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.Comment: 13 page