Stabilization Method of Calculating Resonance Energies: Model Problem

We have applied the stabilization method of calculating resonance energies to the elastic scattering from a one-dimensional model potential containing a barrier. For sufficiently large basis sets, the stabilization method yields good approximations to the inner part of the exact scattering wave functions at energies equal to the eigenvalues of the truncated matrix of the Hamiltonian in both the resonant and nonresonant energy regions. We have calculated good approximations to the exact phase shifts from the square-integrable wave functions produced by the stabilization method. We have derived a simple model to explain the behavior of the eigenvalues as a function of the size of the basis. The degree of stability of the eigenvalues approximating the resonance energy is proportional to the width of the resonance. Both the energy and the width of the resonance can be calculated from the change in the stable eigenvalue as the size of the basis increases.