% % This 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} \def\header#1#2{\hfil\underbar{#1} \hfil\bigskip \headline={{\eightpoint #2\hfill\today}}} % 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}} 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\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{ \toksdef\ta=0 \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{ \advance\nonamectr by1 \edef\temp{{noname\the\nonamectr}} \expandafter\begintheoremlabel\temp } \def\endtheorem{\endtheoremlabel} \def\beginlemmalabel#1{ \toksdef\ta=0 \ta={\typelemma\\{#1}} \immediate\write\thmoutfile{\the\ta} \medbreak\par\noindent{\ifdraftmode{\mcomment{#1}}\bf Lemma \csname thmlabel#1\endcsname :} \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{ \toksdef\ta=0 \ta={\typeproposition\\{#1}} \immediate\write\thmoutfile{\the\ta} \medbreak\par\noindent{\ifdraftmode{\mcomment{#1}}\bf Proposition \csname thmlabel#1\endcsname :} \begingroup \it } \def\endpropositionlabel{\par\medbreak\endgroup} \def\beginproposition{ \advance\nonamectr by1 \edef\temp{{noname\the\nonamectr}} \expandafter\beginpropositionlabel\temp } \def\endproposition{\endpropositionlabel} \def\begincorollarylabel#1{ \toksdef\ta=0 \ta={\typecorollary\\{#1}} \immediate\write\thmoutfile{\the\ta} \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}} \expandafter\begincorollarylabel\temp } \def\endcorollary{\endcorollarylabel} \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 % \def\be#1{ \def\temp{#1} \toksdef\ta=0 \ta={\\{#1}} \immediate\write\eqroutfile{\the\ta}$$ } \def\ee{ \eqno{(\csname eqlabel\temp\endcsname) \ifdraftmode{\rlap{\ {\smallfont\temp}}}}$$ } \def\eel#1{ \eqno{(\csname eqlabel\temp\endcsname #1) \ifdraftmode{\rlap{\ {\smallfont\temp}}}}$$ } \def\lastdisplayline#1#2{ \hfill#2\hfill\llap{(\csname eqlabel#1\endcsname) \ifdraftmode{\rlap{\ {\smallfont#1}}}}\cr \toksdef\ta=0 \ta={\\{#1}} \immediate\write\eqroutfile{\the\ta} } \def\beginproof{\bigbreak\par\noindent{\bf proof:\ }} %\def\endproof{\thinspace\Box\par\medbreak} \def\endproof{\hfill\vrule height .75em width .75em depth 0pt\bigbreak} \def\beginproofof#1{\bigbreak\par\noindent{\bf proof of #1:\ }} %\def\endproofof{\thinspace\Box\par\medbreak} \def\begindefn{\medbreak\par\noindent{\it Definition:\/\ }} \def\enddefn{\par\medbreak} \def\beginexample{\medbreak\par\noindent{\it Example:\/\ }} \def\endexample{\par\medbreak} \def\chead#1{\medbreak\centerline{#1}\medskip} \def\bea#1{$$\eqalign\begingroup} \def\eea{\endgroup $$} \def\beann{$$\eqalign\begingroup} \def\eeann{\endgroup $$} % % These go at the beginning of input % \global\newtoks\eqrtoks \global\newtoks\thmtoks \global\newcount\secnumber \global\newcount\nonamectr \global\newcount\probctr \message{reference macros need to run twice to get references right} \ \vskip1cm \centerline{\bf Resonant Decay Near an Accumulation Point} \vskip1cm \centerline{Christopher King$^{(1)}$ and Roger Waxler$^{(2)}$} \bigskip \noindent{\it $^{(1)}$ Department of Mathematics and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Mass. 02115\hfill\break $^{(2)}$ Department of Mathematics, S.U.N.Y. at Buffalo, Buffalo, N.Y., U.S.A. 14214} \bigskip\bigskip {\narrower\bigskip\noindent{\bf Abstract:}\/ We consider the quantum mechanics of a model system in which meta-stable states arise through perturbation of a sequence of embedded eigenvalues with an embedded accumulation point. It is shown that the embedded eigenvalues become resonances in the perturbed system. These resonances also accumulate, and the position of the accumulation point is unchanged. The positions of the resonances are estimated uniformly up to the accumulation point. The meta-stable states associated with these resonances have the usual approximately exponential decay with time. \bigskip} \vfill\eject \beginsection \header{\S 1: Introduction}{Introduction} Hamiltonians of the form $$ H(\lambda) = \left(\matrix{A & \lambda V\cr \lambda V^{*} & B\cr}\right) $$ where $\lambda$ is a real number, and the operators $A$, $B$ are self-adjoint, have been extensively studied ([Da], [F], [Ho1], [Ho2], [K1], [K2], [W1], [W2]) beginning with the work of Friedrichs on what has become known as the Friedrichs model. In the Friedrichs model $B$ is a rank one operator with a positive eigenvalue $E$ while $A$ has only absolutely continuous spectrum $[0,\infty)$. These Hamiltonians describe simple quantum mechanical models with resonant behavior. In particular, if the discrete spectrum of the operator $B$ has non-empty intersection with the continuous spectrum of $A$ then the Hamiltonian $H(0)$ has eigenvalues embedded in its continuous spectrum. Typically, embedded eigenvalues are unstable under perturbation and disappear for $\lambda\ne0$. Under reasonable conditions on $V$, $H(\lambda)$ is expected to have only absolutely continuous spectrum in a neighborhood of an embedded eigenvalue