Orthogonal polynomials derived from the tridiagonal representation approach

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.

11 pages, 3 tables, 4 figures

References in corpus (3)

• Solution of the nonrelativistic wave equation in the tridiagonal representation approach
• Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment
• The Wilson-Racah Quantum System

Cited by in corpus (11)

• Quantum mechanics with orthogonal polynomials
• Series solutions of Heun-type equation in terms of orthogonal polynomials
• Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
• Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach
• Bound states and the potential parameter spectrum
• Construction of potential functions associated with a given energy spectrum -- An inverse problem. II
• Bound states of a short-range potential with inverse cube singularity
• Exact solvability of two new 3D and 1D nonrelativistic potentials within the TRA framework
• Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment
• Five-parameter potential box with inverse square singular boundaries
• Progressive approximation of bound states by finite series of square-integrable functions