(1)$. Thus, $\num{eig cond}$ has a unique solution. Along with $\num{scam eig funct eq}$ and $\num{eig funct}$ this finishes the proof.\endproof




We will see that the spectrum of $H(\lambda)$ consists of the one negative eigenvalue, $E(\lambda^2)$, from $\thmnum{eigenvalues}$, and absolutely continuous spectrum $[0,\infty)$. Our chief concern is with the absolutely continuous spectrum and the resonant behavior caused by the unperturbed embedded eigenvalues $2j+1$. To this end we construct the spectral measure for $H(\lambda)$, $\derivs{\mu}{\epsilon}(\lambda,\epsilon)$.


>From Stone's formula we have
\be{Stone}
\derivs{\mu}{\epsilon}(\lambda,\epsilon)={1\over2\pi i}\slim_{\zeta\downarrow0}\Big(\big(H(\lambda)-\epsilon-i\zeta\big)^{-1}-\big(H(\lambda)-\epsilon+i\zeta\big)^{-1}\Big).
\ee
To estimate $\num{Stone}$ we will use an integral kernel for $\big(H(\lambda)-z\big)^{-1}$ which will be constructed in $\S 3$ and $\S 4$. Let ${\cal G}(\lambda,z;x,y)$ be this integral kernel; note that it is a 2 by 2 matrix valued function of $x$ and $y$. The equation $\big(H(\lambda)-z\big)\big(H(\lambda)-z\big)^{-1}=1$ implies that
$$
\pmatrix{-{d^2\over dx^2}-z & \lambda u(x) \cr
           \lambda u(x) & -{d^2\over dx^2}+x^2-z \cr}{\cal G}(\lambda,z;x,y)=\pmatrix{1&0\cr0&1\cr}\delta(x-y)
$$
and $\big(H(\lambda)-z\big)^{-1}\big(H(\lambda)-z\big)=1$ that ${\cal G}(\lambda,z;x,y)={\cal G}^t(\lambda,z;y,x)$. Here the superscript $t$ denotes the 2 by 2 matrix transpose. 




Given ${\cal G}(\lambda,z;x,y)$ we have ([A])
$$\eqalign{
{\cal G}(\lambda,&\epsilon+i0^+;x,y)-{\cal G}(\lambda,\epsilon-i0^+;x,y)\cr
&=\int_{-\infty}^\infty\bigg\{{\cal G}(\lambda,\epsilon+i0^+;x,w)\pmatrix{-{d^2\over dw^2}-\epsilon & \lambda u(w) \cr
           \lambda u(w) & -{d^2\over dw^2}+w^2-\epsilon \cr}{\cal G}(\lambda,\epsilon-i0^+;w,y)\cr
&\hskip40pt-\bigg[\pmatrix{-{d^2\over dw^2}-\epsilon & \lambda u(w) \cr
           \lambda u(w) & -{d^2\over dw^2}+w^2-\epsilon \cr}{\cal G}(\lambda,\epsilon+i0^+;w,x)\bigg]^t{\cal G}(\lambda,\epsilon-i0^+;w,y)\bigg\}dw\cr
&=-\int_{-\infty}^\infty\bigg\{{\cal G}(\lambda,\epsilon+i0^+;x,w)\secderiv w{\cal G}(\lambda,\epsilon-i0^+;w,y)\cr
&\hskip120pt-\Big(\secderiv w{\cal G}(\lambda,\epsilon+i0^+;x,w)\Big){\cal G}(\lambda,\epsilon-i0^+;w,y)\bigg\}dw\cr
&=-\Big\{{\cal G}(\lambda,\epsilon+i0^+;x,w)\derivs{{\cal G}(\lambda,\epsilon-i0^+;w,y)}{w}\cr
&\hskip120pt-\derivs{{\cal G}(\lambda,\epsilon+i0^+;x,w)}{w} {\cal G}(\lambda,\epsilon-i0^+;w,y)\Big\}\Big|_{w=-\infty}^\infty
}$$
so that the integral kernel for $\derivs{\mu}{\epsilon}(\lambda,\epsilon)$ is
\be{d mu int kern}\eqalign{
\derivs{\mu}{\epsilon}(\lambda,\epsilon;x,y)=-{1\over2\pi i}\Big\{{\cal G}(\lambda&,\epsilon+i0^+;x,w)\derivs{{\cal G}(\lambda,\epsilon-i0^+;w,y)}{w}\cr
&-\derivs{{\cal G}(\lambda,\epsilon+i0^+;x,w)}{w} {\cal G}(\lambda,\epsilon-i0^+;w,y)\Big\}\Big|_{w=-\infty}^\infty.
}\ee


Using $\num{d mu int kern}$ and the constructions of the following two sections it will be shown that, for $\epsilon\ge0$ and $\lambda$ sufficiently small, $\derivs{\mu}{\epsilon}$ is absolutely continuous with respect to Lebesgue measure. In addition, the continuum eigenfunctions for $H(\lambda)$ will be extracted. The techniques used are constructive and provide systematic small $\lambda$ approximations to the continuum eigenfunctions as well as to $\derivs{\mu}{\epsilon}$. These approximations cannot take the form of power series in $\lambda$, since the nature of the spectrum of $H(\lambda)$, for $\lambda\ne0$, is drastically different from that of $H(0)$. In particular, $\derivs{\mu}{\epsilon}$ cannot be analytic in $\lambda$. 


