On the dimensions of oscillator algebras induced by orthogonal polynomials

Abstract

Abstract. There is a generalized oscillator algebra associated with every class of orthogonal polynomials {Ψn(x)} ∞ n=0 , on the real line, satisfying a three term recurrence relation xΨn(x) = bnΨ n+1 (x) + b n−1 Ψ n−1 (x), Ψ 0 (x) = 1, b −1 = 0. This note presents necessary and sufficient conditions on bn for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator algebras associated with Hermite, Legendre and Gegenbauer polynomials. Some remarks on the dimensions of oscillator algebras associated with multi-boson systems are also presented.

On the dimensions of oscillator-like algebras induced by orthogonal polynomials: Nonsymmetric case

“…In this section we shall provide a class of generalized oscillator and oscillator-like algebras based on [1,2,4]. In particular we shall respond to the claims made in [7,8] about our earlier paper [14].…”

“…In a recent paper, [14], we have considered the dimension of generalized oscillator algebras associated with orthogonal polynomials, on the real line, that are orthogonal with respect to a symmetric probability measure.…”

Orthogonal polynomials and deformed oscillators

“…As a first example we consider the case of Chebyshev polynomials of the first kind T n (x) which was not considered in [29]. The polynomials T n (x) is defined by the relation…”

“…In a recently published paper [29] the authors investigated the conditions under which algebra of the generalized oscillator A, associated with orthogonal polynomials in the manner described above, is finite-dimensional. In [29] considered only case of orthogonal polynomials for a symmetric measure on the real axis (when the Jacobi matrix corresponding to the recurrent relations (1) has zero diagonal).…”

“…In a recently published paper [29] the authors investigated the conditions under which algebra of the generalized oscillator A, associated with orthogonal polynomials in the manner described above, is finite-dimensional. In [29] considered only case of orthogonal polynomials for a symmetric measure on the real axis (when the Jacobi matrix corresponding to the recurrent relations (1) has zero diagonal). In the work [30] we have clarified the formulation of sufficient conditions and extended the results to the case of the Jacobian matrix with nonzero diagonal, corresponding to recurrent relations (2).…”