Orthogonal Polynomials and Generalized Oscillator Algebras

TLDR: In this paper, the authors construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian and obtain the explicit form of the hamiltonians, the energy levels and the explicit forms of the impulse operators.

Abstract: For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. the general approach is illustrated by the examples of the classical orthogonal poly-nominals: Hermite, Jacobi and Laguerre Polynomials. For these polynomials we obtain the explicit form of the hamiltonians, the energy levels and the explicit form of the impulse operators