Uniform semiclassical calculation of resonance energies and widths near a barrier maximum

Abstract

The accuracy of semiclassical techniques for calculating time delay resonance energies and widths at energies near a barrier maximum is investigated. Starting from the semiclassical phase shift that is uniformly valid at the barrier maximum, the corresponding time delay function is derived. The semiclassical time delay resonance energies and widths are in good agreement with accurate quantum results for energies below, at and above the barrier maximum. A new approximate formula for the width is derived that is uniformly valid at the barrier maximum. The relation between the new formula and other approximate formulae in the literature is discussed. To obtain results that are uniformly valid at the barrier maximum it is important not to introduce any additional approximations into the semiclassical theory. The discrepancies recently reported by LeRoy and Liu are shown to arise from additional approximations in their formulae or from the use of a different set of boundary conditions in the semiclassical analysis.