Content-Type: multipart/mixed; boundary="-------------0401210352512" This is a multi-part message in MIME format. ---------------0401210352512 Content-Type: text/plain; name="04-15.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-15.keywords" PT-symmetry, canonical decomposition, perturbation theory,singular values ---------------0401210352512 Content-Type: application/x-tex; name="PTfinale.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="PTfinale.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% Spedito al JPyisA per referee il 14/11/03 %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt]{article} \setlength{\textwidth}{15.5cm} \setlength{\textheight}{22.7cm} \setlength{\topmargin}{-1.0cm} \setlength{\oddsidemargin}{-1mm} \setlength{\evensidemargin}{-1mm} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \renewcommand{\thetheorem}{\thesection.\arabic{theorem}} \renewcommand{\theproposition}{\thesection.\arabic{proposition}} \renewcommand{\thelemma}{\thesection.\arabic{lemma}} \renewcommand{\thedefinition}{\thesection.\arabic{definition}} \renewcommand{\thecorollary}{\thesection.\arabic{corollary}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \pagestyle{myheadings} %\markboth{E.Caliceti} %{Distributional Borel of Vacuum Polarization} \def\ha{Ha\-mil\-to\-nian} \def\R{{\bf R}} \def\Z{\bf Z} \def\N{{\bf N}} \def\T{\bf T} \def\C{{\bf C}} \def\b{\beta} \def\t{\theta} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ra{\rangle} \def\ds{\displaystyle} \def\Jb{\overline{J}} \def\Hb{\overline{H}} \def\re{{\rm Re}} \def\im{{\rm Im}} \def\RS{Ray\-leigh-Schr\"o\-din\-ger} \def\Sc{Schr\"odinger} \def\PT{{\cal P}{\cal T}} \def\P{{\cal P}} \def\T{{\cal T}} \def\l{{\lambda}} \def\r{\rho} \def\S{{\cal S}} \def\op{o\-pe\-ra\-tor} \def\arg{{\rm arg}} \def\Im{{\rm Im}} \def\Cinf{C_0^\infty(\R^d)} % \date{} \begin{document} \baselineskip=21pt % \title{Canonical Expansion of $\PT-$Symmetric Operators and Perturbation Theory} \author{E.Caliceti\footnote{e-mail: caliceti@dm.unibo.it} \\ Dipartimento di Matematica, Universit\`{a} di Bologna \\40127 Bologna, Italy \\ S.Graffi\footnote{On leave from Dipartimento di Matematica, Universit\`{a} di Bologna, Italy; e-mail: graffi@mathcs.emory.edu. graffi@dm.unibo.it} \\ Department of Mathematics and Computer Science\\ Emory University, Atlanta, Ga 30322. U.S.A.} \maketitle \vskip 12pt\noindent \begin{abstract} { \noindent\baselineskip =16pt Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $\;-\hbar^2\Delta+(x_1^2+\ldots+x_d^2)+igW(x_1,\ldots,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we explicitly construct the canonical expansion of $H$ and determine the singular values $\mu_j$ of $H$ through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues $\l_j$ of $H$ by Weyl's inequalities. } \end{abstract} \vskip 12pt\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction and statement of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A Schr\"odinger operator $H=-\Delta+V$ acting on ${\cal H}=L^2(\R^d)$ is called $\PT-$symmetric if it is left invariant