Enlarging the class of exactly solvable nonrelativistic problems

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.