A unifying E2-quasi-exactly solvable model

TL;DR

This paper introduces a new non-Hermitian E2-quasi-exactly solvable model that unifies previous models, analyzes its limits, and explores the complex Mathieu Hamiltonian and exceptional point structures.

Contribution

It constructs a novel E2-quasi-exactly solvable model encompassing known models and analyzes its limits and spectral properties.

Findings

Identifies the finite approximation to the complex Mathieu Hamiltonian.

Analyzes branch cut structures and energy eigenvalue loops near exceptional points.

Computes Stieltjes measure and orthogonal polynomial functionals.

Abstract

A new non-Hermitian E2-quasi-exactly solvable model is constructed containing two previously known models of this type as limits in one of its three parameters. We identify the optimal finite approximation to the double scaling limit to the complex Mathieu Hamiltonian. A detailed analysis of the vicinity of the exceptional points in the parameter space is provided by discussing the branch cut structures responsible for the chirality when exceptional points are surrounded and the structure of the corresponding energy eigenvalue loops stretching over several Riemann sheets. We compute the Stieltjes measure and momentum functionals for the coefficient functions that are univariate weakly orthogonal polynomials in the energy obeying three-term recurrence relations.