\endsection\vfill\eject








\beginsection
\def\header#1#2{
              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint #2\hfill \today}}
              }


\header{\S 3: The Resonant Poles}{Resonant Poles}



For $z\notin\R$ the operator $H-z$ is invertible. We have
\be{matrix scam}
\big(H-z\big)^{-1}=\pmatrix{A(z) & -\lambda\ B(z)\cr -\lambda\ C(z) & \big(h(z)-z\big)^{-1}\cr}
\ee
where 
\be{h(z) def}
h(z)=-{d^2\over dx^2}+x^2-\lambda^2 u(-{d^2\over dx^2}-z)^{-1}u
\ee
and
\be{mat scam def}\eqalign{
A(z)&=(-{d^2\over dx^2}-z)^{-1}+\lambda^2(-{d^2\over dx^2}-z)^{-1}u\big(h(z)-z\big)^{-1}u(-{d^2\over dx^2}-z)^{-1}\cr
B(z)&=(-{d^2\over dx^2}-z)^{-1}u\big(h(z)-z\big)^{-1}\cr
C(z)&=\big(h(z)-z\big)^{-1}u(-{d^2\over dx^2}-z)^{-1}.
}\ee
Define, for $z\notin[0,\infty)$,
\be{V def}
V(z)=u\big(-{d^2\over dx^2}-z\big)^{-1}u.
\ee
Recall that $V(z)$ has an integral kernel, $V(z;x,y)$, given by
$$
V(z;x,y)={1\over2\sqrt{-z}}u(x)\, e^{-\sqrt{-z}\, |x-y|}\, u(y).
$$
This integral kernel allows us to construct two analytic (as bounded operator valued functions of $z$) extensions of $V(z)$, $V_\pm(z)$, given by their integral kernels,
\be{V_pm def}
V_\pm(z;x,y)=\mp{1\over2i\sqrt{z}}u(x)\, e^{\pm i\sqrt{z}\, |x-y|}\, u(y)
\ee
for $z\notin(-\infty,0]$. In this paper square roots are always chosen to have their branch cut on the negative real axis. In particular, for all $\epsilon\in(0,\infty)$,
\be{root def}
\lim_{\eta\downarrow0}\sqrt{-\epsilon\mp i\eta}=\mp i\sqrt\epsilon.
\ee


\beginlemmalabel{res pole} Let $z\notin(-\infty,0]$ and define
$$
h_\pm(z)=-{d^2\over dx^2}+x^2-\lambda^2 V_\pm(z).
$$
Let
$$
{\cal N}=\{z\in\C\, \big|\, \Re z>0\ {\rm and}\ |\Im z|<2\}.
$$
If $\lambda$ is small enough there is a discrete set, ${\cal R}_\pm=\big\{\epsilon^{(\pm)}_j(\lambda^2)\, |\, j\in\{0,1,2,\dots\}\big\}\subset{\cal N}$ such that $h_\pm(z)-z$ is invertible for $z\in{\cal N}\setminus{\cal R}_\pm$. The inverses $\big(h_\pm(z)-z\big)^{-1}$ are meromorphic (as bounded operator valued functions of $z$) in ${\cal N}$ with simple poles at $\epsilon^{(\pm)}_j(\lambda^2)$. Further,
$$
|\Re \epsilon^{(\pm)}_j(\lambda^2)-2j-1-\lambda^2\delta_j|\le\const\ \lambda^4
$$
and
$$
|\Im \epsilon^{(\pm)}_j(\lambda^2)\pm\lambda^2\gamma_j|\le\const\ \lambda^4
$$
where $\delta_j$ and $\gamma_j$ are real constants with
$$
\gamma_j={|\inprod{e^{i\sqrt{2j+1}x}u,\phi_j}|^2\over2\sqrt{2j+1}},
$$
and $\phi_j$ is the eigenfunction of $-{d^2\over dx^2}+x^2$ corresponding to the eigenvalue $2j+1$.
\endlemmalabel




\beginproof For such $z\in{\cal N}$ that $|z|>\lambda^2$ the lemma is a simple consequence of the bound
\be{V_pm(z) bound}\eqalign{
\|V_\pm(z)\|&\le \sqrt{\int |V_\pm(z;x,y)|^2\, dx\, dy}\cr
&\le {1\over2|z|^{1\over2}}\sup_{x,y\in supp(u)}\big|e^{\pm i\sqrt{z}\, |x-y|}\big|\ \int|u(x^\prime)|^2\, dx^\prime\cr
&\le\const\ {e^{K|\Im\sqrt{z}\, |}\over|z|^{1\over2}},
}\ee
where
$$
K=2\sup_{x\in supp(u)} |x|,
$$
and analytic perturbation theory.


