Non-real Eigenvalues for PT-Symmetric Double Wells

TL;DR

This paper investigates how eigenvalues of PT-symmetric double-well Schrödinger operators behave under small perturbations, showing they remain real initially but become complex as perturbations increase.

Contribution

It provides a rigorous analysis of eigenvalue bifurcation from real to complex in PT-symmetric double-well systems under perturbations.

Findings

Eigenvalues remain real for very small PT-symmetric perturbations.

Eigenvalues bifurcate into the complex plane as perturbation strength increases.

The transition from real to complex eigenvalues is characterized mathematically.

Abstract

We study small, PT-symmetric perturbations of self-adjoint double-well Schr\"odinger operators in dimension $n\geq 1$. We prove that the eigenvalues stay real for a very small perturbation, then bifurcate to the complex plane as the perturbation gets stronger.