Solution of the wave equation in a tridiagonal representation space

TL;DR

This paper introduces a method for solving the 1D Schrödinger equation by expanding solutions in a basis that yields a tridiagonal matrix representation, enabling the derivation of solvable potentials and their wavefunctions.

Contribution

It develops a novel approach using orthogonal polynomial expansions in a basis with a tridiagonal wave operator matrix, unifying bound and scattering state solutions.

Findings

Derived classes of solvable potentials.

Expressed wavefunctions as series of orthogonal polynomials.

Unified treatment of bound and scattering states.

Abstract

We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis support an infinite tridiagonal matrix representation of the wave operator. By this requirement, we deduce a class of solvable potentials along with their corresponding bound states and stationary wavefunctions expressed as infinite series in terms of these bases. This approach allows for simultaneous treatment of the discrete (bound states) as well as the continuous (scattering states) spectrum on the same footing. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, illustrate the power of this approach in dealing with 1D quantum problems.