Content-Type: multipart/mixed; boundary="-------------0804290751915" This is a multi-part message in MIME format. ---------------0804290751915 Content-Type: text/plain; name="08-86.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-86.comments" to appear on Rendiconti Lincei - Matematica e Applicazioni ---------------0804290751915 Content-Type: text/plain; name="08-86.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-86.keywords" perturbation theory, PT symmetry, stability of spectra ---------------0804290751915 Content-Type: application/x-tex; name="CGPerL E. Caliceti.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="CGPerL E. Caliceti.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} %\documentclass[a4paper,12pt,reqno]{amsart} %\documentclass[a4paper,draft,reqno]{amsart} \input{amssym.def} \input{amssym} %\usepackage[notref]{showkeys} \setlength{\textwidth}{15.0cm} %%DB margin change%% \setlength{\textheight}{23.0cm} \hoffset=-1.0cm \voffset=-2.0cm \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \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}} \renewcommand{\theexample}{\thesection.\arabic{example}} \font\liten=cmr10 at 8pt %\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\Q{{\mathcal Q}} \def\PT{{\mathcal {PT}}} \def\T{{\mathcal {T}}} \def\R{{\mathcal R}} \def\Hv{\widehat{\mathcal H}} \def\Qv{\widehat{\mathcal Q}} \def\Wv{\widehat{\mathcal W}} \def\H{{\mathcal H}} %\def\S{{\mathcal S}} \def\vf{\varphi} \def\ve{\varepsilon} \def\R{\Bbb R} \def\Z{\Bbb Z} \def\N{\Bbb N} \def\T{{\mathcal T}} \def\C{\Bbb C} \def\A{{\mathcal A}} \def\res{{\mathcal R}} \def\etab{\overline{\eta}} \def\ha{Ha\-mil\-to\-nian} \def\Sc{Schr\"o\-din\-ger} \def\hp{{\hbar}} \def\hpp{{h^{\prime}}} \def\zbar{\overline{z}} \def\wbar{\overline{w}} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\ra{\rangle} \def\ds{\displaystyle} \def\chit{\tilde{\chi}} \def\om{\omega} \def\Om{\Omega} \def\ep{\epsilon} \def\RSPE{Ray\-leigh-\Sc\ per\-tur\-ba\-tion ex\-pan\-sion} \newcommand{\Nb}{\overline{\N}} \newcommand{\Rb}{\overline{\R}} \newcommand{\re}{{\rm Re}\,} \newcommand{\im}{{\rm Im}\,} %\binoppenalty=10000 % per non spezzare le cose tra dollari %\relpenalty=10000 % per non spezzare le cose tra dollari \overfullrule=5pt \def\l{\ell} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% %%%%%%%% begin %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip=19pt \begin{center} {\large\bf\sc A criterion for the reality of the spectrum of $\PT$ symmetric \Sc\ operators with complex-valued periodic potentials} \end{center} \vskip 13pt \begin{center} Emanuela Caliceti and Sandro Graffi \\ {\small Dipartimento di Matematica, Universit\`a di Bologna, 40127 Bologna, Italy \footnote{caliceti@dm.unibo.it, graffi@dm.unibo.it}} \end{center} \begin{abstract} \noindent Consider in $L^2(\R)$ the \Sc\ operator family $H(g):=-d^2_x+V_g(x)$ depending on the real parameter $g$, where $V_g(x)$ is a complex-valued but $PT$ symmetric periodic potential. An explicit condition on $V$ is obtained which ensures that the spectrum of $H(g)$ is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs. \end{abstract} \vskip 1cm % %\date{\today} %\subjclass{???} \keywords{????} % %\maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Approximate solutions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction and statement of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% There is currently an intense and ever increasing activity on an aspect of quantum theory known as $PT$-symmetric quantum mechanics (see e.g.\cite{Be1}, \cite{Be2}, \cite{Be4}, \cite{BBJ}, \cite{Cn1}, \cite{Cn3}, \cite{Zn1}, \cite{Sp}, \cite{Cn2}). Mathematically speaking, in the simplest, one -dimensional case one deals with the stationary \Sc\ equation \be \label{S} H\psi:=(-\frac{d^2}{dx^2}+V)\psi=E\psi\;. \ee where the potential $V(x)$ can be complex-valued but is invariant under the combined action of the linear parity operation $P$, $P\psi(x)=\psi(-x)$, and of the (anti)linear "time-reversal" symmetry, i.e. the complex-conjugation operation $T\psi(x)=\overline{\psi}(x)$; namely, $\overline{V}(-x)=V(x)$. The basic mathematical problem is to determine under what conditions, if any, on the complex $PT$ symmetric potential $V$ the spectrum of the corresponding \Sc\ operator is purely real. Here we deal with this problem in the context of periodic potentials on $\R$, already considered in \cite{Ah}, \cite{BDM},\cite{Ce}, \cite{CR}, \cite{Jo}, \cite{Shin}. Without loss, the period is assumed to be $2\pi$. If $V$ is periodic and real valued it is well known (see e.g. \cite{BS}) that, under mild regularity assumptions, the spectrum is absolutely continuous on $\R$ and band shaped. It is then natural to ask whether or not there exist classes of $PT$ symmetric, complex periodic potentials generating \Sc\ operators with real band spectrum. This question has been examined in \cite{Ah}, \cite{BDM},\cite{Ce}, \cite{CR}, \cite{Jo}, by a combination of numerical and WKB techniques, in several particular examples. It was later proved in \cite{Shin} that the above arguments cannot exclude the occurrence of complex spectra, and actually a condition has been isolated under which $H$ admits complex spectrum consisting of a disjoint union of analytic arcs \cite{Shin}. \par Our first result is the explicit determination a class of $PT$ symmetric, complex periodic potentials admitting real band spectrum. Denote $\T(u)$ the non-negative quadratic form in $L^2(\R)$ with domain $H^1(\R)$ defined by the kinetic energy: \be \label{T} \T(u):=\int_\R |u^\prime|^2\,dx, \qquad u\in H^1(\R) \ee Let $q$ be a real-valued, tempered distribution. Assume: \begin{itemize} \item[(1)] $q$ is a $2\pi$-periodic, $P$ symmetric distribution belonging to $H^{-1}_{loc}(\R)$; \item[(2)] $W(x):\R\to \C$ be\-longs to $L^\infty(\R)$ and is $PT$-symmetric, $\overline{W(-x)}=-W(x)$, . \item [(3)] $q$ generates a real quadratic form $\Q(u)$ in $L^2(\R)$ with domain $H^1(\R)$; \item [(4)] $\Q(u)$ is relatively bounded with respect to $\T(u)$ with relative bound $b<1$, i.e. there are $b<1$ and $a>0$ such that \be \label{rb} \Q(u)\leq b T(u)+a\|u\|^2 \ee \end{itemize} Under these assumptions the real quadratic form \be \label{Q0} \H_0(u):=\T(u)+\Q(u) \quad u\in H^{1}(\R) \ee is closed and bounded below in $L^2(\R)$. We denote $H(0)$ the corresponding self-adjoint operator. This is the self-adjoint realization of the formal differential expression (note the abuse of notation) $$ H(0)=-\frac{d^2}{dx^2}+q(x) $$ Under these circumstances it is known (see e.g. \cite{AGHKH}) that the spectrum of ${H}(0)$ is continuous and band shaped. For $n=1,2,\ldots$ we denote $$ B_{2n}:=[\alpha_{2n},\beta_{2n}], \quad B_{2n+1}:=[\beta_{2n+1},\alpha_{2n+1}] $$ the bands of ${ H}(0)$, and $\Delta_n:=]\beta_{2n},\beta_{2n+1}[$, $]\alpha_{2n+1},\alpha_{2(n+2)}[$ the gaps between the bands. Here: $$ 0\leq \alpha_0\leq \beta_0\leq \beta_1\leq \alpha_1\leq \alpha_2\leq \beta_2\leq\beta_3\leq\alpha_3\leq\alpha_4\leq\ldots. $$ The maximal multiplication operator by $W$ is continuous in $L^2$, and therefore so is the quadratic form $\la u,W u\ra$. It follows that the quadratic form family \be \label{Q} \H_g(u):=\T(u)+\Q(u)+g \la u,Wu\ra, \quad u\in H^{1}(\R) \ee is closed and sectorial in $L^2(\R)$ for any $g\in\C$. \par\noindent We denote ${H}(g)$ the uniquely associated $m-$ sectorial operator in $L^2(\R)$. This is the realization of the formal differential operator family $$ H(g)=-\frac{d^2}{dx^2}+q(x)+gW(x) $$ By definition, ${ H}(g)$ is a holomorphic family of operators of type B in the sense of Kato for $g\in\C$; by (1) it is also $PT$ symmetric for $g\in\R$. Our first result deals with its spectral properties. \begin{theorem} \label{t1} Let all gaps of $H(0)$ be open, namely: $ \alpha_n < \beta_n < \alpha_{n+1}$ $\forall \,n\in\N$, and let there exist $d>0$ such that \be \label{infgap} \frac12\inf_{n\in\N} \Delta_n:=d>0 \ee Then, if \be \label{rc} |g|<\frac{d^2}{2(1+d)\|W\|_\infty}:=\overline{g} \ee there exist $$ 0 \leq \alpha_0(g)< \beta_0(g)< \beta_1(g)<\alpha_1(g)<\alpha_2(g)<\beta_2(g)\ldots $$ such that \be \sigma({H}(g))=\left( \bigcup_{n\in\N}B_{2n}(g)\right)\bigcup \left( \bigcup_{n\in\N}B_{2n+1}(g)\right) \ee where, as above $$ B_{2n}(g):=[\alpha_{2n}(g),\beta_{2n}(g)], \quad B_{2n+1}(g):=[\beta_{2n+1}(g), \alpha_{2n+1}(g)]. $$ \end{theorem} {\bf Remark} \newline The theorem states that for $|g|$ small enough the spectrum of the non self-adjoint operator $H(g)$ remains real and band-shaped. The proof is critically dependent on the validity of the lower bound (\ref{infgap}). Therefore it cannot apply to smooth potentials $q(x)$, in which case the gaps vanish as $n\to\infty$. Actually we have the following \par\noindent {\bf Example} \newline A locally $H^{-1}(\R)$ distribution $q(x)$ fulfilling the above conditions is $$ q(x)=\sum_{n\in\Z}\delta(x-2\pi