  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.
