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

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.In [1] we introduced E 2 -quasi-exactly solvable models in analogy to the notion of sl 2 (C)-quasi-exactly solvability originally proposed by Turbiner [2,3]. The different setting is motivated mathematically by the fact that solutions for E 2 -quasi-exactly solvable models do not belong to the general class of hypergeometric functions which emerge as solutions from an sl 2 (C)-setting. The physical motivation results from the current interest in extending the study of solvable models [4,5,6,7, 1] to non-Hermitian quantum mechanical systems [8,9,10,11]. The E 2 -quasi-exactly solvable models are especially interesting in optical settings [12,13,14,15,16,17,18,19,20] where the fact is exploited that the Helmholtz equation results as a reduction from the Schrödinger equation. Solvable models are rare exceptions in the study of quantum mechanical systems and the model presented here should be added to that list.The starting point for the construction of E 2 -quasi-exactly systems consists of expressing the Hamiltonian operator H of the model in terms of the E 2 -basis operators u, v and J obeying the commutation relationsinstead of the standard sl 2 (C)-generators. We now use the particular realization [21]and demand a specific anti-linear symmetry [22], PT 3 : J → J, u → v, v → u, i → −i, as defined in [18], which for (2) becomes PT 3 : θ → π/2 − θ, i → −i. The operators in (2)

A Unifying E2-Quasi Exactly Solvable Model

“…Here we are concerned with the latter. Lie algebraic versions of Hamiltonians in this context are usually taken to be of sl 2 (C)-type [1,2], but as recently proposed [3,24], they may also be taken to be of a Euclidean Lie algebraic type, thus giving rise to qualitatively new structures.…”

“…in [3] and [24], respectively. Both Hamiltonians are expressed in terms of the E 2 -basis operators u, v and J that obey the commutation relations…”

“…Here we are mainly concerned with the extension of quasi-exactly solvable models [16,17,18,19,3] to non-Hermitian quantum mechanical systems [20,21,22,23] within the above mentioned scheme. So far two different types of E 2 -models have been constructed in [3,24] and the main purpose of this manuscript is to investigate whether it is possible to construct a more general model that unifies the two. We show that this is indeed possible by combining the two models and introducing a new parameter into the system that interpolates between the two.…”

Stability boundaries of a Mathieu equation having PT symmetry

“…The fact that one can control the band diagram by tuning the non-Hermitian character of the lattice is also not a new result and it was suggested earlier by several authors [7,9,11,20,21]. There are also some very interesting connections between the Mathieu equation and the E2 algebra in non-Hermitian models [22,23,24]. However, to my knowledge, a detailed analysis aimed to obtain analytic expressions for the stability boundaries in the (a, ε) plane for a P T symmetric lattice has not been considered.…”

Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

“…QES systems that can be cast into such a form are usually referred to as QES models of Lie algebraic type [3,4]. The relevant underlying algebras are either of sl 2 (C)-type, with their compact and non-compact real forms su(2) and su(1, 1), respectively [5], or of Euclidean Lie algebras type [6,7,8]. The latter class was found to be particularly useful when dealing with certain types of non-Hermitian systems.…”