Siegert pseudostates: completeness and time evolution

Abstract

Within the theory of Siegert pseudostates, it is possible to accurately calculate bound states and resonances. The energy continuum is replaced by a discrete set of states. Many questions of interest in scattering theory can be addressed within the framework of this formalism, thereby avoiding the need to treat the energy continuum. For practical calculations it is important to know whether a certain subset of Siegert pseudostates comprises a basis. This is a nontrivial issue, because of the unusual orthogonality and overcompleteness properties of Siegert pseudostates. Using analytical and numerical arguments, it is shown that the subset of bound states and outgoing Siegert pseudostates forms a basis. Time evolution in the context of Siegert pseudostates is also investigated. From the Mittag-Leffler expansion of the outgoing-wave Green's function, the time-dependent expansion of a wave packet in terms of Siegert pseudostates is derived. In this expression, all Siegert pseudostates--bound, antibound, outgoing, and incoming--are employed. Each of these evolves in time in a nonexponential fashion. Numerical tests underline the accuracy of the method.

References in corpus (4)

• Properties of metastable alkaline-earth-metal atoms calculated using an accurate effective core potential
• Comment on "Evolution of a Quasi-Stationary State"
• Competition among molecular fragmentation channels described with Siegert channel pseudostates
• Comment on ``Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities''

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• Siegert pseudostate perturbation theory: one- and two-threshold cases
• Counterexample to the Bohigas Conjecture for Transmission Through aOne-Dimensional Lattice
• Average Atom Model with Siegert States
• Orbital Variational Adiabatic Hyperspherical Method Applied to Bose-Einstein Condensates
• Cosmological "constant" in a universe born in the metastable false vacuum state