Tan's distributions and Fermi-Huang pseudopotential in momentum space

TL;DR

This paper explicitly derives the momentum space form of Tan's distributions and Fermi-Huang pseudopotential, simplifying the derivation of Tan's universal relations for Fermi gases and clarifying their mathematical consistency.

Contribution

It provides an explicit form of the momentum representation of Tan's distributions and pseudopotential, including operator forms and a momentum cut-off version, clarifying their relation to renormalization.

Findings

Explicit Fourier transform of Tan's distribution obtained

Simplified operator forms for Tan's selectors derived

Equivalence between pseudopotential approach and momentum-space renormalization demonstrated

Abstract

The long-standing question of finding the momentum representation for the s-wave zero-range interaction in three spatial dimensions is here solved. This is done by expressing a certain distribution, introduced in a formal way by S. Tan [Ann. Phys. 323, 2952 (2008)], explicitly. The resulting form of the Fourier-transformed pseudopotential remains very simple. Operator forms for the so-called Tan's selectors which, together with Fermi-Huang pseudopotential, largely simplify the derivation of Tan's universal relations for the Fermi gas, are here derived and are also very simple. A momentum cut-off version of the pseudopotential is also provided, and with this no apparent contradiction with the notion of integrals in Tan's methods is left. The equivalence, even at the intermediate step level, between the pseudopotential approach and momentum-space renormalization of the bare Dirac delta interaction is then shown by using the explicit form of the cut-off pseudopotential.