Content-Type: multipart/mixed; boundary="-------------0407220927195" This is a multi-part message in MIME format. ---------------0407220927195 Content-Type: text/plain; name="04-224.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-224.keywords" Kolmogorov theorem, quantization, quantization formula ---------------0407220927195 Content-Type: application/x-tex; name="CGSsub.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="CGSsub.tex" %\documentclass[11pt]{article} \documentclass[12pt]{article} %\documentclass[a4paper,11pt,reqno]{amsart} %\documentclass[a4paper,draft,reqno]{amsart} %\input{amssym.def} %\input{amssym} \setlength{\textwidth}{15.0cm} \setlength{\textheight}{23.0cm} \hoffset=-1.0cm \voffset=-1.0cm \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} \renewcommand{\thesection} {\arabic{section}} \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}} \newcommand{\begsection}[1]{\setcounter{equation}{0}\section{#1}} \newcommand{\finsection}{\vskip20pt} \newcommand{\hindsp}{\hspace{2em}} \def\C{{\mathcal C}} \def\L{{\mathcal L}} \def\B{{\mathcal B}} \def\D{{\mathcal D}} \def\H{{\mathcal H}} \def\P{{\mathcal P}} \def\R{{\mathcal R}} \def\N{{\mathcal N}} \def\G{{\mathcal G}} %\def\S{{\mathcal S}} \def\vf{\varphi} \def\ve{\varepsilon} %\def\R{\Bbb R} \def\Z{\Bbb Z} %\def\N{\Bbb N} \def\T{\Bbb T} %\def\C{\Bbb C} \def\Sc{Schr\"o\-din\-ger} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ra{\rangle} \def\ds{\displaystyle} \def\om{\omega} \def\Om{\Omega} \def\ep{\epsilon} \def\l{\lambda} \def\s{\sigma} \overfullrule=5pt % macro di questo lavoro \def\bdd{bounded} \def\ef{eigenfunction} \def\ev{eigenvalue} \def\e{equation} \def\eq{equation} \def\fy{family} \def\fu{function} \def\F{Fourier} \def\hol{holomorphic} \def\hm{homogeneous} \def\indep{independent} \def\lhs{left hand side} \def\neigh{neighborhood} \def\nondeg{non-degenerate} \def\op{operator} \def\og{orthogonal} \def\pb{problem} \def\Pb{Problem} \def\pde{partial differential equation} \def\pe{periodic} \def\per{periodic} \def\pert{perturbation} \def\prop{proposition} \def\Prop{Proposition} \def\pol{polynomial} \def\pop{pseudodifferential operator} \def\pseudor{pseudodifferential operator} \def\res{resonance} \def\rhs{right hand side} \def\sa{selfadjoint} %\def\sc{semiclassical} \def\Pb{Problem} \def\pde{partial differential equation} \def\pe{periodic} \def\per{periodic} \def\schr{Schr{\"o}dinger operator} \def\sop{Schr{\"o}dinger operator} \def\st{strictly} \def\stpsh{\st{} plurisubharmonic} \def\strans{^\sigma \hskip -2pt} \def\suf{sufficient} \def\sufly{sufficiently} \def\tf{transformation} \def\Th{Theorem} \def\th{theorem} \def\tf{transform} \def\trans{^t\hskip -2pt} \def\top{Toeplitz operator} \def\uf{uniform} \def\ufly{uniformly} \def\vf{vector field} \def\wrt{with respect to} \def\Op{{\rm Op\,}} \def\st{strictly} \def\stpsh{\st{} plurisubharmonic} \def\strans{^\sigma \hskip -2pt} \def\suf{sufficient} \def\sufly{sufficiently} \def\tf{transformation} \def\Th{Theorem} \def\th{theorem} \def\tf{transform} \def\trans{^t\hskip -2pt} \def\top{Toeplitz operator} \def\uf{uniform} \def\ufly{uniformly} \def\vf{vector