by the $\PT$ operation. While generally speaking $\P$ could be the parity operator with respect to at least one variable, here for the sake of simplicity we consider only the case in which $\P$ is the parity operator with respect to all variables, $(\P u)(x_1,\ldots,x_d)=u(-x_1,\ldots,-x_d)$, and $\T$ the complex conjugation (equivalent to time-reversal symmetry) $(\T u)(x_1,\ldots,x_d):=\overline{u}(x_1,\ldots,x_d)$. The condition \be \label{PT} \overline{V}(-x_1,\ldots,-x_d)=V(x_1,\ldots,x_d) \ee defines the $\PT-$symmetry on the potential $V(x_1,\ldots,x_d)$. The $\PT$-symmetric operators are currently the object of intense investigation because, while not self-adjoint, they admit in many circumstances a real spectrum. Hence the investigation is motivated (at least partially), by an attempt to remove the self-adjointess condition on the observables of standard quantum mechanics (see e.g.\cite{Ah},\cite{Be1},\cite{Be2},\cite{Be3},\cite{Cn1},\cite{Cn2}, \cite{Cn3}, \cite{Zn1},\cite{Zn2}). The simplest and most studied class of $\PT$ symmetric operators is represented by the {\it odd anharmonic oscillators with purely imaginary coupling} in dimension one, namely the maximal differential operators in $L^2(\R)$ \be \label{od} Hu(x) :=[-\frac{d^2}{dx^2} +x^2+igx^{2m+1}], \quad g\in\R,\quad m=1,2,\ldots \ee It has long been conjectured (Bessis Zinn-Justin), and recently proved \cite{Shin}, \cite{Tateo}, that the spectrum $\sigma(H)$ is real for all $g$; there are however examples of one-dimensional $\PT$-symmetric operators with {\it complex} eigenvalues\cite{Cn1}. Now recall that there is a natural additional notion of spectrum associated with a non-normal operator $T$ in a Hilbert space which is by construction real. Any closed operator $T$ admits a {\it polar decomposition} (\cite{Ka}, Chapt. VI.7) $T=U|T|$, where $|T|$ is self-adjoint and $U$ is unitary. The modulus of $T$ is the self-adjoint operator $\ds |T|=\sqrt{T^\ast T}$. The (obviously real and positive) eigenvalues of $|T|$ are called the {\it singular values} of $T$. In this paper we consider the self-adjoint operator $\sqrt{H^{\ast}H}$; its eigenvalues $\mu_j;\; j=0,1,\ldots$, necessarily real and positive, are the by definition the {\it singular values} of $H$. A first immediate question arising in this context is to determine how these singular values are related to the $\PT$-symmetry of $H$. A related question is the explicit construction of the canonical expansion of $H$ (see e.g.\cite{Ka}) in terms of the spectral analysis of $\sqrt{H^{\ast}H}$, which entails the diagonalization of $H$ with respect to a pair of dual bases (which do not form a biorthogonal pair); a further one is the actual computation of the singular values. The determination of the singular values reflects directly on the object of physical interest, namely the eigenvalues $\l_j;\; j=0,1,\ldots$ of $H$. If the eigenvalues and the singular values are ordered according to increasing modulus, the Weyl inequalities (see e.g. \cite{Ho}) indeed yield \begin{eqnarray} \label{Weyl1} \sum_{j=1}^k |\l_j|\leq \sum_{j=1}^k\mu_j,\quad |\l_1\cdots\l_k|\leq \mu_1\cdots\mu_k, \quad k=1,2,\ldots \end{eqnarray} We intend in this paper to give a reply to these questions for the most general class of odd anharmonic oscillator in $\R^d$. Namely, we consider in $L^2(\R^d)$ the \Sc\ operator family \be \label{odM} H(g)u(x):= H_0u(x)+igW(x)u(x),\quad x=(x_1,\ldots,x_d)\in\R^d \ee Here: \begin{enumerate} \item $W$ is a real homogenous polynomial of odd order $2K+1$, $K=1,2\ldots$; $$ W(\l x)=\l^{2K+1}W(x) $$ \item $H_0$ is the \Sc\ operator of the harmonic oscillator in $\R^d$: \be \label{HO} H_0u(x)=-\Delta u(x)+x^2u(x),\quad x^2:=x_1^2+\ldots+x_d^2 \ee \end{enumerate} Under these conditions the operator family $H(g)$, which is obviously $\P T$-symmetric (see below for the mathematical definition), but non self-adjoint, enjoys the following properties (proved in \cite{CGM} for $d=1$ and in \cite{Na} for $d>1$; see below for a more detailed statement): \begin{enumerate} \item The operator $H(g)$, defined as the closure of the minimal differential operator $\dot{H}(g)u=-\Delta u(x)+x^2u(x)+igW(x)$, $u\in C_0^\infty(\R^d)$, generates a holomorphic operator family with compact resolvents with respect to $g$ in some domain ${\cal S}\subset\C$, with $H(g)^\ast=H(\overline{g})$. An operator family $T(g)$ depending on the complex variable $g\in \Omega$, where $\Omega\subset\C$ is open is holomorphic (see \cite{Ka}, VII.1) if the scalar products $\langle u,T(g)v\rangle$ are holomorphic functions of $g\in \Omega$ $\forall\,(u,v)\in T(g)$ and the resolvent $\ds [T(g)-zI]^{-1}$ exist for at least one $g\in\Omega$. \item All eigenvalues of $H_0:=H(0)$ are stable with respect to the operator family $H(g)$. This means (see e.g.\cite{Ka}, VIII.1) that if $\lambda_0$ is any eigenvalue of $H(0)$ of multiplicity $m$, there is $B(\lambda_0)>0$ such that $H(g)$ has exactly $m$ (repeated) eigenvalues $\lambda_j(g): j=1,\ldots,m$ near $\lambda_0$ for $g\in{\cal S}$, $|g|0$ is arbitrary. \item The Rayleigh-\Sc\ perturbation expansion for any eigenvalue $\mu_j(g):j=1,\ldots,m(l)$ of $Q(g)$ near the eigenvalue $\l_l$ of $\P H_0$ for $|g|$ small is Borel summable to $\mu_j(g):j=1,\ldots,m(l)$. \end{enumerate} \end{theorem} {\bf Remark} \par\noindent Let $\mu(g)$ be a singular value near an unperturbed eigenvalue $\l$. The Borel summability (see e.g.\cite{RS}, Chapter XII.5) means that it can be uniquely reconstructed through its divergent perturbation expansion $\ds \sum_{s=0}^\infty\mu_sg^s,\;\mu_o=\l$ in the following way: \be \label{Borel} \mu(g)=\frac1{q}\int_0^\infty\mu_B(gt)e^{-t^{1/q}}t^{-1+1/q}\,dt \ee Here $\ds q=\frac{2K-1}{2}$ and $\mu_B(g)$, the {\it Borel transform of order $q$} of the perturbation series, is defined by the power series $$ \mu_B(g)=\sum_{s=0}^\infty\frac{\mu_s}{\Gamma[q(s+1)] }g^s $$ which has a positive radius of convergence. The proof of (\ref{Borel}) consists precisely in showing that $\mu_B(g)$ has analytic continuation along the real positive axis and that the integral converges for some $0\leq g0$. \par\noindent {\bf Example} \par\noindent The H\'enon-Heiles potential, i.e. the third degree polynomial in $\R^2$ $$ W(x)=x_1^{2} x_2 $$ \section{Proof of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% Let us begin by a more detailed quotation of Theorem 1.1 of \cite{Na}. The results are more conveniently formulated in the variable $\beta=ig$ instead of $g$. Let $\b\in\C$, $0<|{\rm arg}\,\b|<\pi$, and let $\dot{H}(\b)$ denote the minimal differential operator in $L^2(\R^d)$ defined by $-\Delta+x^2+\b W(x)$ on $C_0^\infty(\R^d)$, with $x^2=x_1^d+\ldots+x_d^2$. Then \begin{itemize} \item [(N1)] $\dot{H}(\b)$ is closable. Denote $H(\beta)$ its closure. \item [(N2)] $H(\beta)$ represents a pair of type-A holomorphic families in the sense of Kato for $\ds 0<{\rm arg }\b<{\pi}$ and $\ds -\pi<{\rm arg }\b<0$, respectively, with $H(\b)^\ast=H(\overline{\b})$. Recall that an operator family $T(g)$ depending on the complex variable $g$ belonging to some open set $\Omega\subset\C$ is called type-A holomorphic if its domain $D$ does not depend on $g$ and the scalar products $\langle u,T(g)\rangle$ are holomorphic functions for $g\in D$ $\forall\;(u,v)\in D$. A general theorem of Kato (\cite{Ka}, VII.2) states that the isolated eigenvalues of a type-A holomorphic family are locally holomorphic functions of $g\in D$ with at most algebraic branch points. \item [(N3)] $H(\beta)$ has compact resolvent $\forall\,\b\in\C$, $0<|{\rm arg}\,\b|<\pi$. \item[(N4)] All eigenvalues of $H_0=H(0)$ are stable with respect to the operator family $H(\b)$ for $\b\to 0$, $0<|{\rm arg}\,\b|<\pi$; \item[(N5)] Let $\b\in\C$, $\sigma\in\C$, $0<|{\rm arg}\,\b|<\pi$, $-\pi+{\rm arg}\,\b \leq {\rm arg}\,\sigma \leq {\rm arg}\,\b$, and let $\dot{H}_\sigma(\b)$ denote the minimal differential operator in $L^2(\R^d)$ defined by $-\Delta+\sigma x^2+\b W(x)$ on $C_0^\infty(\R^d)$. Then $\dot{H}_\sigma(\b)$ is sectorial (and hence closable) because its numerical range is contained in the half-plane $\{z\in\C: -\pi+{\rm arg}\,\b \leq {\rm arg}\,\sigma \leq {\rm arg}\,\b\}$; \item[(N6)] Let ${H}_\sigma(\b)$ denote the closure of $\dot{H}_\sigma(\b)$. Let $\sigma\in\C, \sigma\notin ]-\infty,0]$. Then the operator family $\b\mapsto {H}_\sigma(\b)$ is type-A holomorphic with compact resolvents for $\b\in {\cal C}_\sigma:=\{\b\in\C: 0<{\rm arg}\,\b-{\rm arg}\,\sigma <\pi\}$. Moreover if $\b\in\C, {\rm Im}\b >0$, the operator family $\sigma\mapsto {H}_\sigma(\b)$ is type-A holomorphic with compact resolvents in the half-plane ${\cal D}_\beta= \{\sigma\in\C: 0<{\rm arg}\,\b-{\rm arg}\,\sigma <\pi\}$ \end{itemize} Let us now introduce the operator \be \label{dilat} H(\b,\t)=e^{-2\t}\Delta+e^{2\t} x^2+\b e^{2(K+1)\t}W(x):=e^{-2\t}K(\b,\t) \ee For $\t\in\R$ $H(\b,\t)$ is unitarily equivalent to $H(\b)$, ${\rm Im}\b> 0$, via the dilation operator defined by \be \label{dilat1} (U(\t)\psi)(x)=e^{d\t/2}\psi(e^\t x), \qquad \forall\,\psi\in L^2(\R^d) \ee As a consequence of (N6) (see again \cite{Na}, or also \cite{Ca}, where all details are worked out for $d=1$, and where the reader