Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

Abstract

Let $H$ be any $\PT$ symmetric Schrödinger operator of the type $-{\hslash }^2\Delta +(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2({\mathbb{R}}^d)$, where $W$ is any odd homogeneous polynomial and $g\in \mathbb{R}$. 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 ${ł}_j$ of $H$ by Weyl's inequalities.

20 pages