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arXiv · 1806.05726

Generalization of the Fermi Pseudopotential

Trang T. Le·Zach Osman·D. K. Watson·Martin Dunn·B. A. McKinney
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Abstract

Introduced eighty years ago, the Fermi pseudopotential has been a powerful concept in multiple fields of physics. It replaces the detailed shape of a potential by a delta-function operator multiplied by a parameter giving the strength of the potential. For Cartesian dimensions $d>1$, a regularization operator is necessary to remove singularities in the wave function. In this study, we develop a Fermi pseudopotential generalized to $d$ dimensions (including non-integer) and to non-zero wavenumber, $k$. Our approach has the advantage of circumventing singularities that occur in the wave function at certain integer values of $d$ while being valid arbitrarily close to integer $d$. In the limit of integer dimension, we show that our generalized pseudopotential is equivalent to previously derived $s$-wave pseudopotentials. Our pseudopotential generalizes the operator to non-integer dimension, includes energy ($k$) dependence, and simplifies the dimension-dependent coupling constant expression derived from a Green's function approach. We apply this pseudopotential to the problem of two cold atoms ($k\to0$) in a harmonic trap and extend the energy expression to arbitrary dimension.

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Trang T. Le, Zach Osman, D. K. Watson, Martin Dunn, B. A. McKinney. 2018-06-14. Generalization of the Fermi Pseudopotential. https://doi.org/10.1088/1402-4896%2Fab0811

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