Recurrence relations and general solution of the exceptional Hermite equation

TL;DR

This paper explores the recurrence relations and general solutions of the exceptional Hermite equation, linking them to supersymmetric quantum mechanics, and introduces new algebraic methods for constructing states and solutions.

Contribution

It provides explicit general solutions for the exceptional Hermite polynomials and develops Rodrigues formulas and algebraic techniques for state construction.

Findings

Explicit general solutions linked to confluent Heun equation

New Rodrigues formulas for exceptional Hermite polynomials

Algebraic construction of polynomial and non-polynomial states

Abstract

Exceptional orthogonal Hermite and Laguerre polynomials have been linked to the k-step extension of harmonic and singular oscillators. The exceptional polynomials allow the existence of different supercharges from the Darboux-Crum and Krein-Adler constructions of supersymmetric quantum mechanics. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable, and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum is discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We plan to discuss in detail the case of the exceptional Hermite polynomials $X_2^{(1)}$ and to explicitly present new chains obtained by acting with different types of ladder operators. We will exploit a recent result by one of the authors [32], where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators also in an algebraic manner based on further commutation relations between monomials of the generators.