Cubic algebras, induced representations and general solution of the exceptional Laguerre equation $X_1$

TL;DR

This paper develops an algebraic framework using spectrum generating algebras and Darboux-Crum transformations to find general solutions of the exceptional Laguerre differential equation, including polynomial and non-polynomial states.

Contribution

It introduces a novel algebraic approach for solving exceptional Laguerre equations of type I, II, and III, constructing ladder operators, zero modes, and Rodrigues-like formulas for the general solutions.

Findings

Constructed fourth-order ladder operators for exceptional Laguerre polynomials.

Derived all zero modes for raising and lowering operators.

Provided algebraic Rodrigues formulas for the general solutions.

Abstract

We consider the case of exceptional Laguerre polynomials $X_1$ of type I, II and III, their ordinary differential equations and the problem of finding general solution beside the polynomial part. We will develop an algebraic approach based on the Schrodinger form of the problem and associate representations of the underlying spectrum generating algebra. We use the Darboux-Crum transformation to construct ladder operators of fourth order for the case of the exceptional Laguerre polynomials $X_1$ of type I, II and III. We then obtain all zero modes for the lowering and raising operators. We construct the induced representation for the linearly independent solutions, including the polynomial states. Those states forming the general solution are important non only in the construction of wider set of physical states satisfying different boundary conditions, but also in context of getting isospectral deformations. Our approach, allows to provide a completely algebraic construction of the two linearly independent solutions of the ordinary differential equation of the exceptional orthogonal polynomials of Laguerre type $X_1$ ( case I, II and III). The analogue of 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 all the states (polynomial and non-polynomial), in a purely algebraic way, but also to obtain coefficients from the action of the ladder operators also in an algebraic manner. Those results are established via higher commutation relations related to the cubic Heisenberg Weyl algebra.