Efficient calculation of bands of eigenvalues and eigenvectors in one dimension

It is suggested that the most efficient way of calculating large numbers of sequential eigenvalues and eigenvectors lying between specified limits is through the replacement of the second-order differential operator by the second central difference, followed by a matrix formulation of the eigenvalue problem. The capabilities of this neglected method are illustrated by the calculation of shape resonances in scattering and of Franck–Condon factors in the spectroscopy of high vibrational states.