Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

TL;DR

This paper studies PT-symmetric Schrödinger operators, proving their self-adjointness, relating eigenvalues to singular values, and constructing a canonical expansion using Borel summability to estimate eigenvalue locations.

Contribution

It introduces a canonical expansion for PT-symmetric operators and links their eigenvalues to singular values via Borel summability, advancing perturbation theory methods.

Findings

Proved that PT-symmetric operators are self-adjoint after a specific transformation.

Established the eigenvalues coincide with singular values up to a sign.

Constructed a canonical expansion and used Borel summability to analyze eigenvalues.

Abstract

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(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 $\P H$ 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.