Exact Isospectral Pairs of PT-Symmetric Hamiltonians

TL;DR

This paper introduces a method to construct pairs of PT-symmetric Hamiltonians with identical real spectra, linking complex and real differential equations and revealing quantum anomalies that vanish in the classical limit.

Contribution

It provides a novel technique to generate infinite pairs of isospectral PT-symmetric Hamiltonians, demonstrating their spectral equivalence and classical orbit correspondence.

Findings

Constructed infinite pairs of isospectral PT-symmetric Hamiltonians.

Proved the reality of eigenvalues through complex-to-real differential equation equivalence.

Identified quantum anomalies in the real Hamiltonians that disappear in the classical limit.

Abstract

A technique for constructing an infinite tower of pairs of PT-symmetric Hamiltonians, $\hat{H}_n$ and $\hat{K}_n$ (n=2,3,4,...), that have exactly the same eigenvalues is described. The eigenvalue problem for the first Hamiltonian $\hat{H}_n$ of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second Hamiltonian $\hat{K}_n$ of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated with $\hat{K}_n$ is real, the Hamiltonian $\hat{K}_n$ exhibits quantum anomalies (terms proportional to powers of $\hbar$). These anomalies are remnants of the complex nature of the equivalent Hamiltonian $\hat{H}_n$. In the classical limit in which the anomaly terms in $\hat{K}_n$ are discarded, the pair of Hamiltonians $H_{n,classical}$ and $K_{n,classical}$ have closed classical orbits whose periods are identical.