The Lippmann–Schwinger Formula and One Dimensional Models with Dirac Delta Interactions
We show how a proper use of the Lippmann–Schwinger equation simplifies the calculations to obtain scattering states for one dimensional systems perturbed by N Dirac delta equations. Here, we consider two situations. In the former, attractive Dirac deltas perturbed the free one dimensional Schrödinger Hamiltonian. We obtain explicit expressions for scattering and Gamow states. For completeness, we show that the method to obtain bound states use comparable formulas, although not based on the Lippmann–Schwinger equation. Then, the attractive N deltas perturbed the one dimensional Salpeter equation. We also obtain explicit expressions for the scattering wave functions. Here, we need regularisation techniques that we implement via heat kernel regularisation.
KeywordsScattering states Schrödinger and Salpeter one dimensional Hamiltonians Contact perturbations Gamow wave functions Lippmann–Schwinger equation
We dedicate this paper to Professor Véronique Hussin for her contributions to science and her friendship. The present work has been fully financed by TUBITAK from Turkey under the “2221 - Visiting Scientist Fellowship Programme”. We are very grateful to TUBITAK for this support. We also acknowledge Osman Teoman Turgut for clarifying discussions and his interest in the present research. This work was also sponsored by the Ministerio de Economía y Competitividad of Spain (Project No. MTM2014-57129-C2-1-P with EU-FEDER support) and the Junta de Castilla y León (Projects VA057U16, VA137G18 and BU229P18).
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