A new non-Hermitian E2-quasi-exactly solvable model

TL;DR

This paper introduces a novel non-Hermitian E2-quasi-exactly solvable model with unique eigenfunctions involving weakly orthogonal polynomials, and explores its Hermitian limit and connection to the complex Mathieu equation.

Contribution

It presents the first known non-Hermitian E2-quasi-exactly solvable model with factorized recurrence relations and analyzes its properties and limits.

Findings

Eigenfunctions involve weakly orthogonal polynomials

Model reduces to Hermitian case at specific parameter values

Connection established with the complex Mathieu equation

Abstract

We construct a previously unknown $E_2$-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.