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On the inverse problem in quantum scattering theory
Author(s) -
Brändas Erkki,
Engdahl Erik,
Rittby Magnus,
Elander Nils
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560280842
Subject(s) - completeness (order theory) , mathematics , quantum inverse scattering method , complex plane , scattering , inverse , inverse problem , inversion (geology) , scaling , inverse scattering problem , scattering theory , mathematical analysis , pure mathematics , theoretical physics , inverse scattering transform , quantum mechanics , physics , paleontology , geometry , structural basin , biology
The Gel'fand‐Levitan formulation of the inverse problem in quantum scattering theory is discussed with respect to completeness and analytic extensions. The classic Green's function and the associated completeness relation are analyzed within the Titchmarsh‐Wcyl framework. An attractive feature of the Titchmarsh‐Weyl formulation concerns the possibility to invoke complex scaling to a rather general set of potentials in order to expose resonance structures in the complex plane. In addition this procedure allow for an analytic extension of the classic Green's function and the associated completeness relation. The generalized completeness relation can be used to construct the kernels of the Gel'fand‐Levitan integral equation. In addition to supplying a possibility for testing completeness properties of generalized expansions one may also find inversion formulas for potentials that exhibit analytic extensions to some sector in the complex plane. As a test we have analyzed a simple exponential potential which was found to contain a whole string of complex energy resonances with the resulting generalized spectral density being subjected to a particular deflation property.