Orthogonal polynomials derived from the tridiagonal representation approach

TL;DR

This paper introduces two new classes of orthogonal polynomials derived from the tridiagonal representation approach, which are useful for modeling quantum systems with both scattering and bound states, and explores their properties numerically.

Contribution

The work defines new orthogonal polynomials from the tridiagonal approach and highlights their potential applications in quantum physics, encouraging further analytical study.

Findings

Polynomials have mixed continuous and discrete spectra.

Numerical properties of the polynomials are analyzed.

Potential applications in quantum systems with scattering and bound states.

Abstract

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.