For $z\in{\cal N}$ with $|z|\le\lambda^2$ we use
$$\eqalign{
V_\pm(z;x,y)&=\mp{1\over2i\sqrt{z}}u(x)\, u(y)\mp u(x)\, {e^{\pm i\sqrt{z}\, |x-y|}-1\over2i\sqrt{z}}\, u(y)\cr
&=\mp{1\over2i\sqrt{z}}u(x)\, u(y)+V^{(1)}_\pm(z;x,y).
}$$
Then $V^{(1)}_\pm(z)$ is bounded analytic in $z$ for $z\notin(-\infty,0]$ and satisfies the bound $\|V^{(1)}_\pm(z)\|\le\const\ e^{K|\Im\sqrt{z}\, |}$. Thus, for $\lambda$ small enough,
$$
h^{(1)}_\pm(z)-z=-{d^2\over dx^2}+x^2-z-\lambda^2 V^{(1)}_\pm(z)
$$
is invertible for all $z\in{\cal N}$. To see that
$$
h_\pm(z)-z=h^{(1)}_\pm(z)-z\pm{\lambda^2\over2i\sqrt{z}}u\otimes u
$$
is invertible we note that $h_\pm(z)-h^{(1)}_\pm(z)$ is rank one so that $h_\pm(z)-z$ is invertible if and only if $\lambda^2\inprod{u,\big(h^{(1)}_\pm(z)-z\big)^{-1}u}\neq \mp 2i\sqrt{z}$. If so then
\be{small epsilon}\eqalign{
\big(h_\pm(z)-z&\big)^{-1}=\big(h^{(1)}_\pm(z)-z\big)^{-1}\cr
&\mp{\lambda^2\over 2i\sqrt{z}\pm\lambda^2\inprod{u,\big(h^{(1)}_\pm(z)-z\big)^{-1}u}}\, \big(h^{(1)}_\pm(z)-z\big)^{-1}u\otimes u\big(h^{(1)}_\pm(z)-z\big)^{-1}.
}\ee
To finish the proof of the lemma it must be shown that $\lambda^2\inprod{u,\big(h^{(1)}_\pm(z)-z\big)^{-1}u}\neq \mp 2i\sqrt{z}$ for $z\in{\cal N}$ with $|z|\le\lambda^2$. For such $z$
$$\eqalign{
\|\big(h^{(1)}_\pm(z)-z\big)^{-1}-\big(-{d^2\over dx^2}+x^2\big)^{-1}\|&\le\|\big(h^{(1)}_\pm(z)-z\big)^{-1}-\big(-{d^2\over dx^2}+x^2-z\big)^{-1}\|\cr
&\hskip25pt+\|\big(-{d^2\over dx^2}+x^2-z\big)^{-1}-\big(-{d^2\over dx^2}+x^2\big)^{-1}\|\cr
&\le\const\ \lambda^2.
}$$
If $\phi_j$ is the eigenfunction of $-{d^2\over dx^2}+x^2$ corresponding to the eigenvalue $2j+1$ then it follows that
$$\eqalign{
\inprod{u,\big(h^{(1)}_\pm(z)-z\big)^{-1}u}&=\inprod{u,\big(-{d^2\over dx^2}+x^2\big)^{-1}u}+\O(\lambda^2)\cr
&=\sum_{j=0}^\infty {|\inprod{\phi_j,u}|^2\over 2j+1}+\O(\lambda^2).
}$$
Thus, if $\lambda^2\inprod{u,\big(h^{(1)}_\pm(z)-z\big)^{-1}u}= \mp 2i\sqrt{z}$ then $z=-\lambda^4\big(\inprod{u,\big(-{d^2\over dx^2}+x^2\big)^{-1}u}\big)^2+\O(\lambda^6)$. It follows that, for $\lambda$ small enough, $\Re z<0$ and thus $z\notin{\cal N}$.\endproof



Using $\num{matrix scam}$ to estimate $\big(H-z\big)^{-1}$ it is sufficient to estimate $\big(h(z)-z\big)^{-1}$. Let $h_\pm(z)$ be the operators obtained by replacing $u(-{d^2\over dx^2}-z)^{-1}u$ with $V_\pm(z)$ in $\num{h(z) def}$. Let $\Re z>0$ and $|z-2j-1|<{3\over2}$. Using $\thmnum{res pole}$ we can extract the singular parts from $\big(h_\pm(z)-z\big)^{-1}$. We have, for $j>0$ (the generalization to $j=0$ is straightforward),
\be{Laurent exp}
\big(h_\pm(z)-z\big)^{-1}={1\over z-\epsilon^{(\pm)}_j(\lambda^2)}E^{(j)}_\pm(\lambda^2)+R^{(j)}_\pm(\lambda^2,z)
\ee
where
\be{E^{(j)} Cauchy rep}
E^{(j)}_\pm(\lambda^2)={1\over2\pi i}\int_\Gamma \big(h_\pm(w)-w\big)^{-1}\, dw,
\ee
\be{R^{(j)} Cauchy rep}
R^{(j)}_\pm(\lambda^2,z)={1\over2\pi i}\int_\Gamma {1\over w-z}\big(h_\pm(w)-w\big)^{-1}\, dw
\ee
and $\Gamma=\{z\in\C\, \big|\, |z-2j-1|={3\over2}\}$. Note that $R^{(j)}_\pm(\lambda^2,z)$ is analytic in $z$ for $|z-2j-1|<{3\over2}$. Since $V_\pm(z)$ is uniformly bounded for $|z-2j-1|\le{3\over2}$ we have, for $\lambda$ sufficiently small,
\be{E^{(j)} exp}
E^{(j)}_\pm(\lambda^2)=\sum_{N=0}^\infty \lambda^{2N} E^{(j)}_{\pm; N},
\ee
where
$$
E^{(j)}_{\pm; N}={1\over2\pi i}\int_\Gamma \big(-{d^2\over dx^2}+x^2-w\big)^{-1}\Big(V_\pm(w)\big(-{d^2\over dx^2}+x^2-w\big)^{-1}\Big)^N\, dw,
$$
and
\be{R^{(j)} exp}
R^{(j)}_\pm(\lambda^2,z)=\sum_{N=0}^\infty \lambda^{2N} R^{(j)}_{\pm; N}(z),
\ee
where
$$
R^{(j)}_{\pm; N}(z)={1\over2\pi i}\int_\Gamma {1\over w-z}\big(-{d^2\over dx^2}+x^2-w\big)^{-1}\Big(V_\pm(w)\big(-{d^2\over dx^2}+x^2-w\big)^{-1}\Big)^N\, dw.
$$
Note that both $E^{(j)}_{\pm; N}$ and $R^{(j)}_{\pm; N}(z)$ are uniformly bounded for $|z-2j-1|\le{3\over2}$. 






