PT invariant Non-Hermitian Potentials with Real QES Eigenvalues

TL;DR

This paper demonstrates that certain PT-invariant non-Hermitian potentials in quantum mechanics have real quasi-exactly solvable eigenvalues and associated orthogonal polynomial properties when a parameter is an odd integer.

Contribution

It establishes the reality of QES eigenvalues and orthogonal polynomial norms for specific PT-invariant non-Hermitian potentials with odd integer parameters.

Findings

QES eigenvalues are real for odd integer parameters.

Norms and weight functions of associated polynomials are real.

Results apply to both hyperbolic and periodic potentials.

Abstract

We show that at least the quasi-exactly solvable eigenvalues of the Schr\"odinger equation with the potential $V(x) = -(\zeta \cosh 2x -iM)^2$ as well as the periodic potential $V(x) = (\zeta \cos 2\theta -iM)^2$ are real for the PT-invariant non-Hermitian potentials in case the parameter $M$ is any odd integer. We further show that the norm as well as the weight functions for the corresponding weak orthogonal polynomials are also real.