is referred also for the proof of statement( N8) below) we have: \begin{itemize} \item[(N7)] $H(\b,\t)$ defined on $D(H(\b))$ represents a type-A holomorphic family with compact resolvents for $\b$ and $\t$ such that $ s={\rm arg}\b, \; t=\im\t$ are variable in the parallelogram ${\cal R}$ defined as \be \label{par} {\cal R}=\{(s,t)\in\R^2:0<(2K-1)t+s<\pi, 0<(2K+3)t+s<\pi\}, \; \ee Moreover $C_0^\infty$ is a core of $H(\b,\t)$. The spectrum of $H(\b,\t)$ does not depend on $\t$. Note that $(s,t)\in {\cal R}$ entails that the maximal range of $\b$ is $-(2K-1)\pi/4 <\arg\b<(2K+3)\pi/4$ and that the maximal range of $\t$ is $-\pi/4 <\Im\t <\pi/4$; \item[N8)] Let $\b$ and $\t$ be such that $(s,t)\in{\cal R}$. Then: \begin{itemize} \item[(i)] If $\l\notin\sigma (K(0,\t))$, then $\l\in\tilde\Delta$, where: \begin{eqnarray} \label{unif} \tilde\Delta:=\{z\in\C:z\notin \sigma(K(\b,\t)); \|[z-(K(\b,\t)]^{-1}\| \\ \mathrm{is}\; \mathrm{uniformly}\;\mathrm {bounded} \;\mathrm {for}\;|\b|\; \to 0 \} ; \nonumber \end{eqnarray} \item[(ii)] If $\l\in\sigma (K(0,\t))$, then $\l$ is stable with respect to the operator family $K(\b,\t)$. \end{itemize} (N7) and (N8) entail: \item[(N9)] Let $\b\in\C$ with $\ds 0<{\rm arg}(\b)<{\pi}$. Then for any $\delta >0$ and any eigenvalue $\l(g)$ of $H(\b)$ there exists $\rho >0$ such that the function $\l(\b)$, a priori holomorphic for $0<|g|<\rho$, $\ds \delta<{\rm arg}(\b)<{\pi}-\delta$, has an analytic continuation to the Riemann surface sector $\ds \tilde{\S}_{K,\delta}:= \{\b\in\C: 0<|\b|<\rho; -(2K-1)\frac{\pi}{4}+\delta<{\rm arg}(\b)<(2K+3)\frac{\pi}{4}-\delta\}$. \end{itemize} {\bf Remarks} \begin{enumerate} \item The stability statement means the following: if $r>0$ is sufficiently small, so that the only eigenvalue of $K(0,\t)$ enclosed in $\Gamma_r:=\{z\in\C: |z-\l|=r\}$ is $\l$, then there is $B>0$ such that for $|\b|0$ such $H(g)^{-1}$ is uniformly bounded in $\tilde{S}:=\{g\in \S_1\cup\S_2, |g|0$ such that $$ \liminf_{m\to\infty}\|M_hu_m\|\geq a>0, \quad \forall\,h $$ \item[(2)] For some $z\in\tilde{\Delta}_1$: $$ \lim_{h\to\infty}\|[M_h,K(\r)][z-K(\r)]^{-1}\|=0 $$ \item[(3)] $\ds \lim_{h\to\infty\atop \r\downarrow 0}d_h(\l,\r)=+\infty$ $\forall\,\l\in\C$, where: $$ d_h(\l,\r):=\inf \{\|[\l-K(\r)]M_hu\|:u\in D(K(\r)), \|M_hu\|=1\} $$ \end{itemize} \end{itemize} Hence we must verify the analogous properties, denoted $(a^\prime)$, $(b^\prime)$, $(c^\prime)$, for the operator family $T(\r)$. Remark that, as in \cite{Ca}, the verification of (b') requires an argument completely independent of \cite{HV} because the operator family $T(\rho)$ is not sectorial. We have: \par\noindent $(a^\prime)$ From $(a)$ and the continuity of $\P$ we can write $$ \lim_{\r\downarrow 0}T(\r)u=T(0)u,\quad \lim_{\r\downarrow 0}T(\r)^\ast u=T(0)^\ast u,\quad \forall\,u\Cinf $$ $(b^\prime)$ First remark that $0\in\tilde{\Delta}$ by N9) (i) since $0\notin \sigma(K(0,\t))$. Then there is $B>0$ such that $$ \sup_{0\leq |\b|0$ such that $\mu=0$ is not an eigenvalue of $T(\b,\t)$ for $|\b|0$ such that $$ \liminf_{m\to\infty}\|M_hu_m\|\geq a>0,\quad \forall\,h $$ \item[(2')] As proved in \cite{HV}, if (c2) holds for some $z\in\tilde{\Delta}_1$ then it holds for all $z\in\tilde{\Delta}_1$. Thus we can take $z=0\in \tilde{\Delta}\cap \tilde{\Delta}_1$ and we have: \begin{eqnarray*} \lim_{h\to\infty}\|[M_h,T(\r)](\P K(\r))^{-1}\|= \lim_{h\to\infty}\|(M_h\P K(\r)-\P K(\r)M_h)(\P K(\r))^{-1}\| \\ =\lim_{h\to\infty}\|\P [M_h,K(\r)]K(\r))^{-1}\P\|=0\qquad\qquad\qquad\qquad \end{eqnarray*} where the last equality follows from the unitarity of $\P$ and (c2). \item[(3')] Let $\l\in \C$ and $$ d^{\prime}_h(\l,\r):=\inf \{\|(\l-T(\r))M_hu\|:u\in D(T(\r)), \|M_hu\|=1\} $$ Then: \begin{eqnarray*} \|[\l-T(\r)]M_hu\|=\|[\l(1-\P)+\P(\l-K(\r))]M_hu\|\geq \\ \|[\l-K(\r)]M_hu\|-|\l|\|(1-\P)M_hu\|\geq \|[\l-K(\r)]M_hu\|-|\l| \end{eqnarray*} Hence $d^{\prime}_h(\l,\r)\geq d_h(\l,\r)-|\l|$ and by (3) $\ds \lim_{h\to\infty}d^{\prime}_h(\l,\r)=+\infty$. The assertion is now a direct application of \cite{HV}, Theorem 5.4. This concludes the proof of Assertions 1 and 2 of Theorem 1.3. \end{itemize} \vskip 0.3cm\noindent Let us now turn to the proof of Assertion 3, i.e. the Borel summability of the eigenvalues of the operator family $Q(g,\t):=Q(\b,\t)$ for $\b=ig$, $-\pi/4<{\rm arg}g <\pi/4$, $|g|$ suitably small (depending on the unperturbed eigenvalue). To this end, we adapt to the present situation the proof \cite{Na} valid for the \op\ family $H(g,\t):=H(\b,\t), \b=ig$, in turn based on the general argument of \cite{HP}. First remark that if $(\b,\t)$ generates the parallelogram ${\cal R}$ defined in (\ref{par}) then $(g,\t)$ generates the parallelogram \be \label{par1} \widehat{\cal R}=\{(s,t)\in\R^2:-\pi/2<(2K-1)t+s<\pi/2, -\pi/2<(2K+3)t+s<\pi/2\}, \; \ee where now $s={\rm arg}\,g= {\rm arg}\,\b-\pi/2$. From now on, with abuse of notation, we write $(g,\t)\in \widehat{\cal R}$ whenever $(s,t)\in{\cal R}$. Let $\l$ be an eigenvalue of $H_0(\t):=H(0,\t)$ of multiplicity $m(\l):=m$. Denote $P(0,\t)$ the corresponding projection. By the above stability result, this means that if $\Gamma$ is a circumference of radius $\epsilon$ centered at $\l$ there is $C>0$ independent of $(g,\t)\in \widehat{\cal R}$ such that, denoting $R_Q(z,g,\t):=[Q(g,\t)-z]^{-1}$ the resolvent of $Q(g,\t)$: $$ \sup_{z\in\Gamma_0}\|[Q(g,\t)-z]^{-1}\|\leq C, \quad |g|\to 0 $$ and that ${\rm dim}\,\widehat{P}(g,\t)={\rm dim}\,\widehat{P}$ as $|g|\to 0$, $(g,\t)\in \widehat{\cal R}$, ${\rm arg}\,g$ fixed. This time: \be \label{proiettore} \widehat{P}(g,\t):=\frac{1}{2\pi i}\int_{\Gamma}R_Q(z,g,\t)\,dz, \quad \widehat{P}\equiv \widehat{P}(0,\t):= \frac{1}{2\pi i}\int_{\Gamma}R_Q(z,0,\t)\,dz \ee are the projections on the parts of $\sigma(Q(g,\t))$, $\sigma({\P} H(0,\t))$ enclosed in $\Gamma$. We recall that $\sigma(Q(g,\t))$ is independent of $\t$ for all $(g,\t)$ in the stated analyticity region, and that $\widehat{P}(0,\t)=P(0,\t)$. It follows that $Q(g,\t)$ has exactly $m$ eigenvalues (counting multiplicities) in $\Gamma$, denoted once again $\mu_1(g),\ldots,\mu_m(g)$. We explicitly note