To estimate $E^{(j)}_{\pm}(\lambda^2)$ and $R^{(j)}_{\pm}(\lambda^2,z)$ one must evaluate $\num{E^{(j)} exp}$ and $\num{R^{(j)} exp}$. For example, if $\phi_j$ is the eigenfunction of $-{d^2\over dx^2}+x^2$ corresponding to the eigenvalue $2j+1$ and $P_{j\perp}$ is the orthogonal projection onto the orthogonal complement to $\phi_j$ in $\L^2(\R)$ then
$$
E^{(j)}_{\pm;0}=\phi_j\otimes\phi_j.
$$
This is enough to show that if $|z-2j-1|\le{3\over2}$ then
\be{mat scam approx}
\|\big(h_\pm(z)-z\big)^{-1}-{1\over z-\epsilon^{(\pm)}_j(\lambda^2)}\phi_j\otimes\phi_j\|\le\const\ .
\ee
To get an estimate uniformly correct to $\O(\lambda^2)$ we need both
$$\eqalign{
E^{(j)}_{\pm;1}=\inprod{\phi_j,&V_\pm^\prime(2j+1)\phi_j}\phi_j\otimes\phi_j\cr
&-P_{j\perp}\big(-{d^2\over dx^2}+x^2-k^2\big)^{-1}P_{j\perp}V_\pm(2j+1)\phi_j\otimes\phi_j\cr
&-\phi_j\otimes\phi_jV_\pm(2j+1)P_{j\perp}\big(-{d^2\over dx^2}+x^2-k^2\big)^{-1}P_{j\perp}
}$$
and, if $|z-2j-1|\le{3\over2}$,
$$
R^{(j)}_{\pm;0}(z)=P_{j\perp}\big(-{d^2\over dx^2}+x^2-z\big)^{-1}P_{j\perp}.
$$
Then, for such $z$,
$$
\|\big(h_\pm(z)-z\big)^{-1}-{1\over z-\epsilon^{(\pm)}_j(\lambda^2)}\Big(\phi_j\otimes\phi_j+\lambda^2 E^{(j)}_{\pm;1}\Big)-P_{j\perp}\big(-{d^2\over dx^2}+x^2-z\big)^{-1}P_{j\perp}\|\le\const\ \lambda^2.
$$



\endsection\vfill\eject






\beginsection


\def\header#1#2{
              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint #2\hfill \today}}
              }


\header{\S 4: The Spectral Density and the Continuum Eigenfunctions}{Continuum Eigenfunctions}


In this section we find an integral kernel for $\big(H(\lambda)-z\big)^{-1}$ and apply $\num{d mu int kern}$ to obtain an integral kernel for the spectral density $\derivs{\mu}{\epsilon}$. We will then be able to extract expressions for the continuum eigenfunctions for $H(\lambda)$. 






Recall that the integral kernel for $\big(-{d^2\over dx^2}+x^2-z\big)^{-1}$, $g^{(0)}(z;x,y)$, is given by
$$
g^{(0)}(z;x,y)={2^{-{z\over2}-1}\over\pi}\Gamma({3-z\over4})\Gamma({1-z\over4})\, U(-{z\over2},-\sqrt2\, x_<)U(-{z\over2},\sqrt2\, x_>).
$$
Here $U(a,x)$ is the parabolic cylinder function, $x_<=\min\{x,y\}$ and $x_>=\max\{x,y\}$. $U(a,x)$ is analytic in $a$ and satisfies
$$
U(a,x)= x^{-a-{1\over2}}e^{-{1\over4}x^2}\big(1+\O({1\over|x|})\big)
$$
as $x\to\infty$ and
$$
U(a,x)={\sqrt\pi\over2^{{1\over2}a+{1\over4}}\Gamma({3\over4}+{1\over2}a)}+{\sqrt\pi\over2^{{1\over2}a-{1\over4}}\Gamma({1\over4}+{1\over2}a)}\, x+\O(x^2)
$$
for small $x$. Using the fact that $V_\pm(z)$ is an integral operator with compact support convergent expansions for the integral kernels for $E^{(j)}_\pm(\lambda^2)$ and $R^{(j)}_\pm(\lambda^2,z)$ follow. Then for $z$ satisfying $|z-2j-1|\le{3\over2}$ if $j>0$ and $|z-1|\le{3\over2}$, $\Re z\ge{1\over2}$ if $j=0$
$$
g_\pm(z;x,y)={1\over z-\epsilon^{(\pm)}_j(\lambda^2)}E^{(j)}_\pm(\lambda^2;x,y)+R^{(j)}_\pm(\lambda^2,z;x,y)
$$
is the resulting integral kernel for $\big(h_\pm(z)-z\big)^{-1}$. For $0<\Re z<{1\over2}$ we obtain an integral kernel from $\num{small epsilon}$. These integral kernels are symmetric in $x$ and $y$ and all approach zero as any one of their arguments, say $y$, approaches $\pm\infty$. In general, as $y\to\infty$ these integral kernels and their derivatives are all bounded by $f(x)\ e^{-{1\over4}y^2}$. Here $f$ is continuous and positive.