of $H(0)$. Further, if $\Psi$ is an eigenvector of $H(0)$ corresponding to an embedded eigenvalue then, if $\lambda\ne0$ is small enough, $\Psi$ is expected to behave like a quasi-stable state under the dynamics generated by $H(\lambda)$. To discuss the perturbation of an embedded eigenvalue $E$, the notion of a resonance associated with $E$ is often introduced ([WW], see [S1] [S2] for reviews). There are various ways to define a resonance. For our purposes it suffices to use the following formulation, which seems the simplest. Let $\Psi$ be an eigenvector of $H(0)$ corresponding to $E$. The matrix element $\inprod{\Psi,\big(H(\lambda)-z\big)^{-1}\Psi}$ is an analytic function of $z$ in the upper half plane. For $\lambda=0$ the matrix element $\inprod{\Psi,\big(H(0)-z\big)^{-1}\Psi}$ is meromorphic in an open disk centered at $E$ with a simple pole at $z=E$. Suppose that, for $\lambda\ne0$, $\inprod{\Psi,\big(H(\lambda)-z\big)^{-1}\Psi}$ has a meromorphic continuation to an open disk centered at $E$, whose only singularity is a simple pole below the real axis. Then this pole is called a {\it resonance}. Typically, the position of the pole depends on $\lambda$ and approaches $E$ as $\lambda \rightarrow 0$. For well-behaved perturbations $V$, this position is accurately estimated by lowest order perturbation theory. Our interest here is in the behavior of resonances near accumulation points, an issue which does not arise in the Friedrichs model. Indeed to the best of our knowledge, ours is the first model of this type in which results have been obtained. The Hamiltonian for an atom in the quantised radiation field ([BFS], [D], [PS]) has this structure, namely an accumulation point embedded in continuous spectrum, as does the Hamiltonian for a hydrogen atom in a sufficiently strong magnetic field ([FW1], [FW2]). Although our model is considerably simpler than these models, we hope that it can point the way toward an analysis of such problems. For the model that we consider in this paper, the operator $A$ has purely continuous spectrum $[0, \infty)$ while the operator $B$ has discrete spectrum with an accumulation point at the number 1 and continuous spectrum $[1, \infty)$. Note that when $\lambda=0$ the operator $H(0)$ has embedded eigenvalues with an accumulation point at $1$. Our analysis depends on having sufficient control over $V^*(A-z)^{-1}V$ and over the resolvent of $B$. In order to allow a complete and straightforward analysis we make simple choices for $A$, $B$ and $V$. We choose $A=-{d^2\over dx^2}$ so that we have an explicit expression for the resolvent kernel of $A$. Further, we choose $V$ to be a multiplication operator with compact support so that $V^*(A-z)^{-1}V$ has an analytic continuation across the positive real line. Finally, we choose $B=-{d^2\over dx^2}-{1\over x}+1$ with Dirichlet boundary conditions at $x=0$. The operator $B$ is a simple example of a Schr\"odinger operator with an accumulation point, and we have detailed estimates for its eigenvalues, discrete eigenfunctions and continuum eigenfunctions [FW1]. Briefly, our results are as follows. For $|\lambda|>0$ sufficiently small, each embedded eigenvalue of $H(0)$ turns into a resonance of $H(\lambda)$ in the sense discussed above. The position of the resonance agrees to lowest order with the predictions of perturbation theory. Furthermore, the resonances approach the accumulation point of the embedded spectrum in a regular way. Namely, in our model the embedded eigenvalues $E_{J}=1-{1\over4J^2}$ of $H(0)$ approach 1 as $J \rightarrow \infty$, in such a way that $J^{3} \bigl(E_{J+1} - E_{J}\bigr)$ approaches a nonzero limit. The resonances $z_{J}$ also approach 1 as $J \rightarrow \infty$, and in such a way that $J^{3} \bigl(z_{J} - E_{J}\bigr)$ approaches a nonzero limit as J$\to\infty$. Our techniques rely heavily on the explicit form of the operators $A$, $B$ and $V$, and they do not generalise easily to other Hamiltonians. Nevertheless we feel that it is important to have a good supply of ``exactly solvable" models in quantum mechanics to illustrate the basic phenomena that can occur. The paper is organised as follows. In section 2 we define the model, explain our goals more precisely and state our results. In section 3 we explore the difficulties presented by the embedded accumulation point, and show how to overcome them in our model. In section 4 we consider resonances near the embedded accumulation point, and prove that they behave in the regular way described above. \endsection\vfill\eject \beginsection \header{\S 2: Definition of the Model and Statement of Results} {Model and Results} The model we consider is defined by the Hamiltonian $H(\lambda)$, acting in the state space $\L^2\big([0,\infty)\big)\oplus\L^2\big([0,\infty)\big)$. $H(\lambda)$ is given by \be{H def} H(\lambda)=\pmatrix{-{d^2\over dx^2} &\lambda u \cr \lambda