Generalization of the Fermi Pseudopotential

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.