Bender-Wu singularities

TL;DR

This paper investigates the spectral properties of a quantum Hamiltonian with an imaginary double well potential, demonstrating infinite eigenvalue crossings and their accumulation at a critical energy, revealing semiclassical localization effects.

Contribution

It proves the existence of infinite eigenvalue crossings in a family of non-Hermitian quantum Hamiltonians with an imaginary potential, detailing the eigenvalue pairing and accumulation phenomena.

Findings

Eigenvalue crossings occur infinitely often in the spectrum.

Crossings follow specific selection rules for eigenvalue pairs.

Eigenvalues accumulate at a critical energy level.

Abstract

We consider a family of quantum Hamiltonians $H_\hbar=-\hbar^2\,(d^2\!/dx^2) +V(x)$, $x\in\mathbb{R},$ $\hbar>0,$ where $V(x)=i(x^3-x)$ is an imaginary double well potential. We prove the existence of infinite eigenvalue crossings with the selection rules of the eigenvalue pairs taking part in a crossing. This is a semiclassical localization effect. The eigenvalues at the crossings accumulate at a critical energy for some of the Stokes lines.