field} \def\wrt{with respect to} \def\Op{{\rm Op\,}} \def\Re{{\rm Re\,}} \def\Im{{\rm Im\,}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% %%%%%%%% begin %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip=21pt \begin{center} {\large\bf SPECTRA OF $PT-$SYMMETRIC OPERATORS AND PERTURBATION THEORY } \end{center} \vskip 13pt \begin{center} Emanuela Caliceti\footnote[1]{ Dipartimento di Matematica, Universit\`{a} di Bologna, I- 40127 Bologna (Italy) (caliceti@dm.unibo.it)}, Sandro Graffi,\footnote[2]{ Dipartimento di Matematica, Universit\`{a} di Bologna, I- 40127 Bologna (Italy) (graffi@dm.unibo.it)} Johannes Sj\"ostrand\footnote[3]{Centre de Math\'ematiques, \'Ecole Polytechnique, F-91190 Palaiseau Cedex (France) (johannes@math.polytechnique.fr)}, \end{center} \begin{abstract} \noindent Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric Schr\"odinger operators are discussed. \end{abstract} \vskip 1cm % %\date{\today} %\subjclass{???} \keywords{????} % %\maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% %% Approximate solutions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% \begsection{Introduction and statement of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% \setcounter{lemma}{0}% \setcounter{corollary}{0}% \setcounter{definition}{0}% The Schr\"odinger operators invariant under the combined application of a reflection symmetry operator $P$ and of the (antilinear) complex conjugation operation $T$ are called PT-symmetric. A standard class of such operators has the form $H=H_0+iW$ where: \begin{enumerate} \item $H_0$ is a self-adjoint realization of $-\Delta+V$ on some Hilbert space $L^2(\Omega); \Omega\subset\R^n$, $n\geq 1$; $V$ and $W$ are real multiplication operators. \item $V$ are even and odd with respect to $P$, respectively: $PV=V$, $PW=-W$. $P$ is the parity operation $$ (P \psi)(x)=\psi((-1)^{j_1}x_1,\ldots,(-1)^{j_n}x_n), \qquad\psi\in L^2 $$ where $j_i=0,1$; $j_i=1$ for at least one $1\leq i\leq n$; \end{enumerate} If $T$ is the involution defined by complex conjugation: $\ds (T\psi)(x)=\overline{\psi}(x)$, one immediately checks that $(PT)H=H(PT)$. \par\noindent $PT-$symmetric quantum mechanics (see e.g. \cite{Ah},\cite{Be1},\cite{Be2},\cite{Be3}, \cite{Cn1},\cite{Cn2},\cite{Cn3}, \cite{Zn1},\cite{Zn2}) requires the reality of the spectrum of $PT-$symmetric operators, recently proved, for instance, for the one dimensional odd anharmonic oscillators \cite{Tateo}, \cite{Shin}. Imposing boundary conditions along complex directions, ho\-we\-ver, examples of $PT-$ sym\-me\-tric operators with complex eigenvalues have been constructed \cite{DD}. It is therefore an important issue in this context to determine whether or not the spectrum of $PT$-symmetric \Sc\ operators with standard $L^2$ boundary conditions at infinity is real. We deal with this problem only in perturbation theory, but we will obtain criteria both for existence of complex eigenvalues (Theorem 1.1) and for the reality of the spectrum (Theorem 1.2), in even greater generality than the $PT$ symmetry. Let ${\cal H}$ be a Hilbert