On the Inverse Problem in Quantum Scattering Theory

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-Weyl 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 completenesb 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.

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