u & h_0+1\cr} \ee where $h_0$ is the differential operator \be{h_0 def} h_0=-{d^2\over dx^2}-{1\over x}, \ee $\lambda$ is a real number and $u$ is a self-adjoint multiplication operator. $H(\lambda)$ is defined on the domain appropriate to Dirichlet boundary conditions at $0$, namely $$ D\big(H(\lambda)\big)=\big\{\pmatrix{\phi\cr\psi\cr}\, \big|\, \phi(0)=\psi(0)=0,\ \phi\in D(-{d^2\over dx^2})\ \hbox{and}\ \psi\in D(h_0)\big\}. $$ Here $D(A)$ is the domain of the operator $A$. The function $u(x)$ is real valued and is assumed to be continuous with compact support in $[0,\infty)$. That $H(\lambda)$ is self-adjoint follows immediately from the self-adjointness of the Dirichlet problems for $-{d^2\over dx^2}$ and $h_0$ on the half-line, from which it follows that $H(0)$ is self-adjoint, and from the fact that $u$ is bounded, from which it then follows that $H(\lambda)-H(0)$ is bounded. The Hamiltonian $h_0$ is related to the $0$ angular momentum sector of the Hydrogen atom Hamiltonian, see [LL]; its spectrum is $$ \sigma(h_0)=\big\{-{1\over4j^2}\, \big|\, j\in\{1,2,3,\dots\}\big\}\cup [0,\infty). $$ The eigenvalues $-{1\over4j^2}$ are simple. For each $j\in\{1,2,3,\dots\}$ we define $\psi_j$ to be the normalised eigenfunction of $h_0$ with eigenvalue $-{1\over4j^2}$. In section 3 we will write down the explicit expressions for the eigenfunctions $\psi_j$, as well as for the continuum eigenfunctions. Similarly, the Hamiltonian $-{d^2\over dx^2}$ has spectrum $\sigma \big((-{d^2\over dx^2})_D\big)=[0,\infty)$ (here the subscript $D$ indicates Dirichlet boundary conditions). It follows that the operator $H(0)$ has spectrum $[0,\infty)$ and has eigenvalues \be{emb eigs} E_j=1-{1\over4j^2} \ee embedded in the continuous spectrum. Let $$ \Psi_j=\pmatrix{0\cr\psi_j\cr} $$ be the corresponding eigenvectors. Note that the eigenvalues of $H(0)$ have an embedded accumulation point at $1$. As a final comment we note that it follows from the assumption that $u$ has compact support that $H(\lambda)-H(0)$ is not only bounded but is also a relatively compact perturbation of $H(0)$ for all $\lambda$. It follows that the essential spectrum of $H(\lambda)$ is equal to that of $H(0)$, namely $\sigma_{ess}\big(H(\lambda)\big)=[0,\infty)$. Our main goal is to analyse resonances of $H(\lambda)$ associated to the embedded eigenvalues of $H(0)$. Define, for each $J\in\{1,2,3,\dots\}$ and for $\Im z>0$ \be{R_j def} R_J(\lambda;z)=\inprod{\Psi_J,\big(H(\lambda)-z\big)^{-1}\Psi_J}. \ee Let \be{D_j} D_J=\{z\, \big|\, |z-E_J|\le{1\over16 J^3}\}. \ee Note that $R_J(0;z)$ has a meromorphic extension to $D_J$ whose only singularity is a simple pole at $z=E_J$. We will find a meromorphic continuation of $R_J(\lambda;z)$ across the positive real half-line for $\lambda\ne0$. The only singularity of this extension is a simple pole whose position we estimate. Explicitly, there is a $\lambda_0>0$, such that for all $\lambda$ with $|\lambda|<\lambda_0$ and for each $J$, the function $R_J(\lambda;z)$ has a meromorphic extension to $D_J$ whose only singularity is a simple pole. Let $z_J$ be the position of this pole in $D_J$. Then $\Im z_J<0$ and $z_J\to E_J$ as $\lambda\to0$. We refer to $z_J$ as the perturbative resonance of $H(\lambda)$ associated to $E_J$. Further, by Stone's formula (recall that the $\psi_j$ are real), \be{Stone} \inprod{\Psi_J,e^{-itH(\lambda)}\Psi_J}= {1\over\pi}\lim_{\eta\downarrow0}\int e^{-it\epsilon}\Im R_J(\lambda;\epsilon+i\eta)\, d\epsilon. \ee Our results are enough to easily establish approximate exponential decay in time for $\inprod{\Psi_J,e^{-itH(\lambda)}\Psi_J}$ and to identify the lifetime of the meta-stable state $\Psi_J$ with the inverse of the imaginary part of the resonance $z_J$ ([H], [S1], [S2], [WW]). We emphasize that our results are uniform in $J$. Before stating our results we will need some definitions. We write $\sqrt{-z}$ for the branch of the function which is analytic off the positive real axis, and satisfies $\sqrt{-1}=+i$. Let $V(z)$ be the operator on $\L^2\big([0,\infty)\big)$ with integral kernel $V(z;x,y)$, given by \be{V(z) int kern def} V(z;x,y)=u(x){1\over2\sqrt{-z}} \big(e^{-\sqrt{-z}\, |x-y|}-e^{-\sqrt{-z}\, (x+y)}\big)u(y). \ee Then a straightforward calculation shows that for $\Im z>0$ $$ V(z)=u(-{d^2\over dx^2}-z)^{-1}u; $$ recall that we have Dirichlet boundary conditions at $x=0$. Further, it is easy to check, using $\num{V(z) int kern def}$ and the fact that $u$ has compact support, that $V(z)$ is an analytic bounded operator valued function of $z$ in the strip $\Re z>{1\over2}$, $|\Im z|<1$. Finally, the function \be{varphi def} \varphi(x)=\sqrt x\, J_1(2\sqrt x) \ee for $x\ge0$ plays an important role in our analysis. Here $J_1$ is the Bessel function [AS]. Note that $\varphi$ is the zero energy continuum eigenfunction for $h_0$. In particular $h_0\varphi=0$ as a differential equation