We can now use $\num{d mu int kern}$ to estimate $\derivs{\mu}{\epsilon}(\lambda,\epsilon)$. Let the subscript $\pm$ denote the operator obtained on replacing $V(z)$ with $V_\pm(z)$. If $\pm\Im z>0$ then
$$
{\cal G}(\lambda,z;x,y)=\pmatrix{A_\pm(z;x,y) & -\lambda\ B_\pm(z;x,y)\cr -\lambda\ B_\pm(z;y,x) & g_\pm(z;x,y)\cr}
$$
where
$$\eqalign{
A_\pm(z;x,y)=\mp{1\over2i\sqrt{z}} &e^{\pm i\sqrt{z}\, |x-y|}\cr
&-\lambda^2{1\over4z}\int e^{\pm i\sqrt{z}\, |x-r_1|}u(r_1)g_\pm(z;r_1,r_2)u(r_2) e^{\pm i\sqrt{z}\, |r_2-y|}\, dr_1\, dr_2,
}$$
and
$$
B_\pm(z;x,y)=\mp{1\over2i\sqrt{z}}\int e^{\pm i\sqrt{z}\, |x-r|}u(r)g_\pm(z;r,y)\, dr.
$$


Let $\sigma,\delta\in\{\pm\}$ and define
\be{psi_sigma,delta def}
\psi^{(\sigma)}_{\delta}(z;x)=\int e^{-\sigma\delta i\sqrt{z}\, r}u(r)g_\sigma(z;r,x)\, dr.
\ee
Note that $\overline{\psi^{(\sigma)}_{\delta}(z;x)}=\psi^{(-\sigma)}_{\delta}(\bar z;x)$. With this notation we find that, as $x\to\delta\infty$,
$$
A_\sigma(z;x,y)={-\sigma\over2i\sqrt{z}}e^{\sigma\delta i\sqrt{z}\, (x-y)}-\lambda^2{1\over4z}e^{\sigma\delta i\sqrt{z}\, x}\int \psi^{(\sigma)}_{\delta}(z;r)u(r) e^{\sigma i\sqrt{z}\, |r-y|}\, dr
$$
and
$$
B_\sigma(z;x,y)={-\sigma\over2i\sqrt{z}}e^{\sigma\delta i\sqrt{z}\, x}\psi^{(\sigma)}_{\delta}(z;y).
$$
Both $g_\sigma(z;x,y)$ and $B_\sigma(z;y,x)$, as well as their derivatives with respect to $x$, are bounded, for fixed $y$, by $\const\cdot e^{-{1\over4}x^2}$.



Substituting these last expressions in $\num{d mu int kern}$ we find that
$$
\derivs{\mu}{\epsilon}(\lambda,\epsilon;x,y)=\pmatrix{\derivs{\mu_{1,1}}{\epsilon} &\derivs{\mu_{1,2}}{\epsilon}\cr
\derivs{\mu_{2,1}}{\epsilon} &\derivs{\mu_{2,2}}{\epsilon}\cr}
$$
where
$$\eqalign{
\derivs{\mu_{1,1}}{\epsilon}={1\over4\pi \sqrt{\epsilon}}\sum_{\delta=\pm}\Big(e^{-\delta i\sqrt\epsilon\, x}+i{\lambda^2\over2\sqrt\epsilon}&\int e^{i\sqrt\epsilon\, |x-r|}u(r)\psi^{(+)}_{\delta}(\epsilon;r)\, dr\Big)\cr
&\cdot\Big(e^{\delta i\sqrt\epsilon\, y}-i{\lambda^2\over2\sqrt\epsilon}\int e^{-i\sqrt\epsilon\, |y-r|}u(r)\psi^{(-)}_{\delta}(\epsilon;r)\, dr\Big),
}$$
$$
\derivs{\mu_{1,2}}{\epsilon}={-\lambda\over4\pi \sqrt{\epsilon}}\sum_{\delta=\pm}\Big(e^{-\delta i\sqrt\epsilon\, x}+i{\lambda^2\over2\sqrt\epsilon}\int e^{i\sqrt\epsilon\, |x-r|}u(r)\psi^{(+)}_{\delta}(\epsilon;r)\, dr\Big)\psi^{(-)}_{\delta}(\epsilon;y),
$$
$$
\derivs{\mu_{2,1}}{\epsilon}={-\lambda\over4\pi \sqrt{\epsilon}}\sum_{\delta=\pm}\psi^{(+)}_{\delta}(\epsilon;x)\Big(e^{\delta i\sqrt\epsilon\, y}-i{\lambda^2\over2\sqrt\epsilon}\int e^{-i\sqrt\epsilon\, |y-r|}u(r)\psi^{(-)}_{\delta}(\epsilon;r)\, dr\Big)
$$
and
$$
\derivs{\mu_{2,2}}{\epsilon}={\lambda^2\over4\pi \sqrt{\epsilon}}\sum_{\delta=\pm}\psi^{(+)}_{\delta}(\epsilon;x)\psi^{(-)}_{\delta}(\epsilon;y).
$$
It follows that if we let
\be{cont eig functs}
\Psi^{(\pm)}_{\delta}(\lambda,\epsilon;x)={1\over\sqrt{2\pi}}\pmatrix{e^{\mp\delta i\sqrt\epsilon\, x}\pm i{\lambda^2\over2\sqrt\epsilon}\int e^{\pm i\sqrt\epsilon\, |x-r|}u(r)\psi^{(\pm)}_{\delta}(\epsilon;r)\, dr\cr 
-\lambda\psi^{(\pm)}_{\delta}(\epsilon;x)\cr}
\ee
then
\be{spec meas}
\derivs{\mu}{\epsilon}(\lambda,\epsilon;x,y)={1\over2\sqrt\epsilon}\sum_{\delta=\pm}\Psi^{(+)}_{\delta}(\lambda,\epsilon;x)\otimes_{\C^2} \Psi^{(-)\, t}_{\delta}(\lambda,\epsilon;y).
\ee
The tensor product $\otimes_{\C^2}$ in $\num{spec meas}$ is the tensor product in $\C^2$.