space with scalar product denoted $(x|y)$, and $H_0:{\cal H}\to {\cal H}$ be a closed operator with dense domain ${\cal D}\subset{\cal H}$. Let $H_1$ be an operator in ${\cal H}$ with ${\cal D}(H_1 )\supset {\cal D}$. This entails that $H_1$ is bounded relative to $H_0$, i.e. there exist $b>0$, $a>0$ such that $\|H_1\psi\|\leq b\|H_0\psi\|+a \|\psi\|$ $\forall\,\psi\in {\cal D}$. We can therefore define on ${\cal D}$ the operator family $H_\ep:=H_\ep=H_0+\ep H_1$, $\forall\ep\in\C$. \par We assume the following symmetry properties: there exists a unitary involution $J :{\cal H}\to {\cal H}$ mapping ${\cal D}$ to ${\cal D}$, such that \be \label{H1} { JH_0=H_0^\ast J,\quad JH_1=H_1^\ast J } \ee In other words, $J$ intertwines $H_0$ and $H_1$ with the corresponding adjoint operators. Note that: \begin{enumerate} \item The properties $J^2=1$ (involution) and $J^\ast=J^{-1}$ (unitarity) entail $J^\ast=J$, i.e. self-adjointness of $J$; \item The properties (\ref{H1}) entail, if $\ep\in\R$, $JH_\ep=H_\ep^\ast J$; therefore the spectrum $\sigma(H_\ep)$ of $H_\ep$ is symmetric with respect to the real axis if $\ep\in\R$. \item An example of $J$ is the parity operator $P$. \end{enumerate} Let $H_0$ admit a real isolated eigenvalue $\l_0$ of multiplicity $2$ (both algebraic and geometric, i.e. we assume absence of Jordan blocks). Let $e_1,e_2$ be linearly independent eigenvectors, and $E_{\l_0}$ the eigenspace spanned by $ e_1,e_2$. Clearly $JE_{\l_0}:=E_{\l_0}^\ast$ is the eigenspace of $H_0^\ast$ corresponding to the eigenvalue $\overline{\l}_0=\l_0$, and hence the bilinear form $(u^\ast|v), u^\ast\in E_{\l_0}^\ast, v\in E_{\l_0}$ is non degenerate. Therefore we can choose $e_1,e_2$ in $E_{\l_0}$ in such a way that, writing $u=u_1e_1+u_2e_2$, the quadratic form $Q(u,u)=(Ju|u)$ on $E_{\l_0}$ assumes the canonical form \be \label{can} Q(u,u)=\tau_1u_1^2+\tau_2u_2^2, \quad \tau_1=\pm 1, \tau_2=\pm 1 \ee Notice that if $e_1^\ast, e_2^\ast$ is the dual basis, then (\ref{can}) means that $Je_j=\tau_je^\ast_j$. Under these circumstances we want to prove the following \begin{theorem} \label{th1} With the above assumptions and notations, consider the operator family $H_\ep$ for $\ep\in\R$. Denote: \be \label{elm} H_{11}=(H_1e_1|e_1), \quad H_{22}=(H_1e_2|e_2), \quad H_{12}=(H_1e_1|e_2) \ee Then $(e_1|H_1e_1)\in\R$, $(e_2|H_1e_2)\in\R$ and there exists $\ep^\ast >0$ such that, for $|\ep|<\ep^\ast$: \begin{itemize} \item[(i)]If $\tau_1\cdot\tau_2=-1$, and \be \label{cond} 4|H_{12}|^2>(H_{11}-H_{22})^2 \ee $H_\ep$ has a pair of non real, complex conjugate eigenvalues near $\l_0$; \item[(ii)] If $\tau_1\cdot\tau_2=1$ $H_\ep$ has a pair of real eigenvalues near $\l_0$. \end{itemize} \end{theorem} {\bf Remarks} \begin{enumerate} \item The above theorem applies to the $PT$-symmetric operator family $H_\ep=H_0+i\ep W$, where $H_0$ and $iW=H_1$ are as above. Here $J=P$, and hence $PH_0=H_0P$, $P(i\ep W)=-(i\ep W)P=(i\ep W)^\ast P$ so that $JH_\ep=H_\ep^\ast J$. In that case Assumption (\ref{cond}) follows from the weaker assumption $H_{12}\neq 0$ because the $P-$symmetry of $H_0$ and