PT-symmetry and Schr\"odinger operators. The double well case

TL;DR

This paper investigates PT-symmetric semiclassical Schr"odinger operators with double-well potentials, revealing that eigenvalues remain real only for exponentially small perturbations before bifurcating into complex values, with detailed asymptotic analysis.

Contribution

It extends previous work on single-well potentials to the double-well case, providing explicit asymptotic expansions and demonstrating the limits of eigenvalue reality under perturbations.

Findings

Eigenvalues stay real for exponentially small perturbations

Eigenvalues bifurcate into complex domain as perturbation increases

Provides explicit asymptotic expansions for eigenvalues

Abstract

We study a class of PT-symmetric semiclassical Schr\"odinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.