E2-quasi-exact solvability for non-Hermitian models

TL;DR

This paper introduces $E_{2}$-quasi-exact solvability for non-Hermitian models, providing explicit solutions, analyzing exceptional points, and exploring orthogonal polynomial properties in the spectrum.

Contribution

It defines $E_{2}$-quasi-exact solvability and applies it to a non-Hermitian Hamiltonian, deriving explicit solutions and spectral properties, including orthogonal polynomial structures.

Findings

Explicit solutions for a non-Hermitian Hamiltonian system.

Identification of exceptional points via algebraic equations.

Polynomials in eigenfunctions are orthogonal and factorize at higher levels.

Abstract

We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.