 0415 E. Caliceti, S. Graffi
 Canonical Expansion of PTSymmetric Operators and Perturbation
Theory
(44K, Latex)
Jan 21, 04

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Abstract. 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
selfadjoint 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.
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