The functions $\Psi^{(\pm)}_{\delta}(\lambda,\epsilon;x)$ are the continuum eigenfunctions for $H(\lambda)$. They satisfy $\overline{\Psi^{(+)}_{\delta}(\lambda,\epsilon;x)}=\Psi^{(-)}_{\delta}(\lambda,\epsilon;x)$. By Stone's formula they are complete in the sense that
$$
{1\over2}\Big(P_{[a,b]}(x,y)+P_{(a,b)}(x,y)\Big)=\int_a^b {1\over2\sqrt\epsilon}\sum_{\delta=\pm}\Psi^{(+)}_{\delta}(\lambda,\epsilon;x)\otimes_{\C^2}\Psi^{(-)\, t}_{\delta}(\lambda,\epsilon;y)\, d\epsilon
$$
where $P_A(x,y)$ is the integral kernel for the spectral projection of $H(\lambda)$ on $A\subset\R$. It follows immediately that $H(\lambda)$ has purely absolutely continuous spectrum in $(0,\infty)$. Note that $P_{(0,\infty)}=P_{ac}$ is the spectral projection on the absolutely continuous spectrum of $H(\lambda)$.



As usual, it is convenient to parameterize the spectral density by the ``wave number", $k=\sqrt\epsilon$, rather than by the energy, $\epsilon$. Define
$$
\Psi^{(\pm)}(\lambda,k;x)=\Psi^{(\pm)}_{\pm{\rm sgn}\, k}(\lambda,k^2;x)
$$
and
$$
\psi^{(\pm)}(k;x)=\psi^{(\pm)}_{\pm{\rm sgn}\, k}(k^2;x)
$$
where ${\rm sgn}\, k={k\over|k|}$. Explicitly
\be{k space cont eig functs}
\Psi^{(\pm)}(\lambda,k;x)={1\over\sqrt{2\pi}}\pmatrix{e^{-ik x}\pm i{\lambda^2\over2|k|}\int e^{\pm i|k|\, |x-r|}u(r)\psi^{(\pm)}(k;r)\, dr\cr 
-\lambda\psi^{(\pm)}(k;x)\cr}
\ee
and
\be{psi^{(pm)}(k) explicit form}
\psi^{(\pm)}(k)=\big(h_\pm(k^2)-k^2\big)^{-1}e^{-ik\cdot}u(\cdot).
\ee
Here $e^{-ik\cdot}u(\cdot)\in\L^2(\R)$ is the function $e^{\mp ikx}u(x)$. Note that the functions $\Psi^{(\pm)}(\lambda,k)$ are defined in such a way that they are solutions of the Lippmann-Schwinger equations
$$
\pmatrix{1 & \lambda (-{d^2\over dx^2}-k^2\mp i0^+)^{-1}u\cr \lambda u & -{d^2\over dx^2}+x^2-k^2\cr}\Psi^{(\pm)}(\lambda,k)={1\over\sqrt{2\pi}}\pmatrix{e^{-ikx}\cr0\cr}.
$$




With these definitions we have
\be{spec meas in k space}
\derivs{\mu}{k}(\lambda,k)=\Psi^{(+)}(\lambda,k)\otimes\Psi^{(+)}(\lambda,k).
\ee
The notation $\Psi_1\otimes\Psi_2$ is that of a rank one operator in $\L^2\oplus\L^2$ and is defined by
$$
(\Psi_1\otimes\Psi_2)\Phi=\Big(\int \overline{\Psi_2^t(x)}\Phi(x)\, dx\Big)\Psi_1.
$$
The parameter $k$ ranges over $(-\infty,\infty)$. Note that
\be{-=+conj}
\overline{\Psi^{(+)}(\lambda,k;x)}=\Psi^{(-)}(\lambda,-k;x)
\ee
and that
\be{orthog}
\int \overline{\Psi^{(\pm)}(\lambda,q;x)}\Psi^{(\pm)}(\lambda,k;x)\, dx=\delta(q-k).
\ee





We get approximations for $\Psi^{(\pm)}(\lambda,k)$ by approximating the function $\psi^{(\pm)}(k)$. If we fix $j$ and let $k$ satisfy $\min\{{1\over2},2j-{1\over2}\}<k^2<2j+{5\over2}$ then, recalling $\num{Laurent exp}$, we have
$$
\psi^{(\pm)}(k)=\Big({1\over k^2-\epsilon^{(\pm)}_j(\lambda^2)}E^{(j)}_\pm(\lambda^2)+R^{(j)}_\pm(\lambda^2,k^2)\Big)e^{- ik\cdot}u.
$$
Recall that both $\|E^{(j)}_{\pm}(\lambda^2)\|\le\const\ $ and $\|R^{(j)}_{\pm}(\lambda^2,k^2)\|\le\const\ $ uniformly in $k$ for the range of $k$ considered here. Using $\num{mat scam approx}$ we have
$$
\|\psi^{(\pm)}(k)-{\inprod{\phi_j,e^{- ik\cdot}u}\over k^2-\epsilon^{(\pm)}_j(\lambda^2)}\, \phi_j\|\le\const\ .
$$
Thus, to leading order in $\lambda$, we have
$$
\Psi^{(\pm)}(\lambda,k;x)={1\over\sqrt{2\pi}}\pmatrix{e^{-ikx}\pm \lambda^2\,{\inprod{\phi_j,e^{- ik\cdot}u}\over k^2-\epsilon^{(\pm)}_j(\lambda^2)}\int {i\over2|k|}\, e^{\pm i|k|\, |x-r|}u(r)\phi_j(r)\, dr\cr \cr 
\lambda{\inprod{\phi_j,e^{- ik\cdot}u}\over k^2-\epsilon^{(\pm)}_j(\lambda^2)}\, \phi_j\cr}+\O(\lambda)
$$
pointwise in $x$ uniformly in $k$.



