Widths and spacing of Stark ladder levels

The problem of an electron in a one-dimensional finite or semi-infinite periodic Kronig-Penney potential with an applied homogeneous electric field is treated. A very accurate approximation is used in which the relevant $S$ matrix can easily be computed as an analytic function in the complex energy plane. The positions and widths of energy levels are calculated. The existence of a discrete Stark ladderlike structure in the energy spectrum is confirmed. The levels associated with each band are not exactly equally spaced, but at higher energies they approach the equal spacing originally predicted by Wannier. The widths of levels associated with any given band are found to be smaller than the level spacing.