Enlarging the class of exactly solvable nonrelativistic problems

TL;DR

This paper introduces a method to find exactly solvable nonrelativistic quantum problems by using non-diagonal Hamiltonian representations, leading to new potentials, solutions, and orthogonal polynomials.

Contribution

It generalizes existing solvable models by allowing non-diagonal Hamiltonian matrices and discovers new solution spaces and orthogonal polynomials for quantum problems.

Findings

Larger class of exactly solvable potentials identified

New orthogonal polynomials discovered as solutions

Analytic solutions include both discrete and continuous spectra

Abstract

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support a tridiagonal matrix representation of the wave operator. Doing so results in exactly solvable problems with a class of potentials which is larger than, and/or generalization of, what is already known. In addition, we found new representations for the solution space of some well known potentials. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. Some of these solutions are new kind of orthogonal polynomials. The examples given, which are not exhaustive, are for problems in one and three dimensions. The analytic solutions obtained by this approach include the discrete as well as the continuous spectrum of the Hamiltonian.