Tridiagonal Representation Approach in Quantum Mechanics

TL;DR

The paper introduces the Tridiagonal Representation Approach (TRA), an algebraic method inspired by the J-matrix technique, for solving quantum wave equations exactly using orthogonal polynomials, expanding solvable problem classes.

Contribution

It presents a novel algebraic framework that broadens the class of exactly solvable quantum problems through orthogonal polynomial theory.

Findings

Larger class of exactly solvable problems identified

Physical properties derived directly from orthogonal polynomials

Simplified computation of spectra and scattering data

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.