n) $$ the periodic $\delta$ function. Here we have: $$ \Q(u)=\sum_{n\in\Z}|u(2\pi n)|^2=\int_\R q(x)|u(x)|^2\,dx,\quad u\in H^1(\R) $$ This example is known as the Kronig-Penney model in the one-electron theory of solids. Let us verify that condition (\ref{rb}) is satisfied. As is known, this follows from the inequality (see e.g. \cite{Ka}, \S VI.4.10): \begin{eqnarray*} |u(2\pi n)|^2\leq \epsilon\int_{2\pi n}^{2\pi (n+1)}|u^\prime (y)|^2\,dy+\delta \int_{2\pi n}^{2\pi (n+1)}|u (y)|^2 \end{eqnarray*} where $\epsilon$ can be chosen arbitrarity small for $\delta$ large enough. In fact, if $u\in H^1(\R)$ this inequality yields: $$ \int_{\R}q(x)|u(x)|^2\,dx=\sum_{n\in\Z}|u(2\pi n)|^2\leq \epsilon\int_\R |u^\prime (y)|^2\,dy+\delta \int_{R}|u (y)|^2 $$ $$ =\ep \T(u)+\delta \|u\|^2, \qquad u\in H^1(\R) $$ which in turn entails the closedness of $\T(u)+\Q(u)$ defined on $H^1$ by the standard Kato criterion. The closedness and sectoriality of $\H_g(u)$ defined on $H^1(\R)$ is an immediate consequence of the continuity of $W$ as a maximal multiplication operator in $L^2$. For the verification of (\ref{infgap}), see e.g.\cite{AGHKH}. Hence any bounded $PT$-symmetric periodic perturbation of the Kronig-Penney potential has real spectrum for $g\in\R$, $|g|<\overline{g}$, where $\overline{g}$ is defined by (\ref{rc}). . \vskip 0.3cm As a second result, we show that an elementary argument of perturbation theory % direct application of the criterion proved in [CGS] to show existence of complex %eigenvalues of a class of $PT$-symmetric operators allows us to sharpen the result of \cite{Shin} about the existence of complex spectra for $PT$-symmetric periodic potentials. \par\noindent Let indeed $W(x)\in L^\infty(\R;\C)$ be a $2\pi$-periodic function. Then the continuity of $W$ as a multiplication operator in $L^2(\R)$ entails that the \Sc\ operator \be \label{K} { K}(g)u:= -\frac{d^2u}{dx^2}+gWu,\quad u\in D( K):=H^2(\R), \quad g\in\R \ee is closed and has non-empty resolvent set. Consider the Fourier expansion of $W$: $$ W(x)=\sum_{n\in\Z}w_ne^{inx},\quad w_n=\frac{1}{2\pi}\int_{-\pi}^\pi\,W(x)e^{-inx}\,dx $$ which converges pointwise almost everywhere in $[-\pi,\pi]$. Then we have \begin{theorem} \label{complesso} Let $W(x)$ be $PT$-symmetric, namely $\overline{W(-x)} =W(x)$. Then \begin{enumerate} \item $\overline{w}_n=w_n$, $\forall\,n\in\Z$; \item Furthermore, let there exist $k\in\N$, $k$ odd, such that $w_kw_{-k}<0$. \end{enumerate} Then there is $\delta >0$ such that for $|g|<\delta$ the spectrum of ${K}(g)$ contains at least a pair of complex conjugate analytic arcs. \end{theorem} {\bf Remarks} \begin{enumerate} \item This theorem sharpens the results of \cite{Shin} in the sense that its assumptions are explicit because they involve only the given potential $W(x)$, while those of Theorem 3 and Corollary 4 of \cite{Shin} involve some conditions on the Floquet discriminant of the equation ${K}(g)\psi=E\psi$. This requires some a priori information on the solutions of the equation itself. \item Explicit examples of potentials fulfilling the above conditions are: $$ W(x)=i\sin^{2k+1}{nx}, \quad k=0,1\ldots; \quad n\;{\rm odd} $$ For $g=1$ these potentials have been considered in \cite{BDM}, where is is claimed that the spectrum is purely real. A more careful examination by \cite{Shin} shows that the appearance of complex spectra cannot be excluded. \end{enumerate} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% \section{Proof of the statements} \setcounter{equation}{0}% \setcounter{theorem}{0}% Let us first state an elementary remark under the form of a lemma. Incidentally, this also proves Assertion 1 of Theorem 1.2. \begin{lemma} \label{VPT} Let $f(x)\in L^\infty(\R;\C)$ be $2\pi$ periodic, $f(x+2\pi)=f(x)$, $x\in\R$, and $PT$ symmetric, $\overline{f}(-x)=f(x)$. Consider its Fourier coefficients $$ f_n=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)e^{-inx}\,dx $$ Then $\overline{f}_n=f_n$ $\forall\,n\in\Z$. \end{lemma} {\bf Proof} \newline The assertion is an immediate consequence of the Carleson-Fefferman theorem, which states the pointwise convergence of the Fourier expansion $$ \overline{f}(-x)=\sum_{n\in\Z}\overline{f}_ne^{inx}=\sum_{n\in\Z}f_ne^{inx}=f(x) $$ almost everywhere in $[0,2\pi]$. This proves the Lemma. \par To prove Theorem 1.1, let us first recall that by the Floquet-Bloch theory (see e.g. \cite{BS}, \cite{Ea}), $\lambda \in\sigma(H(g))$ if and