dorsal/arxiv
View SchemaSiegert pseudostates: completeness and time evolution
| Authors | Robin Santra, Jeffrey M. Shainline, Chris H. Greene |
|---|---|
| Categories | |
| ArXiv ID | physics/0408051 |
| URL | https://arxiv.org/abs/physics/0408051 |
| DOI | 10.1103/PhysRevA.71.032703 |
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.
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"abstract": "Within the theory of Siegert pseudostates, it is possible to accurately\ncalculate bound states and resonances. The energy continuum is replaced by a\ndiscrete set of states. Many questions of interest in scattering theory can be\naddressed within the framework of this formalism, thereby avoiding the need to\ntreat the energy continuum. For practical calculations it is important to know\nwhether a certain subset of Siegert pseudostates comprises a basis. This is a\nnontrivial issue, because of the unusual orthogonality and overcompleteness\nproperties of Siegert pseudostates. Using analytical and numerical arguments,\nit is shown that the subset of bound states and outgoing Siegert pseudostates\nforms a basis. Time evolution in the context of Siegert pseudostates is also\ninvestigated. From the Mittag-Leffler expansion of the outgoing-wave Green\u0027s\nfunction, the time-dependent expansion of a wave packet in terms of Siegert\npseudostates is derived. In this expression, all Siegert pseudostates--bound,\nantibound, outgoing, and incoming--are employed. Each of these evolves in time\nin a nonexponential fashion. Numerical tests underline the accuracy of the\nmethod.",
"arxiv_id": "physics/0408051",
"authors": [
"Robin Santra",
"Jeffrey M. Shainline",
"Chris H. Greene"
],
"categories": [
"physics.atom-ph"
],
"doi": "10.1103/PhysRevA.71.032703",
"title": "Siegert pseudostates: completeness and time evolution",
"url": "https://arxiv.org/abs/physics/0408051"
},
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