and $\varphi(0)=0$. We also define (with abuse of the notation for the inner product in $\L^2$), for $\Re z>{1\over2}$ and $|\Im z|<1$, \be{ def} \inprod{\varphi,V(z)\varphi}=\int_0^{\infty}\int_0^{\infty} \varphi(x)V(z;x,y)\varphi(y)\, dx\, dy. \ee Although $\varphi$ is not square integrable, $\num{ def}$ exists since $u$ has compact support. \bigskip \noindent{\bf Results:} There is a $\lambda_0>0$ such that for all $\lambda\in (-\lambda_0,\lambda_0)$ and for all $J\in\{1,2,3,\dots\}$ the function $R_J(\lambda;z)$ has a meromorphic extension to $D_J$ whose only singularity is a simple pole at $z=z_J$. Define $\gamma_J\in \C$ by \be{gamma_J def} z_J=E_J+{\gamma_J\over J^3}. \ee It is immediate that $\Im\gamma_J\le0$, since $H(\lambda)$ is self adjoint, and that $|\gamma_J|\le {1\over16}$, since $z_J\in D_J$. For all $J$, $\gamma_J$ satisfies the perturbative estimate $$ |\gamma_J+\lambda^2 J^3\inprod{\psi_J,V(E_J)\psi_J}|\le\const\ \lambda^4. $$ Further, $$ J^3\inprod{\psi_J,V(E_J)\psi_J}={1\over2}\inprod{\varphi,V(1)\varphi}+\O({1\over J^2}). $$ If we then make the additional assumption \be{res assump}\eqalign{ \Im \inprod{\varphi,V(1)\varphi}&=\Big(\int_0^\infty u(x)\varphi(x)\, \sin x\, dx\Big)^2\cr &\ne0 }\ee then $\inprod{\varphi,V(1)\varphi}\ne0$, and there is a $J_0>0$ such that for $J>J_0$, $\Im\gamma_J<0$. In addition $\{\gamma_J\}$ converges as $J \to \infty$. \bigskip We end this section by pointing out that, despite the eigenvalue accumulation, for all $\lambda\in (-\lambda_0,\lambda_0)$ and $J>J_0$ we get the expected approximate exponential decay away from the initial condition $\Psi_J$: $$ |\inprod{\Psi_J,e^{-itH(\lambda)}\Psi_J}-e^{-itz_J}|\le\const\ \lambda^2. $$ This follows from Stone's formula $\num{Stone}$ and the existence of the resonant pole, $z_J$, precisely as in [H]. \endsection\vfill\eject \beginsection \header{\S 3: Perturbation Theory}{Perturbation Theory} In this section we recall the standard method for studying resonances using analytic perturbation theory. We point out why the standard method has a difficulty in this model, due to the eigenvalue accumulation, and we show how this difficulty is overcome. We will have several occasions to use the following observation from linear algebra. Let $V_1$ and $V_2$ be vector spaces and let $$ A=\pmatrix{M&f\cr e&a\cr} $$ be a linear operator on $V_1\oplus V_2$. Here $M:V_1\to V_1$, $f:V_2\to V_1$, $e:V_1\to V_2$ and $a:V_2\to V_2$. Let $M$ be invertible. Then $A$ is invertible iff $$ \alpha=a-eM^{-1}f $$ is invertible. If so then \be{matrix scam} A^{-1}=\pmatrix{M^{-1}+M^{-1}f\alpha^{-1}eM^{-1}&-M^{-1}f\alpha^{-1}\cr -\alpha^{-1}eM^{-1}&\alpha^{-1}\cr}. \ee Now let $z\in D_J$. If $\Im z>0$ then $-{d^2\over dx^2}-z$ is invertible and, setting $M=-{d^2\over dx^2}-z$, we can apply $\num{matrix scam}$ to $\num{H def}$ and $\num{R_j def}$. Thus $R_J(\lambda;z)$ can be rewritten \be{matrix scam 1} R_J(\lambda;z)=\inprod{\psi_J,\big(h_0+1-z-\lambda^2V(z)\big)^{-1}\psi_J}. \ee Define \be{h(z) def} h(z)=h_0+1-z-\lambda^2V(z) \ee and introduce the orthogonal projection $\Pi_J$ onto the eigenspace of $h_0$ with eigenvalue $-{1\over4J^2}$. Writing $\Pi_J^\perp=1-\Pi_J$ we have $\L^2\big([0,\infty)\big)={\rm Ran}\Pi_J^\perp\oplus{\rm Ran}\Pi_J$. In general, given an operator $A$ in $\L^2\big([0,\infty)\big)$ for which the restriction $\Pi_J^\perp A\Pi_J^\perp$ is invertible on ${\rm Ran}\Pi_J^\perp$, we will denote by $A_{\perp,J}^{-n}$ (for $n=1,2$) the operator on $\L^2\big([0,\infty)\big)$ which equals $(\Pi_J^\perp A\Pi_J^\perp)^{-n}$ on ${\rm Ran}\Pi_J^\perp$, equals zero on ${\rm Ran}\Pi_J$, and leaves invariant the subspaces ${\rm Ran}\Pi_J^\perp$ and ${\rm Ran}\Pi_J$. We will show that $\Pi_J^\perp h(z)\Pi_J^\perp$ is invertible on ${\rm Ran}\Pi_J^\perp$ and that $h(z)_{\perp,J}^{-1}$ is an analytic bounded operator valued function of $z$, for $z\in D_J$. Thus, another application of $\num{matrix scam}$, now with $M^{-1}=h(z)_{\perp,J}^{-1}$, will show that $R_J(\lambda;z)$ is analytic in $z$ unless \be{pert res cond} E_J-z-\lambda^2\inprod{\psi_J,V(z)\psi_J}-\lambda^4 \inprod{\psi_J,V(z)h(z)_{\perp,J}^{-1} V(z)\psi_J}=0. \ee If $\num{pert res cond}$ is not satisfied then \be{matrix scam 2} R_J(\lambda;z)={1\over \inprod{\psi_J,h(z)\psi_J}-\lambda^4 \inprod{\psi_J,V(z)h(z)_{\perp,J}^{-1} V(z)\psi_J}}. \ee If $\num{pert res cond}$ is satisfied for some $z=z_J$ then it is immediate that \be{gamma_J pert est} |z-E_J+\lambda^2\inprod{\psi_J,V(z)\psi_J}|\le\const\ \lambda^4. \ee where $\const$ may depend on $J$. It follows from the self-adjointness of $H$ that $\num{pert res cond}$ has no solutions if ${\rm Im} z > 0$. Solutions with ${\rm Im} z < 0$ are called resonances for $H$ while real solutions are embedded eigenvalues. In studying $h(z)_{\perp,J}^{-1}$ and $\num{pert res cond}$ the chief difficulty to be overcome is that of obtaining