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PT-symmetry, canonical decomposition, perturbation theory,singular values
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%\markboth{E.Caliceti}
%{Distributional Borel of Vacuum Polarization}
\def\ha{Ha\-mil\-to\-nian}
\def\R{{\bf R}}
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\def\N{{\bf N}}
\def\T{\bf T}
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\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}}
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\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}}
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\def\T{{\cal T}}
\def\l{{\lambda}}
\def\r{\rho}
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\def\op{o\-pe\-ra\-tor}
\def\arg{{\rm arg}}
\def\Im{{\rm Im}}
\def\Cinf{C_0^\infty(\R^d)}
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\begin{document}
\baselineskip=21pt
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\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
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\section{Introduction and statement of the results}
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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 $0\leq g****0$.
\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}
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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)
$$
which provides a diagonalization for an operator $H$ with compact resolvent
with respect
to the pair of orthonormal bases $\{\psi_k\}$ and $\{\psi^\prime_k\}$,
requires a priori the computation of $\mu_k$ and $\psi_k$ as solutions
of the spectral problem
\begin{equation}
\label{ccc}
H^\ast(g) H(g)\psi=\mu^2\psi
\end{equation}
which represents an eigenvalue problem more complicated than
$H(g)\phi=\lambda\phi$.
The result of this paper means that the eigenvalue
problem (\ref{ccc}) can be replaced by the more tractable one
$$
H(g)\psi=\mu\P\psi
$$
which can be solved by perturbation theory and Borel summability.
\vfill\eject
\vskip 0.3cm\noindent
{\bf Acknowledgment}
\par\noindent
We thank Francesco Cannata for his interest in this work and several
useful suggestions.
\vskip 1cm\noindent
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\end{document}
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