Cubic algebras, induced representations and general solution of the exceptional Laguerre equation X1

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 solutions beside the polynomial part. We will develop an algebraic approach based on the Schrödinger 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 not only in the construction of a wider set of physical states satisfying different boundary conditions but also used in the context of getting isospectral deformations as they allow often to overcome obstruction as several Wronskian constructions of Hamiltonian lead to only formal Darboux transformations. 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 analogues of the 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 by means of higher commutation relations related to the cubic Heisenberg–Weyl algebra. The zero modes are associated with eigenstates, but also generalised eigenstates.