that, unlike the $m=1$ case, when the unperturbed eigenvalue is degenerate, the analyticity of the \op\ family does not a priori entail the same property of the eigenvalues $\mu_1(g),\ldots,\mu_m(g)$, so that the analysis of \cite{Na},\cite{HP} is necessary. Following [\cite{HP}, Sect.5] set: $$ {\cal M}(g,\t):=Ran(\widehat{P}_Q(g,\t)); \qquad \widehat{D}(g,\t) :=\widehat{P}(0,\t)\widehat{P}(g,\t)\widehat{P}(0,\t) $$ Under the present conditions $\widehat{D}(g,\t)$ is invertible on ${\cal M}(0):=Ran(\widehat{P}(0,\t))$. Hence the present problem can be reduced to a finite-dimensional one in ${\cal M}(0,\t)$ by setting \begin{eqnarray*} E(g,\t)&:=&\widehat{D}(g,\t)^{-1/2}N(g,\t)\widehat{D}(g,\t)^{-1/2}; \\ N(g,\t)&:=&\widehat{P}(0,\t)\widehat{P}(g,\t)[Q(g,\t)-\l]\widehat{P}(g,\t) \widehat{P}(0 ,\t) \end{eqnarray*} \par\noindent As in [\cite{HP}, Thms 4.1, 4.2] the \RS\ series for each eigenvalue $\mu_s(g): s=1,\ldots,m$ near $\l$ is Borel summable upon verification of the two following assertions: there exist $\eta(\delta)>0$ and a sequence of linear \op s $\{E_i(0,\t)\}$ in ${\cal M}(0,\t)$ such that \begin{itemize} \item[(i)] $E(g,\t)$ is an \op -valued analytic function for $(g,\t)\in\widehat{\cal R}$; As we know, this entails that $E(g)$ is is an \op -valued analytic function in the sector $$ {\cal S}_{K,\delta}:=\{g\in\C: 0<|g|<\eta(\delta); -(2K-1)\frac{\pi}{2}+\delta<{\rm arg}\,(g)<(2K+3)\frac{\pi}{2}-\delta\} $$ \item[(ii)] $E(g,\t)$ fulfills a strong asymptotic condition in $\widehat{\cal R}$ (and thus, in particular, for $g\in{\cal S}_{K,\delta}$) and admits $\ds \sum_{i=0}^\infty E_i(0,\t)g^i$ as asymptotic series; namely, there exist $A(\delta)>0$, $C(\delta)>0$ such that \be \label{Sac} ||R_N(g)\|:=\|E(g,\t)-\sum_{i=0}^{N-1} E_i(0,\t)g^i\|\leq AC^N\Gamma((2K-1)N/2)|g|^N \ee as $|g|\to 0$, $(g,\t)\in \widehat{\cal R}$, $g\in {\cal S}_{K,\delta}$; \item[(iii)] $\qquad E_i(0,\t)=E^\ast_i(0,\t)$, $\quad i=0,1,\ldots$, $\quad\t\in\R$. \end{itemize} Given the stability result (Assertion 2 of the present Theorem 1.3) the proof of (i) and (iii) is identical to that of \cite{Na}, Lemma 2.5 (i) and is therefore omitted. We prove assertion (ii). Under the present conditions the Rayleigh-\Sc\ perturbation expansion is generated by inserting in (\ref{proiettore}) the (formal) expansion of the resolvent $R_Q(z,g,\t):=[Q(g,\t)-z]^{-1}$: \be \label{Neumann} R_Q(z,g,\t)=R_Q(z,g,\t)\sum_{p=0}^{N-1}[igWR_\P(z,0,\t)]^p+R_Q(z)[igWR_\P(z ,0,\t)]^N \ee and performing the contour integration. Moreover (see once more \cite{HP}, Section 5.7), to prove (\ref{Sac}) it is enough to prove the analogous bound on $\widehat{D}(g,\t)$ and $N(g,\t)$. Since $\widehat{D}(g)= \widehat{{P}}(0,\t)\widehat{{P}}_Q(g,\t)\widehat{P}(0,t)$, we have, inserting (\ref{Neumann}) \begin{eqnarray*} D_N(g,\t)&:=&D(g,\t)-\sum_{i=0}^{N-1} D_i(0,\t)g^i \\ &=&\widehat{P}(0,\t)\frac{1}{2\pi i} \int_{\Gamma_0}R_Q(z,g,\t)[W(x)R_\P(z,0,\t)]^N\widehat{{P}}(0,\t) \end{eqnarray*} By the analyticity and uniform boundedness of the resolvent $R_Q(z,g,\t)$ in $\widehat{\cal