the $P-$antisymmetry of $W$ entail $H_{11}=H_{22}=0$. Indeed, we have $Pe_j=\tau_je_j$ and $$ H_{jj}=(iWe_j|e_j)=(iPWe_j,Pe_j)=(-iWPe_j|Pe_j)=-(iWe_j|e_j)=-H_{jj} $$ \item The physical relevance of Theorem 1.1 is best illustrated by an elementary e\-xam\-ple. Let ${\cal H}=L^2(\R^2)$ and $H_0:{\cal H}\to {\cal H}$ be the (self-adjoint) two dimensional harmonic oscillator with frequencies $\om_1, \om_2$: $$ H_0u=-\frac12\Delta u+\frac12(\om_1^2x_1^2+\om_2^2x_2^2)u $$ We have $\ds \sigma(H_0)=\{E_{k_1,k_2}\}:=\{k_1\om_1+k_2\om_2+\frac{\om_1}{2}+\frac{\om_2}{2}\}, k_i=0,1,2\ldots, i=1,2$. Let again $H_\epsilon =H_0+i\epsilon W$, $\epsilon \in\R$, with $$ W(x)=\frac{x_1^2x_2}{1+x_1^2+x_2^2} $$ Then $W$ is bounded relative to $H_0$, and $PW=-W$ if $Pu(x_1,x_2)=u(x_1,-x_2)$ or $Pu(x_1,x_2)=u(-x_1,-x_2)$. Set $\om_1=1, \om_2=2$, $k_1=2, k_2=0$; i.e., we consider the eigenvalue $E_{2,0}=E_{0,1}$. Then for $|\ep|>0$ small enough $H_\ep$ has a pair of complex conjugate eigenvalues near $E_{2,0}$. To see this, remark that $E_{2,0}=E_2(\om_1)+E_0(\om_2)=E_0(\om_1)+E_1(\om_2)$, where $\l_i(\om_i)=(k+1/2)\om_i$ are the eigenvalues of the one-dimensional harmonic oscillators with frequencies $\om_i, i=1,2$. $E_{2,0}$ has multiplicity $2$. A basis of eigenfunctions is given by $$ \psi_1(x_1,x_2)=e_2(x_1)f_0(x_2); \qquad \psi_2(x_1,x_2)=e_0(x_1)f_1(x_2) $$ Here $e_0$, $e_2$ are the eigefunctions corresponding to $E_0(1)$ and $E_2(1)$, respectively; $f_0$, $f_1$ are the eigenfunctions corresponding to $E_0(2)$ and $E_1(2)$, respectively; note that $e_0$, $e_2$ and $f_0$ are even while $f_1$ is odd. To first order perturbation theory, the two eigenvalues $\Lambda_{j}(\ep): j=1,2$ of $H_\ep$ near $E_{2,0}$ are given by $$ \Lambda_{j}(\ep)=E_{2,0}+i\ep \lambda_{j} $$ where $\lambda_{j}: j=1,2$ are the eigenvalues of the $2\times 2$ matrix $$ W_{l,k}=\left(\begin{array}{ll} (W\psi_1|\psi_1) & (W\psi_1|\psi_2) \\ (W\psi_2|\psi_1) & (W\psi_2|\psi_2) \end{array}\right) $$ Now $\psi_1$ is even, $\psi_2$ is odd, and $W$ is odd. Therefore $\tau_1\cdot \tau_2=-1$. Moreover: $(W\psi_1|\psi_1) = (W\psi_2|\psi_2)=0$, $(W\psi_2|\psi_1) = (W\psi_1|\psi_2) :=w>0$ Therefore $\lambda_{j}=\pm w$ and $\Lambda_{j}(\ep)=E_{2,0}\pm i\ep w$. Hence the conditions of Theorem 1.1 (i) are satisfied and for $\ep$ small enough $H_\ep$ has a pair complex conjugate eigenvalues near $E_{2,0}$. \item By essentially the same proof, the result of Theorem \ref{th1} remains true under the following more general conditions: under the above assumptions on $H_0$ and $H_1$ let $H_0$ admit two real, simple eigenvalues $E_1,E_2$. Let $d:=E_2-E_1$ be their relative distance; $D:={\rm dist}[(\sigma(H_0)\setminus\{E_2,E_1\}),\{E_2,E_1\}]$ their distance from the rest of the spectrum; $e_1, e_2$ the corresponding eigenvectors, all other notation being the same. Then if $d/D$ is small enough the same conclusion of Theorem \ref{th1} holds provided $\ds |\ep H_{12}|>\frac{d}{2D}$. \item Example: {\it Odd perturbations of quantum mechanical double wells: existence of complex eigenvalues.