  We choose a complete set of square integrable functions as basis for the
expansion of the wavefunction in configuration space such that the matrix
representation of the nonrelativistic time-independent wave operator is
tridiagonal and symmetric. Consequently, the matrix wave equation becomes a
symmetric three-term recursion relation for the expansion coefficients of the
wavefunction in this basis. The recursion relation is then solved exactly in
terms of orthogonal polynomials in the energy. Some of these polynomials are
not found in the mathematics literature. The asymptotics of these polynomials
give the phase shift of the continuous energy scattering states and the
spectrum for the discrete energy bound states. Depending on the space and
boundary conditions, the basis functions are written in terms of either the
Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number
of potential functions that yield exact solutions of the wave equation.
Nonetheless, the class of exactly solvable problems in this approach is larger
than the conventional class (see Table 12). We also give very accurate results
for cases where the wave operator matrix is not tridiagonal but its elements
could be evaluated either exactly or numerically with high precision.