only if the equation $H(g)\psi=\lambda \psi$ has a non-constant bounded solution. In turn, all bounded solutions have the (Bloch) form \be \label{Bloch} \psi_p(x;\lambda,g) =e^{ipx}\phi_p(x;\lambda,g) \ee where $p\in]-1/2,1/2]:=B$ (the Brillouin zone) and $\phi_p$ is $2\pi$-periodic. It is indeed immediately checked that $\psi_p(x;\lambda,g)$ solves $H(g)\psi=\lambda\psi$ if and only if $\phi_p(x;\lambda,g) $ is a solution of $$ H_p(g) \phi_p(x;\lambda,g)=\lambda \phi_p(x;\lambda,g). $$ Here $H_p(g)$ is the operator in $L^2(0,2\pi)$ given by \be \label{Hpg} H_p(g)u=\left(-i\frac{d}{dx}+p\right)^2u+qu+gWu, \quad u\in D(H_p(g)) \ee with periodic boundary conditions; its realization will be recalled below. More precisely, denote $S^1$ the one-dimensional torus, i.e. the interval $[-\pi,\pi]$ with the endpoints identified. By Assumptions (1) and (2) above the restriction of $q$ to $S^1$, still denoted $q$ by a standard abuse of notation, belongs to $H^{-1}(S^1)$ and generates a real quadratic form $\Q_p(u)$ in $L^2(S^1)$ with domain $H^{1}(S^1)$. By Assumption (3) $\Q_p(u)$ is relatively bounded, with relative bound zero, with respect to \begin{eqnarray} %\label{Qp} %\\ \T_p(u) := \int_{-\pi}^{\pi}[-iu^\prime +pu][i\overline{u}^\prime+p\overline{u}]\,dx, \quad D(\T_p(u))=H^1(S^1) %\nonumber %\Q_p(u) &=&\int_{\R}[|-iu^\prime+p|^2+q|u|^2+igW|u|^2]\,dx,\quad u\in H^{1}(S^1) \end{eqnarray} so that the real semibounded form $\H^0_p(u):=\T_p(u)+ \Q_p(u)$ defined on $H^1(S^1)$ is closed. The corresponding self-adjoint operator in $L^2(S^1)$ is the self-adjoint realization of the formal differential expression (note again the abuse of notation) $$ H_p(0)=-\frac{d^2}{dx^2}-2ip\frac{d}{dx}+p^2+q $$ As above, the form $\H_p(g)(u):=\H^0_p(u)+\la u,W u\ra$ defined on $H^1(S^1)$ is closed and sectorial in $L^2(S^1)$. Let ${ H}_p(g)$ the associated $m$-sectorial operator in $L^2(S^1)$. On $u\in D({ H}_p(g))$ the action of the operator ${ H}_p(g)$ is specified by (\ref{Hpg}); moreover, $H_p(g)$ has compact resolvent. \par\noindent Let $$ \sigma(H_p(g)):=\{\lambda_n(g;p):\;n=0,1,\ldots\} $$ denote the spectrum of $H_p(g)$, with $p\in ]-1/2,1/2]$, $|g|<\overline{g}$. By the above remarks we have $$ \sigma(H(g))=\bigcup_{p\in ]-1/2,1/2]}\sigma(H_p(g))=\bigcup_{p,n}\lambda_n(g;p) $$ To prove the reality of $ \sigma(H(g))$, $|g|<\overline{g}$, it is therefore enough to prove the reality of all eigenvalues $\lambda_n(g;p):\;n=0,1,\ldots; p\in ]-1/2,1/2]; n=0,1,\ldots$. \par To this end, let us further recall the construction of the bands for $g=0$: it can be proved that under the present conditions all eigenvalues $\lambda_{n}(0;p)$ are simple $\forall\,p\in[-1/2,1/2]$; the functions $\lambda_{n}(0;p)$ are continuous and even in $[-1/2,1/2]$ with respect to $p$, so that one can restrict to $p\in [0,1/2]$; the functions $\lambda_{2k}(0;p)$ are strictly increasing on $[0,1/2]$ while the functions $\lambda_{2k+1}(0;p)$ are strictly decreasing, $k=0,1\ldots$. Set: $$ \alpha_k=\lambda_k(0,0),\quad \beta_k=\lambda_k(0,1/2) $$ Then: $$ \alpha_0<\beta_0<\beta_1<\alpha_1<\alpha_2< \beta_2<\beta_3 \ldots $$ The intervals $[\alpha_{2n},\beta_{2n}]$ and $[\beta_{2n+1},\alpha_{2n+1}]$ coincide with the range of $\lambda_{2n}(0,p)$, $\lambda_{2n+1}(0,p)$, respectively, and represent the bands of $\sigma(H(0))$; the intervals $$ \Delta_n:=]\beta_{2n},\beta_{2n+1}[, \quad ]\alpha_{2n+1},\alpha_{2(n+2)}[ $$ the gaps between the bands. \par\noindent The monotonicity of the functions $\lambda_n(0,p)$ and Assumption (\ref{infgap}) entail \be \label{infband} \inf_n \min_{p\in [0,1/2]} |\lambda_n(0,p)-\lambda_{n+1}(0,p)|\geq 2d \ee Let us now state the following preliminary result: \begin{proposition} \label{prop1} \vskip 2pt\noindent \begin{enumerate} \item Let $g\in \overline{{\mathcal D}}$, where $\overline{\mathcal D}$ is the disk $\{g\,:\, |g|<\overline{g}\}$. For any $n$, there is a function $\lambda_n(g,p):\overline{\mathcal D}\times [0,1/2] \to \C$, holomorphic in $g$ and continuous in $p$, such that $\lambda_n(g,p)$ is a simple eigenvalue of ${\mathcal H}_p(g)$ for all $(g,p)\in \overline{\mathcal D}\times [0,1/2]$. \item $$ \sup_{g\in \overline{\mathcal D}, p\in[0,1/2]}|\lambda_n(g,p)-\lambda_n(0,p)|<\frac{d}{2} $$ \item If $g\in\R\cap \overline{\mathcal D}$ all eigenvalues $\lambda_n(g,p)$ are real; \item If $g\in\R\cap \overline{\mathcal D}$ then $\sigma({\mathcal H}_p(g))\equiv \{\lambda_n(g,p)\}_{n=0}^\infty$. \end{enumerate} \end{proposition} Assuming the validity of this Proposition the proof of Theorem 1.1 is immediate. \par\noindent {\bf Proof of Theorem 1.1} \newline Since the functions $\lambda_n(g;p)$ are real and continuous for $g\in\R\cap \overline{\mathcal D}$, $p\in [0,1/2]$, we can define: \begin{eqnarray} \label{estremi} \alpha_{2n}(g):=\min_{p\in [0,1/2]}\lambda_n(g;p); \quad \beta_{2n}(g):=\max_{p\in [0,1/2]}\lambda_{2n}(g;p)\qquad \\ \alpha_{2n+1}(g):=\max_{p\in [0,1/2]}\lambda_{2n+1}(g;p); \quad \beta_{2n+1}(g):=\min_{p\in [0,1/2]}\lambda_{2n+1}(g;p)\end{eqnarray} Then: $$ \sigma(H(g))=\bigcup_{n=0}^\infty B_n(g) $$ where the bands $B_n(g)$ are defined, in analogy with the $g=0$ case, by: \begin{eqnarray*} B_{2n}(g)&:=&[\alpha_{2n}(g),\beta_{2n}(g)] \\ B_{2n+1}(g)&:=&[\beta_{2n+1}(g),\alpha{2n+1}(g)] \end{eqnarray*} By Assertion 2 of Proposition 2.2 we have, $\forall\;n=0,1,\ldots$, $\forall\,g\in\overline{D}$: \begin{eqnarray*} \alpha_{2n}(g)-\frac{d}{2}=\lambda_{2n}(0,0)-\frac{d}{2}\leq \lambda_{2n}(0,p)-\frac{d}{2} \\ \leq \lambda_{2n}(g,p)\leq \lambda_{2n}(0,1/2)+\frac{d}{2}=\beta_{2n}+\frac{d}{2}\quad \end{eqnarray*} whence $$ \alpha_{2n}(g)-\frac{d}{2}\leq \lambda_{2n}(g,p)\leq \beta_{2n}+\frac{d}{2},\quad \forall\,n,\;\forall\,g\in\overline{\D}. $$ This yields: $$ B_{2n}(g)\subset \left[\alpha_{2n}-\frac{d}{2} , \beta_{2n}+\frac{d}{2}\right] $$ By an analogous argument: $$ B_{2n+1}(g)\subset \left[\beta_{2n+1}-\frac{d}{2}, \alpha_{2n+1}+\frac{d}{2}\right] $$ Therefore the bands are pairwise disjoint, because the gaps $$ \Delta_n(g):= ]\beta_{2n}(g),\alpha_{2n}(g)[, \quad ]\alpha_{2n+1}(g),\beta_{2n+1}(g)[ $$ are all open and their width is no smaller than $d$. In fact, by (\ref{infgap}) we have: $$ |\alpha_n-\alpha_{n+1}|\geq 2d,\quad |\beta_n-\beta_{n+1}|\leq 2d . $$ This concludes the proof of the Theorem \vskip 0.3cm We now prove separately the assertions of Proposition 2.2 \par\noindent {\bf Proof of Proposition 2.2, Assertions 1 and 2} \newline Since the maximal multiplication operator by $W$ is continuous in $L^2(S^1)$ with norm $\|W\|_\infty$, the operator family $\H_p(g)$ is a type-A holomorphic family with respect to $g\in\C$, uniformly with respect to $p\in[0,1/2]$. Hence we can direct apply regular perturbation theory (see e.g.\cite{Ka}): the perturbation expansion near any eigenvalue $\lambda_n(0;p)$ of $\H(0,p)$ exists and is convergent in $\overline{\mathcal D}$ for $g\in \overline\D$ to a simple eigenvalue $\lambda_n(g;p)$ of $H(g;p)$: \be \label{ps} \lambda_n(g;p)=\lambda_n(0,p)+\sum_{s=1}^\infty\lambda_n^s(0;p)g^s, \quad g\in \overline\D \ee The convergence radius $r_n(p)$ is no smaller than $\overline{g}$. Hence $\overline{g}$ represents a lower bound for $r_n(p)$ independent of $n$ and $p$. Moreover $\lambda_n^s(0;p)$ is continuous for all $p\in[0,1/2]$, and hence the same is true for the sum $\lambda_n(g;p)$. This proves Assertion 1. \newline To prove Assertion 2, recall that the coefficients $\lambda_n^s(0;p)$ fulfill the majorization (see \cite{Ka}, \S II.3) \be \label{psm} |\lambda_n^s(0;p)|\leq \left(\frac{2\|W\|_\infty}{\inf_{k}\min_{[0,1/2]}|\lambda_k(0,p)-\lambda_{k\pm 1}(0,p)|}\right)^s\leq \left(\|W\|_{\infty}/d\right)^s \ee Therefore, by (\ref{ps}): \begin{eqnarray*} |\lambda_n(g;p)-\lambda_n(0,p)| &\leq& \frac{|g|\left(\|W\|_{\infty}/d\right)}{1-\left(2|g|\|W\|_{\infty}/d\right)}=\frac{|g|\|W\|_{\infty}}{d-|g|\|W\|_{\infty}} < \frac{d}{2} \end{eqnarray*} whence the stated majorization on account of (\ref{rc}). \par\noindent {\bf Proof of Proposition 2.2, Assertion 3} \newline As is known, and anyway very easy to verify, the $PT$ symmetry entails that the eigenvalues of a $PT$-symmetric operator are either real or complex conjugate. By standard regular perturbation theory (see e.g. \cite{Ka}, \S VII.2) any eigenvalue $\lambda_n(0;p)$ of $H_p(0)$ is stable with respect to $H_p(g)$; since $\lambda_n(0,p)$ is simple, for $g$ suitably small there is one and only one eigenvalue $\lambda_n(g,p)$ of $H_p(g)$ near $\lambda_n(0,p)$, and $\lambda_n(g,p)\to \lambda_n(0,p)$ as $g\to 0$. This excludes the existence of the complex conjugate eigenvalue $\overline{\lambda}_n(g,p)$ distinct from $\lambda_n(g,p)$. Thus for $g\in\R$, $|g|$ suitably small, $\lambda_n(g,p)$ is real. This entails the reality of series expansion (\ref{ps}) for $g$ small and hence $\forall\,g\in\overline{\D}$. This in turn implies the reality of $\lambda_n(g,p)\forall\,g\in \overline{\D}$. \par\noindent {\bf Proof of Proposition 2.2, Assertion 4} \par\noindent We repeat here the argument introduced in \cite{CGS2},\cite{CGC} to prove the analogous result in different contexts. We describe all details to make the paper self contained. We have seen that for any $r\in \N$ the \RSPE\ associated with the eigenvalue $\lambda_r(g;p)$ of $H_p(g)$ which converges to $\lambda_r(0;p)$ as $g\to 0$, has radius of convergence no smaller than $\overline{g}$. Hence, $\forall g\in \R$ such that $|g|<\overline{g}$, $H_p(g)$ admits a sequence of real eigenvalues $\lambda_r(g;p), r\in \N$. We want to prove that for $|g|<\overline{g}, g\in\R$, $H_p(g)$ has no other eigenvalues. Thus all its eigenvalues are real. To this end, for any $r\in\N$ let ${\cal Q}_r$ denote the circle centered at $\lambda_r(0;p)$ with radius $d$. Then if $g\in\R$, $|g|<\overline{g}$, and $\lambda(g)$ is an eigenvalue of $H_p(g)$: $$ \lambda(g)\in\bigcup_{r\in\N}\,{\cal Q}_r . $$ In fact, denoting $$ R_0(z):= (H_p(0) - z)^{-1} $$ for any $\ds z\notin \bigcup_{r\in\N}\,{\cal Q}_r$ we have \be \label{Stima1} \|gWR_0(z)\|\leq |g|\|W\|_\infty \|R_0(z)\|<\overline{g} \|W\|_\infty[{\rm dist}(z,\sigma(H_0))]^{-1}\leq \frac{\overline{g} \|W\|_\infty}{d}< 1 . \ee The last inequality in (\ref{Stima1}) follows directly from the definition (\ref{rc}) of $\overline{g}$. Thus, $z\in\rho (H_p(g))$ and $$ R(g,z):= (H_p(g) - z)^{-1} = R_0(z)[1 + gWR_0(z)]^{-1}\,. $$ Now let $g_0\in\R$ be fixed with $|g|<\overline{g} $. Without loss of generality we assume that $g_0>0$. Let $\lambda(g_0)$ be a given eigenvalue of $H_p(g_0)$. Then $\lambda(g_0)$ must be contained in the interior (and not on the boundary) of ${\cal Q}_{n_0}$ for some $n_0\in\N$. Moreover if $m_0$ is the multiplicity of $\lambda(g_0)$, for $g$ close to $g_0$ there are $m_0$ eigenvalues (counting multiplicities) $\lambda^{(s)}(g), s=1.\dots ,m_0$, of $H_p(g)$ which converge to $\lambda(g_0)$ as $g\to g_0$ and each function $\lambda^{(s)}(g)$ represents a branch of one or several holomorphic functions which have at most algebraic singularities at $g=g_0$ (see \cite{Ka}, Thm. VII.1.8). Let us now consider any one of such branches $\lambda^{(s)}(g)$ for $0g\|W\|_{\infty}$ implies $z\notin \sigma(H_p(g))$, i.e. if $z\in \sigma(H_p(g))\cap\Gamma_{t}$ then $t\leq g\|W\|_{\infty}0$. We will show that it can be continued up to $g=0$, and in fact up to $g=-\overline{g}$. From what has been established so far the function $\lambda(g)$ is bounded as $g\to g_1^+$. Thus, by the well known properties on the stability of the eigenvalues of the analytic families of operators, $\lambda(g)$ must converge to an eigenvalue $\lambda(g_1)$ of $H_p(g_1)$ as $g\to g_1^+$ and $\lambda(g_1)$ is contained in the circle centered at $\lambda_{n_0}(0;p)$ and radius $g_1\|W\|_{\infty}$. Repeating the argument starting now from $\lambda(g_1)$, we can continue $\lambda(g)$ to a holomorphic function on an interval $]g_2, g_1]$, which has at most an algebraic singularity at $g=g_2$. We build in this way a sequence $g_1>g_2>\dots >g_n>\dots $ which can accumulate only at $g=-\overline{g}$. In particular the function $\lambda(g)$ is piecewise holomorphic on $]-\overline{g}, \overline{g}[$. But while passing through $g=0$, $\lambda(g)$ coincides with the eigenvalue $\lambda_r(g;p)$ generated by an unperturbed eigenvalue $\lambda_r(0;p)$ of $H_p(0)$ (namely $\lambda_{n_0}(0;p)$), which represents a real analytic function defined for $g\in ]-\overline{g},\overline{g}[$. Thus, $\lambda(g_0)$ arises from this function and is therefore real. This concludes the proof of Assertion $4$. %%%%%%%%%%%%%%%%%%%% \vskip 0.3cm\noindent {\bf Proof of Theorem 1.2} \newline Consider the operator $K_p(g)$ acting in $L^2(S^1)$, defined on the domain $H^2(S^1)$. By the Floquet-Bloch theory recalled above, we have again $$ \sigma(K(g))=\bigcup_{p\in]-1/2,1/2]}\sigma(K_p(g)). $$ It is then enough to prove that there is $\eta>0$ such that $K_p(g)$ has complex eigenvalues for $p\in]1/2 -\eta,1/2]$. Since $K_p(g)$ is $PT$-symmetric, eigenvalues may occur only in complex-conjugate pairs. The eigenvalues of $K_{p}(0)$ are $\lambda_n(0,p)=(n+p)^2: n\in\Z$. The eigenvalue $\lambda_0(0,p)=p^2$ is simple $\forall\,p\in[0,1/2]$; any other eigenvalues is simple for $p\neq 0$, $p\neq 1/2$ and has multiplicity $2$ for $p=0$ or $p=1/2$ because $n^2=(-n)^2$ and $(n+1/2)^2=(-n-1+1/2)^2$, $n=0,1,\ldots$. For $p=1/2$ a set of orthonormal eigenfunctions corresponding to the double eigenvalue $(n+1/2)^2=(-n-1+1/2)^2$, $n=0,1,\ldots$ is given by $\{u_n, u_{-n-1}\}$, where: $$ u_n:=\frac1{\sqrt{2\pi}}e^{inx}, \quad n\in\Z $$ Remark that, for $00$ such that $$ {\rm Im}\,\lambda_k^\pm(g,p)\neq 0, \quad 1/2-\eta \leq p\leq 1/2 $$ It follows (see e.g.