PT-Symmetry, Canonical Decomposition, Perturbation Theory

Abstract

Let H be any PT-symmetric Schrödinger operator of the type H=-ħ ² Δ+x ² +igW(x), where W is a real polynomial, odd under reflection of all coordinates, g∈R, acting on L ² ( R ^d ). The proof is outlined of the following statements: PH is self-adjoint and its eigenvalues coincide with the eigenvalues of √(H*H). Moreover the eigenvalues of √(H*H), known as the singular values of H, can be computed via perturbation theory by Borel summability.