Time-Dependent Quantum-Mechanical Approaches to the Continuous Spectrum: Scattering Resonances in a Finite Box

Several novel aspects of scattering resonances are studied. An expression, valid for a finite box, relating the continuum phase shift with the energy shift and unperturbed level separation is proposed and applied to obtain the resonance parameters. The effect of the resonance on propagating a wavepacket in imaginary time is studied. It is observed that the resonance strongly affects the cumulants of the energy distribution. In particular, a local minimum of the first derivative of the energy with respect to time (proportional to the second cumulant) serves to estimate the resonance energy and wavefunction. Once the estimate is known, the autocorrelation function is used to evaluate the resonance width. Alternatively, a new iterative approach is developed that is capable of selectively yielding an arbitrary band of energy eigenvalues and eigenfunctions on a grid. This method is applied to give those energy levels that are of interest for the discrete computation of the resonant phase shift, i.e., those close to resonance. Exact (analytical) and approximate results are in good agreement for a particular separable potential model in one dimension. These methods can be extended to realistic potentials in higher dimensions.