} \par\noindent Let ${\cal H}=L^2(\R)$, $\ds H_0(\hbar)=-\hbar^2\frac{d^2}{dx^2}+x^2(1+x)^2$, $\ds D(H_0)=H^2(\R)\cap L^2_4(\R)$, $W(x)\in L^\infty_{loc}(\R)$, $|W(x)|\leq Ax^4$, $|x|\to\infty$, $W(1-x)=-W(x)$. Here $L^2_4(\R)=\{u\in L^2(\R)\,|\,x^4u\in L^2(\R)\}$. In this case it is known that $W$ is bounded relative to $H_0$; moreover $\ds d={\cal O}(e^{-1/c\hbar})$, $D={\cal O}(\hbar)$, $w={\cal O}(1)$ if $E_1,E_2$ are the two lowest eigenvalues, $\psi_1,\psi_2$ the corresponding eigenvectors and $w$ is defined as in Point 2 above. Hence the conditions of Theorem 1 are fulfilled in the semiclassical regime provided $W$ is continuous at zero with $W(0)\neq 0$ and that $|(e_1|We_2)|\geq 1/C$ and thus there exist $A>0, B>0, C>0$ such that $H_\ep(\hbar):=H_0+i\ep W$ will have at least a pair of complex conjugate eigenvalues for $\ds Ae^{-B/\hbar^2}<\ep w<< C\hbar$. Equivalently, we may consider the double well family $\ds H_0(g)=-\frac{d^2}{dx^2}+x^2(1+gx)^2$ defined on the same domain. Here $\ds d={\cal O}(e^{-1/g^2})$, $D={\cal O}(1)$, $w={\cal O}(1)$. The same argument holds for the general case $H_0=-\hbar^2\Delta +V(x)$, where $V:\R^n\to\R$ is smooth, has two equal quadratic minima and diverges positively as $|x|\to\infty$; $W(x)\in L^\infty_{loc}(\R^n)$, $|W(x)|\leq A V(x)$ as $|x|\to\infty$ because the estimate for $d$ is the same as above\cite{HS}. \end{enumerate} The second result concerns the opposite situation, a criterion ensuring the reality of the spectrum. In this case the natural assumption is the simplicity of the spectrum of $H_0$ in addition to its reality. Therefore for the sake of simplicity we assume $H_0$ self-adjoint. \begin{theorem} \label{th2} Let the self-adjoint operator $H_0$ be bounded below (without loss of generality, positive), and let $H_1$ be continuous. Let $H_0$ have discrete spectrum, $\sigma(H_0)=\{0\leq \l_0 <\l_1 \ldots < \l_l< \ldots \}$, with the property \be \label{dist} \delta:=\inf_{j\geq 0}\;[\l_{j+1}-\l_j]/2 >0. \ee Assume that all eigenvalues are simple. Then $\sigma(H(\ep))\in\R$ if $\ep\in\R$, $\ds |\ep|<\frac{\delta}{\|H_1\|}$. \end{theorem} \vskip 0.2cm\noindent {\bf Example} \par\noindent Here again ${\cal H}=L^2(\R)$; $\ds H_0=-\frac{d^2}{dx^2}+V(x)$, $\ds D(H_0)=H^2(\R)\cap D(V)$. $V(x)=kx^{2m}$, $k>0$, $m\geq 1$; $W(x)\in L^\infty(\R)$, $W(-x)=-W(x)$. We have: $\sigma(H_0)=\{\l_l\}, n=0,1,\ldots$; $$ \l_n \sim k^{\frac{1}{2m}}n^{\frac{2m}{m+1}}, \quad n\to\infty $$ Each eigenvalue $\l_n$ is simple. Clearly $\ds \delta\geq 1$. Denote now $H_\ep:=H_0+i\ep W$ the operator family in $L^2(\R)$ defined by $\ds H_\ep= H_0+H_1$, $H_1=i\ep W$, $D(H_\ep)=D(H_0)$. Then $H_\ep$ has real discrete spectrum for $|\ep|<\|W\|_\infty^{-1}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 1.5cm\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% \setcounter{lemma}{0}% \setcounter{corollary}{0}% \setcounter{definition}{0}% {\bf Proof of Theorem 1.1} \par\noindent The proof is based on perturbation theory and consists in two steps. In the first one we show that the $2\times 2$ matrix generated by restricting the perturbation $H_1$ to $E_{\l_0}$ is antihermitian in case (i) of Theorem 1.1 and Hermitian in case (ii). In the second step we show by the method of the Grushin reduction (see, e.g.