estimates which are uniform in $J$. As $J$ increases the unperturbed eigenvalues $E_J$ accumulate. For large $J$ we have $$ E_{J+1}-E_J={1\over2J^3}+\O({1\over J^4}). $$ As a consequence, for $z \in D_J$, \be{(h_0+1-z)_{perp,J}^{-1} norm} \|(h_0+1-z)_{\perp,J}^{-1}\|\sim 2J^3 \ee and for a generic bounded perturbation $U$, $\Pi_J^\perp (h_0+1-z+\lambda^2U)\Pi_J^\perp$ is not invertible for all $z\in D_J$ unless $\lambda^2\|U\|J^3$ is small enough. For any $\lambda\ne0$ this fails for $J$ sufficiently large. It was noticed in [FW1] that this difficulty can be overcome if the perturbation $U$ has an integral kernel $U(z;x,y)$ which approaches zero rapidly enough as either $x$ or $y$ goes to infinity. In the case at hand, because of our assumption that the function $u(x)$ has compact support, $V(z;x,y)$ has compact support in $x$ and $y$. Let $\chi$ be the multiplication operator corresponding to the function \be{chi def} \chi(x)=\cases{1 & if $x\in{\rm supp}(u)$\cr 0 & if $x\notin{\rm supp}(u)$.\cr} \ee Note that, for each $J\in\{1,2,3,\dots\}$, the operator $\chi(h_0+1-z)_{\perp,J}^{-1}\chi$ is an analytic bounded operator valued function of $z$, for $z\in D_J$. We will show that $\chi(h_0+1-z)_{\perp,J}^{-1}\chi$ is also bounded uniformly in $J$, thus overcoming the difficulty expressed by $\num{(h_0+1-z)_{perp,J}^{-1} norm}$. As in [FW1] we introduce the eigenfunction expansion for $h_0$. The discrete eigenfunctions of $h_0$ are [FW1 Appendix B] $$ \psi_j(x)=2^{-{1\over2}}j^{-{3\over2}}xe^{-{1\over2j}x}M(1-j,2,{x\over j}) $$ while the continuum eigenfunctions of $h_0$ are $$ \psi_p(x)=(1-e^{-{\pi\over|p|}})^{-{1\over2}}xe^{-ip x}M(1+{i\over2p},2,2ip x). $$ Here $M(a,b,z)$ is the confluent hypergeometric function [AS], $j\in\{1,2,3,\dots\}$ and the functions $\psi_p(x)$ are normalized with respect to the measure $2p dp$ on $[0,\infty)$. It follows from the identity $M(a,b,z)=e^zM(b-a,b,-z)$ [AS equ. 13.1.27] that the functions $\psi_p(x)$ are real valued. Both $\psi_j$ and $\psi_p$ are solutions to the differential equation $$ ({d^2\over dx^2}+{1\over x}+\delta)\psi(x)=0; $$ for $\psi_j$ we set $\delta=-{1\over4j^2}$, for $\psi_p$ we set $\delta=p^2$. It is easily checked that the following limits hold pointwise: $$ \lim_{j\to\infty}2^{1\over2}j^{3\over2}\psi_j(x)= \lim_{p\to0}\psi_p(x)=\varphi(x). $$ Here $\varphi(x)$ is the zero energy eigenfunction of $h_0$ given by $\num{varphi def}$. We will need some information about the rates of convergence of these limits. The following estimates can be read off from the results of Appendix B of [FW1]. For $x$ in any bounded subset of $[0,\infty)$ \be{psi_j asymp} |\psi_j(x)-2^{-{1\over2}}j^{-{3\over2}}\varphi(x)|\le\const\ j^{-{7\over2}} \ee and \be{psi_p asymp} |\psi_p(x)-\varphi(x)|\le\const\ p^2. \ee \beginlemmalabel{main estimates} There is a real number $K<\infty$ such that for all $J\in\{1,2,3,\dots\}$ and for all $z\in D_J$ \be{cutoff res bnd} \|\chi(h_0+1-z)_{\perp,J}^{-1}\chi\|\le K \ee and \be{cutoff res deriv bnd} \|\chi(h_0+1-z)_{\perp,J}^{-2}\chi\|\le K\, J^3. \ee \endlemmalabel \beginproof Let $\theta_1,\theta_2\in\L^2\big([0,\infty)\big)$, fix $J\in\{1,2,3,\dots\}$ and let $P_1$ be the orthogonal projection onto the spectral subspace of $h_0$ contained in $[-1,1]$. Then, for all $z\in D_J$, \be{mom split}\eqalign{ \inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}\chi\theta_2}= &\inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}P_1\chi\theta_2}\cr &+\inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}(1-P_1)\chi\theta_2}. }\ee Since $$ \|(h_0+1-z)_{\perp,J}^{-1}(1-P_1)\|\le1 $$ the second term in the expression $\num{mom split}$ is analytic in $z$ and $$ |\inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}(1-P_1)\chi\theta_2}| \le\|\theta_1\|\|\theta_2\| $$ uniformly for $z\in D_J$. Using the eigenfunction expansion for $h_0$ the first term in $\num{mom split}$ can be rewritten as follows: \be{spec decomp}\eqalign{ \inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}P_1\chi\theta_2}= &\sum_{j\ne J}{1\over E_j-z}\inprod{\theta_1,\chi\psi_j} \inprod{\chi\psi_j,\theta_2}\cr &+\int_0^1{1\over p^2+1-z}\inprod{\theta_1,\chi\psi_p} \inprod{\chi\psi_p,\theta_2}\, 2pdp. }\ee We now apply the estimates $\num{psi_j asymp}$ and $\num{psi_p asymp}$. We have, for $\sigma\in\{1,2\}$, $$ |\inprod{\theta_\sigma,\chi\psi_j}-2^{-{1\over2}}j^{-{3\over2}} \inprod{\theta_\sigma,\chi\varphi}|\le\const\ j^{-{7\over2}}\|\theta_\sigma\| $$ and $$ |\inprod{\theta_\sigma,\chi\psi_p}-\inprod{\theta_\sigma,\chi\varphi}| \le\const\ p^2\|\theta_\sigma\|. $$ Noting that, for all $z\in D_J$, \be{sing sum est} {1\over2}\sum_{j\ne J}{j^{-r}\over E_j-z}=\cases{-2\ln J+\O(1) & if $r=3$\cr \O(1) & if $r>3$\cr} \ee and \be{sing int est} \int_0^1{p^r\over p^2+1-z}\, 2pdp=\cases{2\ln J+\O(1) & if $r=0$\cr \O(1) & if $r>0$\cr} \ee we find that $$ |\inprod{\theta_1,\chi(h_0+1-z)_{\perp,J}^{-1}P_1\chi\theta_2}| \le\const\ \|\theta_1\|\|\theta_2\|. $$ The bound $\num{cutoff res bnd}$ follows. Proving $\num{cutoff res deriv bnd}$ is similar. The only difference is that we use the estimates \be{deriv sums} {1\over2}\sum_{j\ne J}{j^{-3-r}\over |E_j-z|^2}\le\const\ \cases{J^{3-r}& if $r<3$\cr 1 & if $r\ge3$\cr} \ee and \be{deriv ints} \int_0^1{p^r\over |p^2+1-z|^2}\, 2pdp\le\const\ \cases{J^{2-r} & if $r<2$\cr \ln J & if $r=2$\cr 1 & if $r>2$\cr} \ee for all $z\in D_J$.