PT-symmetric eigenvalues for homogeneous potentials

TL;DR

This paper investigates PT-symmetric eigenvalues in one-dimensional Schrödinger equations with homogeneous potentials, revealing how the real spectrum transitions to non-real as the potential's degree varies, and analyzing the asymptotic behavior of eigenvalues.

Contribution

It provides a rigorous proof of the spectral transition phenomenon observed numerically by Bender and Boettcher, including the asymptotic limits of non-real eigenvalues.

Findings

Real spectrum becomes non-real as potential degree changes

Eigenvalues tend to specific limits at infinity

Confirmed numerical observations with analytical proofs

Abstract

We consider one-dimensional Schr\"odinger equations with homogeneous potential, under appropriate PT-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as the degree of the potential changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues as they tend to infinity.