\cite{HS1}) that for $\ep$ suitably small the control of the above $2\times 2$ matrix is enough to establish the result. Let $\{e_1,e_1\}$ be once more a basis in $E_{\l_0}$ such that (1.2) holds, and denote by $e_1^\ast,e_2^\ast$ the dual basis in the dual subspace $E_{\l_0}^\ast=JE_{\l_0}$. Clearly $Je_j=\tau_j e_j^\ast$, $\tau_j=\pm 1$. We denote $\Pi_0$ the spectral projection from ${\cal H}$ to $E_{\l_0}$. Explicitly: \be \label{Pi} \Pi_0u=(u|e_1^\ast)e_1+(u|e_2^\ast)e_2 \ee Consider now the rank $2$ operator family $\Pi_0 H_\ep \Pi_0$ acting on $E_{\l_0}$. The representing $2\times 2$ matrix is: \be \label{matrice} H(\ep)_{j,k}=\l_0I+\ep H^1_{j,k}, \quad H^1_{j,k}=(H_1e_k|e^\ast_j),\;j,k=1,2 \ee Now $JH_0=H_0^\ast J$, $J\Pi_0=\Pi_0^\ast J$. We also have $JH_1=H_1^\ast J$. Therefore: $$ (JH_1e_k|e_j)=(H_1e_k|Je_j)=\tau_j(H_1e_k|e^\ast_j)=\tau_jH^1_{j,k} $$ and in the same way $$ (JH_1e_k|e_j)=(H_1^\ast Je_k|e_j)=(Je_k|H_1e_j)=\tau_k(e_k^\ast|H^1e_j)=\tau_k v\overline{(H_1e_j|e^\ast_k)}=\tau_k\overline{H^1_{k,j}} $$ Summing up: $$ \tau_jH^1_{j,k}=\tau_k\overline{H^1_{k,j}} $$ Therefore, if $\tau_1\tau_2=1$ the matrix $H(\ep)_{j,k}$ is hermitian for $\ep\in\R$ and its eigenvalues are real; if instead $\tau_1\tau_2=-1$ the matrix $H(\ep)_{j,k}$ has a real diagonal part and an antihermitian off diagonal part for $\ep\in\R$ and its eigenvalues are complex conjugate. This completes the first step. \newline We want now to construct an approximate inverse of $H_\ep-z$ near $\l_0$ by solving a Grushin problem. In this context it is equivalent to the Feshbach reduction, and provides a convenient formalism for it. To this end, define the operators $R_+, R_-$, ${\cal P}_0(z)$ in the following way: \begin{eqnarray} \label{R} R_+:{\cal H}\to{\C}^2, \;R_+u(j)=(u\vert e_j^\ast),\; j=1,2; \\ R_-:{\C}^2\to{\cal H}, \; R_-u_-=\sum_{j=1}^2u_-(j)e_j,\qquad \\ \label{2.4} %\ekv{7} {{\cal P}_0(z)=\pmatrix{ H_0-z &R_-\cr R_+ &0}:{\cal D}\times {\C}^2\to {\cal H}\times {\C}^2.} \end{eqnarray} Note that we have identified $E_{\l_0}$ with its representative $\C^2$, and that $R_+R_-=I$, the $2\times 2$ identity matrix. \par\noindent The associated Grushin system is \be \label{Gr} \left\{\begin{array}{l} (H_0-z)u+R_-u_-=f \\ R_+u =f_+ \end{array}\right. \ee where $u\in{\cal D},f\in{\cal H}$, $u_-,f_+\in\C^2$. $z\in\C$ belongs to a neighborhood of $\l_0$ at a positive distance from $\sigma(H_0)\setminus\{\l_0\}$. After determining $u_-$ in such a way that $f-R_-u_-\in (1-\Pi_0){\cal H}$ the first equation can be solved for $u(z)\in (1-\Pi_0){\cal H}$ and hence the problem is reduced to the the rank $2$ equation $R_+u(z)=f$. To solve explicitly, remark that, for every $z$ in the complex complement of $\sigma (H_0)\setminus\{ \l_0\}$, ${\cal P}_0(z)$ has the \bdd{} inverse, \be \label{2.5} %\ekv{8} { {\cal E}_0(z)=\pmatrix{E^0(z)&E_+^0(z)\cr E_-^0(z) &E_{-+}^0(z)}, } \ee with \begin{eqnarray} \label{2.6} %\eekv{9} {E^0(z)=(H_0-z)^{-1}(1-\Pi ),\quad E_+^0(z)=R_-,} \\ \nonumber {E_-^0(z)=R_+,\quad E_{-+}^0(z)=(z-\l_0)I.