\endsection\vfill\eject






\beginsection


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              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint #2\hfill \today}}
              }


\header{\S 5: The Scattering Matrix}{S-Matrix}


In this section we use the results of $\S 3$ to obtain the scattering matrix for $H(\lambda)$, acting in $\L^2(\R)\otimes\L^2(\R)$, with respect to $-{d^2\over dx^2}$, acting in $\L^2(\R)$. We first define the wave operators $\Omega_\pm:\L^2(\R)\to\L^2(\R)\otimes\L^2(\R)$. For $\phi\in\L^2(\R)$ let $\hat\phi$ be the Fourier transform of $\phi$,
$$
\hat\phi(k)={1\over\sqrt{2\pi}}\int \phi(x)e^{ikx}\, dx.
$$
Then $\Omega_\pm$ is given by
\be{wave operator def}
\big[\Omega_\pm\phi\big](x)=\int \hat\phi(k)\Psi^{(\pm)}(\lambda,k;x)\, dk
\ee
and its adjoint, $\Omega_\pm^*:\L^2(\R)\otimes\L^2(\R)\to\L^2(\R)$, is given by
$$\eqalign{
\big[\Omega_\pm^*\Phi\big](x)&={1\over\sqrt{2\pi}}\int \Big(\int \overline{\Psi^{(\pm)\, t}(\lambda,k;y)}\, \Phi(y)\, dy\Big)e^{-ikx}\, dk\cr
&={1\over\sqrt{2\pi}}\int \Big(\int \Psi^{(\mp)\, t}(\lambda,k;y)\, \Phi(y)\, dy\Big)e^{-ikx}\, dk.
}$$
We now show, using standard arguments [RS], that $\Omega_\pm$ are the wave operators
$$
\Omega_\pm=\slim_{t\to\mp\infty}e^{itH(\lambda)}\pmatrix{1\cr0\cr} e^{it{d^2\over dx^2}}.
$$



\beginpropositionlabel{wave operators} For all $\phi\in\L^2(\R)$
$$
\lim_{t\to\mp\infty}\|\left[\pmatrix{1\cr0\cr}e^{it{d^2\over dx^2}}-e^{-itH(\lambda)}\Omega_\pm\right]\phi\|=0.
$$
\endpropositionlabel

\beginproof We have
\be{eta def}\eqalign{
\eta^{(\sigma)}(\phi;t,x)&=\left(\left[\pmatrix{1\cr0\cr}e^{it{d^2\over dx^2}}-e^{-itH(\lambda)}\Omega_\sigma\right]\phi\right)(x)\cr
&=\int e^{-itk^2}\left[\pmatrix{1\cr0\cr}{1\over\sqrt{2\pi}}e^{-ikx}-\Psi^{(\sigma)}(\lambda,k;x)\right]\, \hat\phi(k)\, dk.
}\ee
It suffices to consider $\phi$ for which $\hat\phi\in C_0^\infty\big(\R\setminus\{0\}\big)$ since such $\phi$ form a dense subset of $\L^2(\R)$ and $\eta^{(\sigma)}(\phi;t)$ is a bounded linear function of $\phi$.


Let $M>0$ be large enough so that ${\rm supp}(u)\subset (-M,M)$ and let $\chi$ be the characteristic function of the interval $(-M,M)$,
\be{chi def}
\chi(x)=\cases{1 & if $x\in(-M,M)$ \cr 0 & if $x\notin(-M,M)$.\cr}
\ee
Changing integration variable in $\num{eta def}$ from $k$ to $\kappa=t^{1\over2}k$ we see that $\eta^{(\sigma)}(\phi;t,x)\to0$ as $t\to\pm\infty$ uniformly in $x$. It follows that $\|\eta^{(\sigma)}(\phi;t)\chi\|\to0$ as $t\to\pm\infty$.




If $x\notin(-M,M)$ then
\be{eig funct asyptotics}
\eta^{(\sigma)}(\phi;t,x)=\int e^{-itk^2} \pmatrix{\lambda^2 f^{(\sigma)}(\lambda,{\rm sgn}\, x;k)e^{\sigma i|k|\, |x|} \cr \lambda\psi^{(\sigma)}(k;x)\cr}\, \hat\phi(k)\, dk
\ee
where ${\rm sgn}\, x={x\over|x|}$ and
\be{f^{(+-)}def}\eqalign{
f^{(\sigma)}(\lambda,\pm;k)&={\sigma i\over2|k|}\int e^{\mp\sigma i|k|r}u(r)\psi^{(\sigma)}(k;r)\, dr\cr
&={\sigma i\over2|k|}\inprod{e^{\mp\sigma i|k|\cdot}u,\big(h_\sigma(k^2)-k^2\big)^{-1}e^{-ik\cdot}u}.
}\ee
Recalling that $\psi^{(\sigma)}(k;x)$ is real analytic in $k$ for $k\ne0$ and that $|\psi^{(\sigma)}(k;x)|\le\const\ e^{-{1\over4}x^2}$ it follows that
$$
\lim_{t\to\pm\infty}\|\int e^{-itk^2} \psi^{(\sigma)}(k)\, \hat\phi(k)\, dk\|=0.
$$
Noting that $-tk^2\pm|k||x|>0$ when $\mp t>0$ it follows from
$$\eqalign{
|\int e^{-itk^2} e^{\pm i|k|\, |x|}\, \hat\phi(k)\, dk|&= |\int e^{-itk^2\pm i|k|\, |x|}\, {d\over dk}\Big({i\over 2tk\mp {\rm sgn}(k)\, |x|}\hat\phi(k)\Big)\, dk|\cr
&\le K(\phi){1\over \big|t\mp |x|\big|};
}$$
here $K(\phi)$ is a constant, recall that $\hat\phi\in C_0^\infty\big(\R\setminus\{0\}\big)$; that
$$
\lim_{t\to\mp\infty}\|\eta^{(\pm)}(\phi;t)(1-\chi)\|=0.
$$
This establishes the proposition.\endproof





