Modeling atom-atom interactions at low energy by Jost-Kohn potentials
arXiv:1902.02177 · doi:10.1088/1361-6455/ab2378
Abstract
More than 65 years ago, Jost and Kohn [R. Jost and W. Kohn, {Phys. Rev.} {\bf 87}, 977 (1952)] derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the -wave scattering length being negative. For , they calculated another class of short-range model potentials [R. Jost and W. Kohn, { Dan. Mat. Fys. Medd} {\bf 27}, 1 (1953)] using a method based on an adaptation from Gelfand-Levitan theory [I. M. Gel'fand and B. M. Levitan, { Dokl. Akad. Nauk. USSR} {\bf 77}, 557-560 (1951)] of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost-Kohn potentials can account for zero-energy resonances. The -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; and below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check, we carry out some simple calculations to show that Jost-Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.
13 pages 5 figures. In this version, two new references are added and an error in the format of the paper is corrected
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