Continuum eigenfunction expansions and resonances: A simple model

A one-dimensional square-well model of an atom in an electric field is solved exactly. This problem is instructive for two reasons: (1) The Hamiltonian has a continuous spectrum, but the eigenfunctions do not reduce asymptotically to plane waves. Therefore the general Titchmarsh–Kodaira theory of eigenfunction expansions is needed to normalize the eigenfunctions properly. The normalization is expressed by a spectral density function, ρ(E). (2) ρ(E) exhibits ’’bumps’’ at values of the energy at which the electron wave function’s amplitude inside the well is particularly large. These are resonant states of the system. The gradual sharpening of the resonances into discrete bound states as the electric field is turned off is demonstrated.