Note that $\Omega_\pm^*\Omega_\pm=1$ and that $\Omega_\pm(-{d^2\over dx^2})=H(\lambda)\Omega_\pm$. Further,  asymptotic completeness, $\Ran\Omega_+=\Ran\Omega_-$, follows from the relation $\overline{\Psi^{(+)}(\lambda,k;x)}=\Psi^{(-)}(\lambda,k;x)$ so that the scattering matrix $S=\Omega_-^*\Omega_+:\L^2(\R)\to\L^2(\R)$ exists. We have
$$
[S\phi](x)={1\over\sqrt{2\pi}}\int S(q,k)\hat\phi(k)e^{iqx}\, dk\, dq
$$
where
$$
S(q,k)=\int \overline{\Psi^{(-)\, t}(\lambda,q;x)}\, \Psi^{(+)}(\lambda,k;x)\, dx.
$$
The relation $\Omega_\pm(-{d^2\over dx^2})=H(\lambda)\Omega_\pm$ implies that $S$ commutes with $-{d^2\over dx^2}$. It follows that $S(q,k)=S(k,k)\delta(q-k)+S(-k,k)\delta(q+k)$.







The most straightforward way to compute $S(\pm k,k)$ is to note that, as differential equations, $\big(H(\lambda)-k^2\big)\Psi^{(-)}(\lambda,\pm k;x)=\big(H(\lambda)-k^2\big)\Psi^{(+)}(\lambda,k;x)=0$ so that
\be{+ as - lin comb onshell}
\Psi^{(+)}(\lambda,k;x)=\alpha(k)\, \Psi^{(-)}(\lambda,k;x)+\beta(k)\, \Psi^{(-)}(\lambda,-k;x).
\ee
By the orthogonality relation $\num{orthog}$ we then have $S(k,k)=\alpha(k)$ and $S(-k,k)=\beta(k)$. For $x\notin {\rm supp}\, u$ we have
$$\eqalign{
{1\over\sqrt{2\pi}}e^{-ikx}+\lambda^2f^{(+)}(\lambda,{\rm sgn}\, x;k)e^{i|k|\, |x|}=&\alpha(k)\Big({1\over\sqrt{2\pi}}e^{-ikx}+\lambda^2f^{(-)}(\lambda,{\rm sgn}\, x;k)e^{-i|k|\, |x|}\Big)\cr
&+\beta(k)\Big({1\over\sqrt{2\pi}}e^{ikx}+\lambda^2f^{(-)}(\lambda,{\rm sgn}\, x;-k)e^{-i|k|\, |x|}\Big).
}$$
>From the case $kx<0$ we find
$$
\alpha(k)=1+\lambda^2\sqrt{2\pi}f^{(+)}(\lambda,-{\rm sgn}\, k;k),
$$
and from $kx>0$
$$
\beta(k)=\lambda^2\sqrt{2\pi}f^{(+)}(\lambda,{\rm sgn}\, k;k).
$$
In this one dimensional setting $\alpha(k)-1$ is called the reflection coefficient and $\beta(k)$ the transmission coefficient.


Define the operator $T(z)$ by
\be{T matrix}
T(z)=u\big(h_+(z)-z\big)^{-1}u;
\ee
$T(z)$ has the integral kernel $T(z;x,y)=u(x)g_+(z;x,y)u(y)$. The operator $-\lambda^2T(z)$ is the so called $T$ matrix ([RS]) for this model. If
$$
T(z;q,k)={1\over2\pi}\int\int e^{iqx}T(z;x,y)e^{-iky}\, dx\, dy
$$
then it follows from $\num{f^{(+-)}def}$ that
\be{S matrix}
S(q,k)=\delta(q-k)+{i\lambda^2\over\sqrt{2\pi}}T(k^2;q,k)\delta(q^2-k^2).
\ee
Using $\num{Laurent exp}$ in $\num{T matrix}$ we see that there is anamolously large scattering when $k^2$ is close to one of the resonant energies $2j+1$. We find from $\num{mat scam approx}$ that for, say, $|k^2-2j-1|<{1\over2}$
$$
T(k^2;q,k)={1\over k^2-\epsilon^{(+)}_j(\lambda^2)}\inprod{e^{iq\cdot}u,\phi_j}\inprod{\phi_j,ue^{ik\cdot}}+\O(1).
$$
In particular, recalling the estimate for $\epsilon^{(+)}_j$ from $\thmnum{res pole}$, near a resonance the scattering cross sections $|\alpha(k)|^2$ and $|\beta(k)|^2$ have the expected Breit-Wigner form,
$$
|\alpha(k)|^2={\lambda^4|\inprod{\phi_j,ue^{ik\cdot}}|^4\over (k^2-2j-1-\lambda^2\delta_j)^2+\lambda^4\gamma_j^2}+\O(\lambda^2)
$$
and
$$
|\beta(k)|^2={\lambda^4|\inprod{\phi_j,ue^{-ik\cdot}}|^4\over (k^2-2j-1-\lambda^2\delta_j)^2+\lambda^4\gamma_j^2}+\O(\lambda^2).
$$




\endsection\vfill\eject








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              \hfil\underbar{#1} \hfil\bigskip
              \headline={{\eightpoint\hfill #2}}
              }

\header{Bibliography}{Bibliography}
{\noindent\frenchspacing

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}





\end