Applying Bogomolny's quantization method to generic classical systems

Semiclassical treatment of Feshbach resonances by transfer matrices

A semiclassical method is presented for the calculation of Feshbach resonance positions and widths. This approach, based on semiclassical transfer matrices, relies only on relatively short trajectory fragments, thus avoiding problems associated with the long trajectories needed in more straightforward semiclassical techniques. Complex resonance energies are obtained from an implicit equation that is developed to compensate for the inaccuracy of the stationary phase approximation underlying the semiclassical transfer matrix applications. Although this treatment requires calculation of transfer matrices for complex energies, an initial value representation method makes it possible to extract such quantities from ordinary real-valued classical trajectories. This treatment is applied to obtain positions and widths for resonances in a model two-dimensional system, and the results are compared to those obtained from accurate quantum mechanical calculations. The semiclassical method successfully captures the irregular energy dependence of resonance widths that vary over a range of more than two orders of magnitude. An explicit semiclassical expression for the width of narrow resonances is also presented and serves as a simpler, useful approximation for many cases.