\endproof An immediate consequence of $\num{cutoff res bnd}$ is the invertibility of $\Pi_J^\perp h(z)\Pi_J^\perp$ on ${\rm Ran}\Pi_J^\perp$ for all $z\in D_J$. If $\lambda$ is small enough the expansion \be{expansion of h(z)} h(z)_{\perp,J}^{-1}=(h_0+1-z)_{\perp,J}^{-1}\sum_{N=0}^\infty \Big(\lambda^2 V(z)(h_0+1-z)_{\perp,J}^{-1}\Big)^N \ee converges uniformly on $D_J$, since it can be written as an expansion in powers of $V(z)$ times the operator $\num{cutoff res bnd}$. It follows that $h(z)_{\perp,J}^{-1}$ is an analytic bounded operator valued function of $z$ in $D_J$, that \be{cutoff h perp inv bnd} \|\chi h(z)_{\perp,J}^{-1}\chi\|\le\const\ \ee and, using $\num{cutoff res deriv bnd}$, that \be{cutoff h perp inv^2 bnd} \|\chi h(z)_{\perp,J}^{-2}\chi\|\le\const\ J^3. \ee Thus $\num{matrix scam 2}$ provides an analytic continuation of $R_J(\lambda;z)$ across the real line to the set $\{z\in D_J\, |\, z\ \hbox{is not a solution of}\ \num{pert res cond}\}$. To see that $\num{pert res cond}$ has a unique solution and that this solution is a simple zero note that, using $\num{psi_j asymp}$, $$\eqalign{ |\deriv z \inprod{\psi_J,V(z)\psi_J}|&=|\inprod{\psi_J,V^\prime(z)\psi_J}|\cr &\le\const\ J^{-3} }$$ and, using $\num{cutoff h perp inv bnd}$, $\num{cutoff h perp inv^2 bnd}$ and $\num{psi_j asymp}$ again, that $$\eqalign{ |\deriv z \inprod{\psi_J,V(z)h(z)_{\perp,J}^{-1} V(z)\psi_J}|\le&| \inprod{\psi_J,V^\prime(z)h(z)_{\perp,J}^{-1} V(z)\psi_J}|\cr &+|\inprod{\psi_J,V(z) h(z)_{\perp,J}^{-1}h^\prime(z)h(z)_{\perp,J}^{-1} V(z)\psi_J}|\cr &+|\inprod{\psi_J,V(z)h(z)_{\perp,J}^{-1} V^\prime(z)\psi_J}|\cr \le&\const\ . }$$ Now apply the contraction mapping theorem. Let $z_J$ be the unique solution of $\num{pert res cond}$ in $D_J$. It follows from the comments above that $R_J(\lambda;z)$ is meromorphic in $z$ for $z\in D_J$ with a simple pole at $z=z_J$. Further, the estimate $\num{gamma_J pert est}$ is uniform in $J$. To estimate the large $J$ behavior of $z_J$ we can use $\num{psi_j asymp}$ and $\num{psi_p asymp}$ in $\num{pert res cond}$. Recalling the definitions $\num{emb eigs}$ of $E_J$ and $\num{gamma_J def}$ of $\gamma_J$ we have $$\eqalign{ \inprod{\psi_J,V(E_J+{\gamma_J\over J^3})\psi_J}&= {1\over 2J^3}\inprod{\varphi,V(E_J+{\gamma_J\over J^3})\varphi}+ \O({1\over J^5})\cr &={1\over 2J^3}\inprod{\varphi,V(1)\varphi}+\O({1\over J^5}). }$$ Further, using $\num{cutoff h perp inv bnd}$, $$ |\inprod{\psi_J,V(E_J+{\gamma_J\over J^3})h (E_J+{\gamma_J\over J^3})_{\perp,J}^{-1} V(E_J+{\gamma_J\over J^3})\psi_J}|\le\const\ {1\over J^3}. $$ Thus \be{gamma_J large J pert est} \gamma_J=-{1\over2}\lambda^2\Big(\inprod{\varphi,V(1)\varphi}+ \O({1\over J^2})\Big)+\O(\lambda^4). \ee \endsection\vfill\eject \beginsection \header{\S 4: The Large $J$ Limit}{Large $J$} In this section we examine the position of the resonant pole $z_{J}$ as $J \rightarrow \infty$ and show that $J^3(z_J-E_J)$ has a limit. This pole is given by the unique solution of $\num{pert res cond}$ in the disk $D_{J}$. Recall the notation introduced in $\num{gamma_J def}$, namely $z_{J} = E_J + \gamma_J J^{-3}$. Let us write $$ z_J(\gamma)=E_J+{\gamma\over J^3} $$ for $|\gamma| \leq {1\over16}$ and define the following function of $\gamma$ and $J$: \be{G_J def} G_{J}(\gamma) = - {\lambda}^{2} J^{3} \inprod{\psi_{J}, V\big(z_J(\gamma)\big) \psi_{J} } - {\lambda}^{4} J^{3} \inprod{ \psi_{J}, V\big(z_J(\gamma)\big) h\big(z_J(\gamma)\big)_{\perp, J}^{-1} V\big(z_J(\gamma)\big) \psi_{J} }. \ee It follows from $\num{pert res cond}$ that ${\gamma}_{J}$ is the unique fixed point of the function $G_{J}$ in the disk $$ D_{0} = \{ \gamma \,\big| \, |\gamma| \le {1\over16}\}. $$ \beginpropositionlabel{convergence of gamma} The sequence ${\gamma}_{J}$ converges in the disk $D_{0}$ as $J \rightarrow \infty$. \endpropositionlabel \beginproof The number ${\gamma}_{J}$ is the solution of the equation $$ {\gamma}_{J} = G_{J}({\gamma}_{J}). $$ Recall that $G_{J}$ is a contraction on $D_{0}$ for $|\lambda|$ sufficiently small. Some elementary estimates now show that to prove convergence of the sequence $\{{\gamma}_{J}\}$, it is sufficient to prove that the sequence $\{G_{J}(\gamma)\}$ converges uniformly for $\gamma \in D_{0}$. Using the estimates $\num{psi_j asymp}$ and $\num{psi_p asymp}$, and $\thmnum{main estimates}$, this reduces to showing convergence of $\inprod{ \varphi, V(1) h\big(z_J(\gamma)\big)_{\perp, J}^{-1} V(1) \varphi }$ uniformly in $\gamma$. Using the fact that $u$ has compact support, and recalling $\num{chi def}$, this is equivalent to the uniform convergence of the sequence of bounded operators $\chi h\big(z_J(\gamma)\big)_{\perp, J}^{-1} \chi$. By the results of $\thmnum{main estimates}$, and using the convergent expansion $\num{expansion of h(z)}$, it is sufficient to prove uniform convergence of $\chi(h_0 +{1\over 4 J^2}-{\gamma \over J^3})_{\perp, J}^{-1} \chi$. As in the proof of $\thmnum{main estimates}$ we may restrict to the spectral subspace of $h_0$ with spectrum less than $1$. Again let $P_1$ be the orthogonal projection on this spectral subspace of $h_0$. On the orthogonal complement to this subspace the operator $(h_0+{1\over 4 J^2}-{\gamma \over J^3})_{\perp, J}^{-1}$ is analytic in $\gamma$ and uniformly bounded by $1$ on $D_0$. In fact we have the following equality of norm limits: \be{large J large p} \lim_{J\to\infty} (1-P_1)(h_0+{1\over 4 J^2}-{\gamma \over J^3})_{\perp, J}^{-1}=\lim_{\eta\to0}(1-P_1)(h_0+i\eta)^{-1}. \ee Again using the eigenfunction expansion for $h_0$ we have $$ \chi P_1(h_0+{1\over 4 J^2}-{\gamma \over J^3})_{\perp, J}^{-1}\chi= \sum_{j \neq J}{\chi\psi_j\otimes\psi_j\chi\over{1\over 4J^2}- {1\over 4j^2}-{\gamma\over J^3}} + \int_0^1 {\chi\psi_p\otimes\psi_p\chi\over p^2+ {1\over 4J^2}-{\gamma\over J^3}}\, 2p dp. $$ Using the estimates $\num{psi_j asymp}$ and $\num{psi_p asymp}$ leads us back to $\num{spec decomp}$. Explicitly, define $$ \psi_j^{(1)}=\psi_j-2^{-{1\over2}}j^{-{3\over2}}\varphi, $$ $$ \psi_p^{(1)}=\psi_p-\varphi $$ and introduce the sum and integral \be{worst case} G_J^{(0)}(\gamma) = {1\over2}\sum_{j \neq J}j^{-3}{1\over{1\over 4 J^2}- {1 \over 4 j^2}-{\gamma \over J^3}}+\int_0^1{1\over p^2+ {1\over 4J^2}-{\gamma \over J^3}}\, 2p dp. \ee Applying $\num{psi_j asymp}$, $\num{psi_p asymp}$, $\num{sing sum est}$ and $\num{sing int est}$ we have the existence of the limit (in norm) \be{large J low p}\eqalign{ \lim_{J\to\infty}\Big(\chi P_1&(h_0+{1\over 4 J^2}- {\gamma \over J^3})_{\perp, J}^{-1}\chi-G_J^{(0)}(\gamma)\, \chi\varphi\otimes\varphi\chi\Big)\cr =-&4\sum_{j=1}^\infty j^2 \big({1\over\sqrt{2j^3}}\, \chi\psi_j^{(1)}\otimes\varphi\chi+{1\over\sqrt{2j^3}}\, \chi\varphi\otimes\psi_j^{(1)}\chi + \chi\psi_j^{(1)}\otimes\psi_j^{(1)}\chi\big)\cr &+2\int_0^1{1\over p}\big(\chi\psi_p^{(1)}\otimes\varphi\chi+ \chi\varphi\otimes\psi_p^{(1)}\chi+ \chi\psi_p^{(1)}\otimes\psi_p^{(1)}\chi\big)\, dp. }\ee Note that convergence is uniform in $\gamma$, and the limit is independent of $\gamma$. The only possible $\gamma$ dependence in the large $J$ limit is from the term $G_J^{(0)}(\gamma)$. We now show that $G_J^{(0)}(\gamma)$ has a limit as $J\to\infty$. It is easy to show that the integral in $\num{worst case}$ satisfies the estimate $$ \int_{0}^{1} {1\over p^{2} + {1 \over 4 J^{2}} - {\gamma \over J^{3}}}\, 2p dp = 2 \ln J + \ln 4 + O(J^{-1}). $$ To estimate the sum in $\num{worst case}$ we introduce the number $$ \alpha = \bigl(1 - {\gamma \over J}\bigr)^{-{1 \over 2}} $$ and rewrite the sum as follows: $$\eqalign{ {1 \over 2} \sum_{j \neq J} j^{-3} \bigl({1\over 4J^{2}} &- {1\over 4j^{2}}-{\gamma \over J^{3}}\bigr)^{-1}= 2 {\alpha}^{2} J^{2} \sum_{j \neq J} {1 \over j( j^{2} - {\alpha}^{2} J^{2})}\cr &=\sum_{j \neq J}\Big(-{2\over j}+{1 \over j+\alpha J}+ {1 \over j- \alpha J}\Big)\cr &={2\over J}-{1\over (1+\alpha)J}+ \lim_{K\to\infty}\Big(-2\sum_{j=1}^K {1\over j}+ \sum_{j=1}^K {1\over j+\alpha J}+ \sum_{1 \le j \le K \atop j \ne J}{1 \over j- \alpha J}\Big). }$$ For $|z| \leq {1 \over 2}$, and $N \geq 1$ define $$ \phi_z(N)=\sum_{n=1}^{N} {1 \over n+z}. $$ Note that $$ \lim_{N \rightarrow \infty} \big({\phi}_{z}(N) - \ln N\big)= C-z\sum_{n=1}^{\infty} {1\over n(n+z)}, $$ where $C$ is Euler's constant, and that convergence is uniform in $z$. Further, let $$ \epsilon = (\alpha-1)J; $$ it is easy to see that $|\epsilon| \leq |\gamma|$ for all $J \geq 1$, and that $\epsilon \rightarrow {\gamma \over 2}$ as $J\rightarrow\infty$. We then have $$ \sum_{j=1}^K {1\over j}=\phi_0(K), $$ $$\eqalign{ \sum_{j=1}^K {1\over j+\alpha J}&=\sum_{j=J+1}^{K+J+1} {1\over j+\epsilon}\cr &=\phi_\epsilon(K+J+1)-\phi_\epsilon(J) }$$ and (for $K > J$) $$\eqalign{ \sum_{1 \le j \le K \atop j \ne J}{1 \over j- \alpha J}&= \sum_{1 \le j \le K \atop j \ne J}{1 \over (j-J)- \epsilon}\cr &={1\over J+\epsilon}-\phi_\epsilon(J)+\phi_{-\epsilon}(K-J). }$$ We thus arrive at the following identity: $$\eqalign{ {1 \over 2} \sum_{j \neq J} j^{-3} \bigl({1\over 4J^{2}}-{1\over 4j^{2}}-{\gamma \over J^{3}}\bigr)^{-1} = {2\over J}&+{1\over J+\epsilon}-{1\over 2J+\epsilon}- 2{\phi}_{\epsilon}(J)\cr &+\lim_{K\to\infty}\Big(-2{\phi}_{0}(K)+{\phi}_{\epsilon}(J+K+1)+ {\phi}_{-\epsilon}(K-J)\Big)\cr ={2\over J}&+{1\over J+\epsilon}-{1\over 2J+\epsilon}- 2{\phi}_{\epsilon}(J)+ 