} \end{eqnarray} where $I$ is the $2\times 2$ identity matrix. The spectral problem within $E_{\l_0}$ is thus reduced to the inversion of $E_{-+}^0(z)$, and obviously its solution is represented by $\l_0, e_0, e_1$. \par Now restrict the attention to the set of complex $z$ with ${\rm dist\,}(z,\{ \l_0\} )<1/2R$, where \be \label{normaR} R:=\|E^0(\l_0)\|=\|(1-\Pi_0)(H_0-\l_0)^{-1}\| \ee so that by the geometrical series expansion \be \label{normaE} \|E^0(z)\|\leq \frac{R}{1-|z-\l_0|R} \ee Consider the operator from ${\cal D}\times \C^2$ to ${\cal H}$ defined as \be \label{2.8} %\ekv{10} { {\cal P}_\epsilon (z)=\pmatrix{H_\epsilon -z &R_- \cr R_+ &0}. } \ee associated to the Grushin system \be \label{Gr!} \left\{\begin{array}{l} (H_\ep-z)u+R_- u_-=f \\ R_+u=f_+ \end{array}\right. . \ee Then \be \label{2.9} %\ekv{11} {{\cal P}_\epsilon (z){\cal E}_0(z)=1+\pmatrix{i\epsilon H_1E^0(z) &i\epsilon H_1E_+^0(z)\cr 0 &0}=:1+{\cal K}.} \ee It is routine to check that ${\cal P}_\epsilon (z)$ has the inverse \be \label{2.10} %\ekv{12} {{\cal E}_\epsilon (z)=\pmatrix{E^\epsilon (z) &E_+^\epsilon (z)\cr E_-^\epsilon (z) &E_{-+}^\epsilon (z)},} \ee with \begin{eqnarray} \label{2.10bis} %(13) E^\ep(z)&=&\sum_{n=0}^\infty ({\epsilon \over i})^nE^0(H_1E^0)^n, \\ E^\ep_+(z)&=&\sum_{n=0}^\infty ({\epsilon \over i})^n(E^0H_1)^nE_+^0 \\ \label{2.10ter} E^\ep_-(z)&=&\sum_{n=0}^\infty ({\epsilon \over i})^nE_-^0(H_1E^0)^n, \\ \label{2.10quater} E^\ep_{-+}(z)&=&E_{-+}^0+\sum_{n=1}^\infty ({\epsilon \over i})^nE_-^0(H_1E^0)^{n-1}H_1E_+^0. \end{eqnarray} where all the series will be proved to have a positive convergence radius (convergence means here uniform, or, equivalently, in the norm operator sense). We also recall the well known fact that $z$ is an eigenvalue of $H_\ep$ precisely when ${\rm det}\,E^\ep_{-+}(z)=0$. \par We next derive the appropriate symmetries for the inverse \op{}s \cite{HS1}. From $JH_\ep=H¬_\ep^\ast J$ we get: \begin{eqnarray*} JR_-u_-&=&\sum_{j=1}^2 u_-(j)J e_j = \sum_{j=1}^2 (\tau u_-)(j)e_j^\ast, \quad \tau:=\left(\begin{array}{ll} \tau_1 & 0 \\ 0 & \tau_2\end{array}\right) \\ R_+^\ast u_-&=&\sum_{j=1}^2 u_-(j)e_j^\ast \end{eqnarray*} where the second equation follows from $$ (R_+u|u_-)=\sum_{j=1}^2 \overline{u_-(j)}(u|e_j^\ast) $$ We thus conclude: $$ JR_-u_-=R_+^\ast\tau u_-,\quad R_-^\ast J=\tau R_+ $$ Therefore: \begin{eqnarray*} \left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right) \left(\begin{array}{ll} H_\ep -z & R_- \\ R_+ & 0\end{array}\right)= \left(\begin{array}{ll} J(H_\ep -z) & JR_- \\ \tau R_+ & 0\end{array}\right) \\ = \left(\begin{array}{ll} (H_\ep^\ast -z)J & R_+^\ast\tau \\ R_-^\ast J & 0\end{array}\right)= \left(\begin{array}{ll} (H_\ep^\ast -z) & R_+^\ast \\ R_-^\ast & 0\end{array}\right)\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right) \end{eqnarray*} whence \be \label{17} \left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right){\cal P}_\ep(z)={\cal P}_\ep(\overline{z})^\ast\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right) \ee Since ${\cal