R}$ (and hence in particular for $g\in\S_{K,\delta}$), it is enough to prove the estimate \be \label{stima1} \sup_{z\in\Gamma_j}\|[igWR_\P(z,0,\t)]^N \widehat{P}_0\|\leq AC^N\Gamma((2K-1)N/2)|g|^N \ee In turn, since $\widehat{P}(0,\t)= P(0,\t)$, by the Combes-Thomas argument (see \cite{HP}, Sect. 5 for details) to prove (\ref{stima1}) it is enough to to find a function $f:\R^d\to\R$ such that \be \label{stima2} \|e^fP(0,\t)\|<+\infty;\qquad \sup_{x\in\R^d}|W(x)e^{-f/N}|\leq N^{\frac{2K-1}{2}} \ee Now a basis in $Ran(P_j)$ is given by $m$ functions of the type $$ {\cal Q}(e^{\t/2}x_1,\ldots,e^{\t/2}x_d)e^{-e^{\t/2}|x|^2} $$ where ${\cal Q}$ is a polynomial of degree at most $m$. Therefore both estimates are fulfilled by choosing $\ds f=\alpha |x|^2$ with $\alpha=\alpha(\t) <1/2$. This condition is always satsfied if $(g,\t)\in \widehat{\cal R}$ because $|{\rm Im}\,\t|<\pi/4$. This concludes the proof of the Theorem. \vskip 0.3cm\noindent {\bf Remark} \par\noindent The summability statement just proved, called Borel summability for the sake of simplicity, is more precisely the Borel-Leroy summability of order $q:=(K-1)/2$. \section{Conclusion} Even though the object of main physical interest are the eigenvalues of $H(g)$ rather than its singular values $\mu_k(g)$ determined in this paper, the singular values yield a property that the eigenvalues cannot in general yield since the operator $H(g)$ is not normal: namely, a diagonal form. If an operator is physically interesting a diagonalization of it is clearly useful. To examine this point in more detail, consider once again the canonical expansion (\ref{canonical}) of Corollary 1.2: $$ \label{canonexp} H(g)u=\sum_{k=0}^{\infty}\mu_k(g)\langle u,\psi_k\rangle\P\psi_k, \quad u\in D(H(g)) $$ Since both vector sequences $\{\psi_k\}$ and $\{\P\psi_k\}$ are orthonormal we have \be \label{dd} \langle \P\psi_k,H(g)\psi_l\rangle=\mu_k(g)\delta_{k,l} \ee Moreover the orthonormal sequences $\{\psi_k\}$ and $\{\P\psi_k\}$ are complete in the Hilbert space. Hence formula (\ref{dd}) is an actual diagonalization of $H(g)$. The basis $\{\psi_k\}$ acts in the domain, and the basis $\{\P\psi_k\}$ in the range. A complete diagonalization of the $\PT$-symmetric but non-normal operator $H(g)$ has been therefore obtained: the singular values $\mu_k(g)$ and the eigenvectors $\psi_k$ (and thus also the vectors $\P\psi_k$) are indeed uniquely defined by perturbation theory through the Borel summability. More precisely, the general formula (\ref{cc1}) $$ Hu=\sum_{k=0}^{\infty}\mu_k\langle u,\psi_k\rangle\psi^\prime_k, \quad u\in D(H) $$ which provides a diagonalization for an operator $H$ with compact resolvent with respect to the pair of orthonormal bases $\{\psi_k\}$ and $\{\psi^\prime_k\}$, requires a priori the computation of $\mu_k$ and $\psi_k$ as solutions of the spectral problem \begin{equation} \label{ccc} H^\ast(g) H(g)\psi=\mu^2\psi \end{equation} which represents an eigenvalue problem more complicated than $H(g)\phi=\lambda\phi$. 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