Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

Abstract

Let $H$ be any $\PT$ symmetric Schrödinger operator of the type $ -\hbar^2Δ+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $¶H$ 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 $μ_j$ 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.