Exact solutions for the three-dimensional schrödinger equation with quasi-local potentials obtained from a three-dimensional Gel'fand-Levitan equation. Examples of totally reflectionless scattering
In an early paper on the inverse scattering problem for the three-dimensional Schrödinger equation using a Gel'fand-Levitan equation, Kay and Moses introduced nonlocal potentials which in the present paper are called “quasi-local.” These potentials are diagonal in the radial variable, but are integral operators in the angular variables and represent a generalization of the usual local potential. In the early paper, we were unable to give explicit potentials for which the Schrödinger equation could be solved, since we did not solve the corresponding Gel'fand-Levitan equation. In the present paper, we introduce another Gel'fand-Levitan equation for which many solutions can be found. Each solution yields a quasi-local potential for which the corresponding threedimensional Schrödinger equation can be solved in closed form. As far as the author knows, these are the first potentials, local or nonlocal, other than separable potentials, for which the Schrödinger equation can be solved in closed form. They should be useful in testing hypotheses of formal scattering theory.
In the present paper, examples of quasi-local potentials are given which support point eigenvalues and for which there is no scattering whatever. These potentials are analogues of the reflectionless potentials of the one-dimensional problem.
Finally, we indicate how the scattering operator can be found from a wave operator satisfying any boundary or initial or final value conditions and the corresponding completeness relation. This result enables us to obtain the scattering operator from spectral data for the three-dimensional problem of the present paper.
KeywordsLocal Potential Wave Operator Radial Variable Schr6dinger Equation Radial Equation
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Footnotes and References
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