2{\epsilon}^{2}\sum_{n=1}^{\infty} {1\over n(n^{2}-{\epsilon}^{2})}. }$$ Substituting these expresions for the sum and integral in $\num{worst case}$, we get the following result: $$ G_{J}^{(0)}(\gamma) = 2 {\epsilon}^{2} \sum_{n=1}^{\infty} {1 \over n(n^{2} - {\epsilon}^{2})} - 2\bigl({\phi}_{\epsilon}(J) - \ln J\bigr) + \ln 4 + O(J^{-1}) $$ uniformly in $\gamma$. Letting $J \rightarrow \infty$ gives \be{G_infty^{(0)} def}\eqalign{ G_\infty^{(0)}(\gamma)&=\lim_{J \rightarrow \infty} G_{J}^{(0)}(\gamma)\cr &={{\gamma}^{2} \over 2} \sum_{n=1}^{\infty} {1 \over n(n^2-{{\gamma}^2 \over 4})}+\gamma \sum_{n=1}^{\infty}{1\over n(n+{\gamma \over 2})} - 2C + \ln 4. }\ee Therefore we conclude that $\lim_{J \rightarrow \infty} G_{J}(\gamma)$ exists for every $\gamma \in D_{0}$, and that $\{G_J\}$ converges uniformly in $D_{0}$. \endproof \medskip To approximate $$ \gamma_\infty=\lim_{J\to\infty}\gamma_J $$ we let $$ A(\gamma)=\lim_{J\to\infty} \chi(h_0 + {1\over 4 J^2}-{\gamma \over J^3})_{\perp, J}^{-1} \chi. $$ >From $\num{large J large p}$, $\num{large J low p}$ and $\num{G_infty^{(0)} def}$ we find that $A(\gamma)$ exists and is given by the sum of these three terms. Note that $$\eqalign{ A(\gamma)-A(0)&=\Big({{\gamma}^{2} \over 2} \sum_{n=1}^{\infty} {1 \over n(n^2-{{\gamma}^2 \over 4})}+\gamma \sum_{n=1}^{\infty} {1\over n(n+{\gamma \over 2})}\Big)\, \chi\varphi\otimes\varphi\chi\cr &={\pi^2\gamma\over6}\ \chi\varphi\otimes\varphi\chi+\O(\gamma^3). }$$ Recalling $\num{expansion of h(z)}$ we have $$ \chi h\big(z_J(\gamma)\big)_{\perp, J}^{-1}\chi= A(\gamma)\sum_{N=0}^\infty \lambda^{2N}\Big(V(1)A(\gamma)\Big)^N. $$ Sustituting in $\num{G_J def}$ it is straightforward to generate an expansion for $\gamma_\infty$ in powers of $\lambda^2$. The first few terms are $$ \gamma_\infty=-{\lambda^2\over 2}\inprod{\varphi,V(1)\varphi}- \lambda^4\inprod{\varphi,V(1)A(0)V(1)\varphi} +\O(\lambda^6). $$ \endsection\vfill\eject \header{Bibliography}{Bibliography} {\noindent\frenchspacing \item{[BFS]} V. Bach, J. Fr\"ohlich, and I. M. Sigal, {\it Mathematical Theory of Nonrelativistic Matter and Radiation}, Lett. Math. Phys. {\bf 34}, 183-201 (1995).\bigskip \item{[D]} P.A.M. Dirac, {\it Quantum Mechanics}, Oxford University Press (1978).\bigskip \item{[Da]} E.B. Davies, {\it Dynamics of a Multilevel Wigner-Weisskopf Atom}, Journal of Mathematical Physics {\bf 15}, 2036-2041 (1974).\bigskip \item{[F]} K.O. Friedrichs, {\it On the Perturbation of Continuous Spectra}, Comm. Pure Appl. Math {\bf 1}, 361-406 (1948).\bigskip \item{[FW1]} R. Froese and R. Waxler, {\it The Spectrum of a Hydrogen Atom in an Intense Magnetic Field}, Reviews in Math. Physics {\bf 6}, 699-832 (1994).\bigskip \item{[FW2]} R. Froese and R. Waxler, {\it Ground State Resonances of a Hydrogen Atom in an Intense Magnetic Field}, Reviews in Math. Physics {\bf 7}, 311-361 (1995).\bigskip \item{[Ho1]} J. Howland: {\it Spectral Concentration and Virtual Poles, II}, Transactions of the A.M.S. {\bf 162}, 141-156 (1971);\bigskip \item{[Ho2]} J. Howland: {\it The Livsic Matrix in Perturbation Theory}, Journal of Mathematical Analysis and Applications {\bf 50}, 415-437 (1975).\bigskip \item{[H]} W. Hunziker, {\it Resonances, Metastable States and Exponential Decay Laws in Perturbation Theory}, Commun. Math. Phys. {\bf 132}, 177-188 (1990).\bigskip \item{[K1]} C. King: {\it Exponential Decay Near Resonance, without Analyticity}, Letters in Math. Phys. {\bf 23}, 215-222 (1991);\bigskip \item{[K2]} C. King: {\it Scattering Theory for a Model of an Atom in a Quantized Field}, Letters in Math. Phys. {\bf 25}, 17-28 (1992).\bigskip \item{[LL]} L. D. Landau and E. M. Lifshitz, {\it Quantum Mechanics (3$^{rd}$ edition)}, Pergamon (1977).\bigskip \item{[PS]} J. Parker and C.R. Stroud, Jr., {\it Coherence and Decay of Rydberg Wave Packets}, Phys. Rev. Letters {\bf 56} 716-719 (1986)\bigskip \item{[RS]} M. Reed and B. Simon: {\it Methods of Mathematical Physics vol. 2: Fourier Analysis, Self Adjointness}, Academic Press (1975);\hfill\break {\it Methods of Mathematical Physics vol. 4: Analysis of Operators}, Academic Press (1978).\bigskip \item{[S1]} B. Simon: {\it Resonances in N-Body Quantum Systems with Dilation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory}, Annals of Math. {\bf 97}, 247-274 (1973);\bigskip \item{[S2]} B. Simon: {\it Resonances and Complex Scaling: a Rigorous Overview}, Int. Journal of Quant. Chem. {\bf 14}, 529-542 (1978).\bigskip \item{[W1]} R. Waxler, {\it The Time Evolution of a Class of Meta-stable States}, Commun. Math. Phys. {\bf 172}, 535-549 (1995).\bigskip \item{[W2]} R. Waxler, {\it Generalized Eigenfunction Expansion Near Resonances}, preprint\bigskip \item{[WW]} V. Weisskopf and E. Wigner, {\it Berechnung der Nat\"urlichen Linienbreite auf Grund der Diracshen Lichttheorie}, Zeitschrift f\"ur Physik {\bf 63}, 54-73 (1930).\bigskip } \end