\cite{BS}, \cite{Ea}) that the complex arcs ${\mathcal E}^\pm_k:={\rm Range}(\lambda_k^\pm(g,p)): p\in[1/2-\eta(g), 1/2]$ belong to the spectrum of ${ K}(g)$. This concludes the proof of the theorem. %%%%%%%%%%% %%%%%%%%%%%% %\newpage \vskip 1.5cm\noindent \begin{thebibliography}{DGHLLL} %\vskip 0.5cm\noindent {\small \bibitem[AGHKH]{AGHKH} S.Albeverio, F.Gesztesy, R.H\/eght-Krohn, H.Holden {\it Solvable models in quantum mechanics} , Springer-Verlag 1988 \bibitem[Ah]{Ah} Z.Ahmed, {\it Energy band structure due to a complex, periodic, $PT$ invariant potential}, Phys.Lett. A {\bf 286}, 231-235 (2001) \bibitem[Be1]{Be1} C. M. Bender, S. Boettcher, and P. N. Meisinger, {\it PT-Symmetric Quantum Mechanics} Journal of Mathematical Physics {\bf 40},2201-2229 (1999) \bibitem[Be2]{Be4} C.M.Bender, {\it Making Sense of Non-Hermitian Hamiltoniians} (hep-th/0703096) \bibitem[BBM]{Be2} C. M. Bender, M. V. Berry, and A. Mandilara {\it Generalized PT Symmetry and Real Spectra}. J.Phys. A: Math. Gen. {\bf 35}, L467-L471 (2002) \bibitem[BBJ]{BBJ} C. M. Bender, D. C. Brody, and H. F. Jones {\it Must a Hamiltonian be Hermitian?}, American Journal of Physics, {\bf 71}, 1039-1031 (2003) \bibitem[BDM]{BDM} C. M. Bender, G.V.Dunne, and N.Meisinger, {\it Complex periodic potentials with real band spectra}, Phys.Lett. A {\bf 252}, 272-276 (1999) \bibitem[BS]{BS} F.Berezin and M.S.Shubin {\it The \Sc\ equation}, Kluwer 1991 \bibitem[Ce]{Ce} J.M.Cerver\`o, {\it $PT$-symmetry in one-dimensional periodic potentials}, Phys.Lett. A {\bf 317}, 26-31 (2003) \bibitem[CG]{CGC} E.Caliceti and S.Graffi {\it On a class of non self-adjoint quantum non-linear oscillators with real spectrum}, J.Nonlinear Math.Phys. {\bf 12}, 138-145 (2005) \bibitem[CR]{CR}J.M.Cerver\`o and J.M.Rodriguez, {\it The band spectrum of periodic potentials with $PT$ symmetry}, J.Phys.A (2003) \bibitem[CGS]{CGS2} {\it Spectra of $\PT$-symmetric operators and perturbation theory}, J.Physics A, Math\&Gen, {\bf 38}, 185-193 (2005) \bibitem[CJN]{Cn3} F.Cannata, M.V.Ioffe, D.N.Nishniadinze, {\it Two-dimensional SUSY Pseudo-Hermiticity without Separation of Variables}. Phys. Lett. {\bf A310}, 344-352 (2003) \bibitem[CJT]{Cn1} F.Cannata, G.Junker and J.Trost, {\it Schr\"odinger operators with complex potential but real spectrum}, Phys Lett {\bf A246} 219-226 (1998) \bibitem[Ea]{Ea} M.S.P.Eastham, {\it The Spectral Theory of Periodic Differential Equations}, Scottish Academic Press, 1973 \bibitem[Jo]{Jo} H.F.Jones, {\it The energy spectrum of complex periodic potentials of Kronig-Penney type}, Phys.Lett. A {\bf 262}, 242-244 (1999) \bibitem[LZ]{Zn1} G. Levai and M. Znojil, {\it Systematic search for PT symmetric potentials with real energy spectra} J. Phys. A: Math. Gen. {\bf 33} (2000) 7165. \bibitem[Ka]{Ka} T.Kato, {\it Perturbation Theory for Linear Operators}, 2nd Edition, Springer-Verlag, 1976 \bibitem[Sh]{Shin} K.C.Shin, {\it On the shape of the spectra of non self-adjoint periodic \Sc\ operators}, J.Phys.A (2004) \bibitem[Sp]{Sp} J. Phys.A, Math\&Gen, {\bf 39},n,32 (2006) (Special Issue on $\PT$-Symmetric Quantum Mechanics) \bibitem[ZCBR]{Cn2} M.Znojil, F.Cannata, B.Bagchi, R.Roychoudhury, {\it Supersymmetry without Hermiticity within $PT$ symmetric quantum mechanics}. Phys. Lett. {\bf B483}, 284 (2000) } \end{thebibliography} \end{document} %\bibitem[BBMe]{Be1} \bibitem{Be1} C. M. Bender, S. Boettcher, and P. N. Meisinger, {\it PT-Symmetric Quantum Mechanics} Journal of Mathematical Physics {\bf 40},2201-2229 (1999) \bibitem[3]{Be2} C. M. Bender, M. V. Berry, and A. Mandilara {\it Generalized PT Symmetry and Real Spectra}. J.Phys. A: Math. Gen. {\bf 35}, L467-L471 (2002) \bibitem[4]{Be3} C. M. Bender, D. C. Brody, and H. F. Jones {\it Must a Hamiltonian be Hermitian?