Local Perturbation of the Discrete Schrödinger Operator and a Generalized Chebyshev Oscillator

Abstract

We discuss the conditions under which a special linear transformation of the classical Chebyshev polynomials (of the second kind) generate a class of polynomials related to “local perturbations” of the coefficients of a discrete Schrödinger equation. These polynomials are called generalized Chebyshev polynomials. We answer this question for the simplest class of “local perturbations” and describe a generalized Chebyshev oscillator corresponding to generalized Chebyshev polynomials.