E}(z)={\cal P}(z)^{-1}$, taking right and left inverses we get $$ {\cal E}(\overline{z})^\ast\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right)=\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right){\cal E}(z) $$ that is \be \label{18} \left(\begin{array}{ll} E(\overline{z})^\ast & E_-(\overline{z})^\ast \\ E_+(\overline{z})^\ast & E_{-+}(\overline{z})^\ast\end{array}\right)\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right)=\left(\begin{array}{ll} J & 0 \\ 0 & \tau\end{array}\right) \left(\begin{array}{ll} E({z}) & E_+({z}) \\ E_-({z}) & E_{-+}({z})\end{array}\right) \ee In particular: $$ E_{-+}(\overline{z})^\ast\tau=\tau E_{-+}({z}) $$ We can thus conclude that, for $z\in\R$, if $\tau_1\cdot\tau_2=1$ the $2\times 2$ matrix $E_{-+}({z})$ is Hermitian, and antihermitian off the diagonal with real diagonal elements if if $\tau_1\cdot\tau_2=-1$. \newline It remains to be proved the norm convergence of the expansions (\ref{2.10bis},\ref{2.10ter},\ref{2.10quater}). We have, by the relative boundedness condition $\|H_1\psi\|\leq b\|H_0\psi\|+a\|\psi\|$ and (\ref{normaE}): \begin{eqnarray*} \|H^1E^0\|&=&\|H^1(H_0-z)^{-1}(1-\Pi_0)\|\leq \\ &\leq& b\|H_0(H_0-z)^{-1}(1-\Pi_0)\|+a\|(H_0-z)^{-1}(1-\Pi_0)\| \\ &\leq & b\|(H_0-z)(H_0-z)^{-1}(1-\Pi_0)\|+ \\ &+& b|z|\|(H_0-z)^{-1}(1-\Pi_0)\|+a\|(H_0-z)^{-1}(1-\Pi_0)\| \\ &\leq& b\|1-\Pi_0\|+\frac{(b|z|+a)R}{1-|z-\l_0|R}0$ because $|z|(H_{11}-H_{22})^2$ there cannot be real zeros for $\ep$ suitably small. We can thus conclude that ${\rm det}E^\ep_{-+}(z)$ is zero for $z=\Lambda_\pm(\ep)$, $$ \Lambda_\pm(\ep)=\frac12[{H_{11}+H_{22}}\pm i\ep \sqrt {4|H_{12}|^2-(H_{11}-H_{22})^2}]+O(\ep^2) $$ and this concludes the proof of the Theorem. \vskip 1.5cm\noindent %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% {\bf Proof of Theorem \ref{th2}} Let us first recall that under the present assumptions $H_\ep$ is a type-A holomorphic family of operators in the sense of Kato (see \cite{Ka}, Chapter VII.2) with compact resolvents $\forall\,\ep\in\C$. Hence $\sigma(H_\ep)=\{\l_l(\ep)\}: l=0,1,\ldots$. In particular: \begin{itemize} \item[(i)] the \ev s $\l_l(\ep)$ are locally holomorphic functions of $\ep$ with only algebraic singularities; \item[(ii)] the \ev s $\l_l(\ep)$ are stable, namely given any eigenvalue $\l(\ep_0)$ of $H_{\ep_0}$ there is exactly one \ev\ $\l(\ep)$ of $H_\ep$ such that $\ds \lim_{\ep\to\ep_0}\l(\ep)=\l(\ep_0)$; \item[(iii)] the Rayleigh-\Sc\ perturbation expansion for the eigenprojections and the eigenvalues near any eigenvalue $\l_l$ of $H_0$ has convergence radius $\ds \delta_l/\|H_1\|$ where $\delta_l$ is half the isolation distance of $\l_l$. \end{itemize} Remark that since $\delta_l\geq \delta\;\forall\,l$, all the series will be convergent for all $\ep\in \Om_{r_0}$; $\Om_{r_0}:=\{\ep\in\C: |\ep|< r_0\}$, where $r_0:=\delta/\|H_1\|$ is a uniform lower bound for all convergence radii. \par\noindent Assume now without loss of generality, to simplify the notation, $\|H_1\|=1$. By hypothesis $|\l_l-\l_{l+1}|\geq 2\delta>0\,\forall\,l\in\N$. First remark that if $\ep\in\R$, $|\ep|