Tridiagonal representation approach in quantum mechanics

Abstract

We present an algebraic approach for finding exact solutions of the wave equation. The approach, which is referred to as the Tridiagonal Representation Approach (TRA), is inspired by the J-matrix method and based on the theory of orthogonal polynomials. The class of exactly solvable problems in this approach is larger than the conventional class. All properties of the physical system (energy spectrum of the bound states, phase shift of the scattering states, energy density of states, etc.) are obtained in this approach directly and simply from the properties of the associated orthogonal polynomials.

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

“…P z are orthogonal polynomials on the real line with 2 0 ( ) f z as their positive definite weight function [1,14,15].…”

“… . Therefore, the un-normalized solution of the differential equation (1) for this class reads as follows…”

“…The "Tridiagonal Representation Approach" (TRA) is an algebraic method that was established recently as a physics tool to solve quantum mechanical problems [1]. It was inspired by the Jmatrix method of scattering [2][3][4].…”

“…However,  should always be a negative real number. In section 2, we substitute the series (2) with the basis elements (3) into the differential equation (1) and identify the scenarios where three-term recursion relations for the expansion coefficients   n f are produced. In sections 3 to 6, we obtain the recursion coefficients for each scenario and identify the orthogonal polynomials that satisfy the corresponding three-term recursion relation.…”

“…Finally, in section 7, we provide a physical application of our findings where we transform the Schrödinger equation into Eq. (1) and hence identify the corresponding potential function, energy and wavefunction.…”