}, American Journal of Physics, {\bf 71}, 1039-1031 (2003) \bibitem[5]{Cn1} F.Cannata, G.Junker and J.Trost, {\it Schr\"odinger operators with complex potential but real spectrum}, Phys Lett {\bf A246} 219-226 (1998) \bibitem[6]{Cn2} M.Znojil, F.Cannata, B.Bagchi, R.Roychoudhury, {\it Supersymmetry without Hermiticity within $PT$ symmetric quantum mechanics}. Phys. Lett. {\bf B483}, 284 (2000) \bibitem[7]{Cn3} F.Cannata, M.V.Ioffe, D.N.Nishniadinze, {\it Two-dimensional SUSY Pseudo-Hermiticity without Separation of Variables}. Phys. Lett. {\bf A310}, 344-352 (2003) \bibitem[8]{Zn1} G. Levai and M. Znojil, {\it Systematic search for PT symmetric potentials with real energy spectra} J. Phys. A: Math. Gen. {\bf 33} (2000) 7165. \bibitem[9]{Be4} C.M.Bender, {\it Making Sense of Non-Hermitian Hamiltoniians} (hep-th/0703096) \bibitem[10]{Sp} J. Phys.A, Math\&Gen, {\bf 39},n,32 (2006) (Special Issue on $\PT$-Symmetric Quantum Mechanics) \bibitem[11]{PGP} E.Prodan, S.R.Garcia, and M.Putinar, {\it Norm estimates of complex symmetric operators applied to quantum systems}. J. Phys.A, Math\&Gen, {\bf 39}, 389-400 (2006) \bibitem[12]{Shin} K.C.Shin, {\it On the reality of the eigenvalues for a class of $PT$-symmetric oscillators}. Comm. Math. Phys. {\bf 229}, 543-564 (2002) \bibitem[13]{Tateo} P.Dorey, C.Dunning, R.Tateo, {\it Spectral Equivalences, Bethe ansatz equations, and reality properties in ${PT}$-symmetric quantum mechanics}. J.Phys. A {\bf 34} 5679-5704, (2001). \bibitem[14]{DD} P.E.Dorey, C.Dunning, R.Tateo, {\it Supersymmetry and the spontaneous breakdown of $PT$ symmetry}, J.Phys. A {\bf 34}, L391-L400 (2001) \bibitem[15]{CGS} E.Caliceti, S.Graffi, J.Sj\"ostrand, {\it Spectra of $\PT$-symmetric operators and perturbation theory}, J.Physics A, Math\&Gen, {\bf 38}, 185-193 (2005) \bibitem[16]{CG} E.Caliceti, S.Graffi, {\it On a class of non self-adjoint quantum non-linear oscillators with real spectrum}, J.Nonlinear Math.Phys. {\bf 12}, 138-145 (2005) \bibitem[17]{KS} R.Kretschmer and L.Szymanovski, {\it Pseudo-Hermiticity in infinite dimensional Hilbert spaces} quant-ph/0305123 (2003) \bibitem[18]{Mo}A.Mostafazadeh, {\it Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian } J.Math.Phys. {\bf 43}, 205-212 (2002); {\it Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum}, ibidem, 2814-2816 (2002); {\it Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries}, ibidem 3944-3951 (2002) \bibitem[19]{We} S.Weigert, {\it Completeness and orthonormality in PT-symmetric quantum systems}, Phys.Rev. {\bf A 68}, 06211-06215 (2003) \bibitem[20]{Mo5} A.Mostafazadeh, {\it Pseudo-Hermiticity for a class of non-diagonalizable Hamiltonians } J.Math.Phys. {\bf 43}, 6342-6352 (2002); \bibitem[21]{ACS} A.A.Andrianov, F. Cannata, A.V. Sokolov {\it Non-linear supersymmetry Ê for non-Hermitian, non-diagonalizable Hamiltonians: I. General Ê properties}, Nuclear Physics {\bf B 773}, 107-136 (2007) \bibitem[22]{CJN} F. Cannata, M.V. Joffe, D.N. Nishnianidze: {\it Pseudo Hermiticity of Ê an exactly solvable two-dimensional model} arXiv: 0704.2219v1 [hep- th] \bibitem[23]{SS} G.Scolarici and L.Solombrino, {\it On the pseudo-Hermitian nondiagonalizable Hamiltonians}, J. Math. Phys. {\bf 44}, 4450-4459 (2003). \bibitem[24]{GP1} S.R.Garcia, and M.Putinar, {\it Complex symmetric operators and applications}, Trans. Amer. Math. Soc. {\bf 358} , 1285-1315, (2006). \bibitem[25]{GP2} S.R.Garcia, and M.Putinar, {\it Complex symmetric operators and applications}, II, Trans. Amer. Math. Soc. {\bf 359}, 3913-3931, (2007). \bibitem[26]{Ca} E.Caliceti, {\it Real spectra of $\PT$-symmetric operators and perturbation theory}, Czech.J.Physics {\bf 54}, 1065-1068 (2004) \bibitem[27]{Ba} V.Bargmann, {\it On a Hilbert space of analytic functions and an associated integral transform}, Comm.Pure Appl.Math. {\bf 14}, 187-214 (1961) \bibitem[28]{BD} C.M.Bender, G.V.Dunne, {\it Large-Order Perturbation Theory for a Non-Hermitian PT-Symmetric Hamiltonian}, J.Math.Phys. {\bf 40}, 4616-4621 (1999) \bibitem[29]{BMW} C.M.Bender, E.J.Weniger, {\it Numerical Evidence that the Perturbation Expansion for a Non-Hermitian PT-symmetric Hamiltonian is Stieltjes}, J.Math.Phys. {\bf 42}, 2167-2183 (2001) \bibitem[30]{RS} M.Reed, B.Simon, {\it Methods of Modern Mathematical Physics}, Vol. IV, Academic Press 1978 - - F. Cannata, M.V. Joffe, D.N. Nishnianidze: "Pseudo Hermiticity of Ê an exactly solvable two-dimensional model" arXiv: 0704.2219v1 [hep- th] (17 Apr 2007) Complex symmetric operators and applications (with Mihai Putinar) (PDF) (DVI) (PS) Trans. Amer. Math. Soc. 358 (2006), 1285-1315. Complex symmetric operators and applications II (with Mihai Putinar) (PDF) (DVI) (PS) Trans. Amer. Math. Soc. 359 (2007), 3913-3931. ---------------0804290751915--