A novel algebraic solution to Schrödinger equations

We obtain approximations to some bound-state solutions of Schrödinger equations with a variety of central potentials V(r) without any direct calculation of the matrix elements of V(r). The method involves solutions of two algebraic eigenvalue problems, one for a real tridiagonal matrix, and one for a more general real symmetric matrix. The eigenvalues, calculated from matrices of dimension less than 64, generally converge rapidly towards their correct limiting values.