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

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 in [S. Tan, Ann. Phys. (NY) 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 cutoff version of the pseudopotential is also provided, and with this no apparent contradiction to 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 cutoff pseudopotential.