Tan's adiabatic sweep theorem from the variational theorem for the scattering length

TL;DR

This paper derives Tan's adiabatic sweep theorem using a variational approach based on the Hellmann-Feynman theorem, providing a new perspective on universal many-body systems and their energy relations.

Contribution

It introduces a novel derivation of Tan's theorem via a variational theorem for the scattering length, extending its applicability to bosonic and fermionic systems.

Findings

Derived Tan's adiabatic sweep theorem using variational methods.

Obtained mean kinetic and interaction energies for bosonic systems.

Established virial theorem for homogeneous and trapped bosons.

Abstract

It is shown that variation of the one-particle dispersion in a universal many-body system enables us to obtain Tan's adiabatic sweep theorem and its generalization. The derivation is based on the Hellmann-Feynman theorem and the variational theorem for the scattering length suggested in our previous paper [Cherny and Shanenko, Phys. Rev. E 62, 1646 (2000)]. As an example, the universal effects in the system of spinless bosons are considered. With the help of the variational theorem, we obtain the mean kinetic and interaction energies and derive the virial theorem for the homogeneous and trapped bosons. The results can easily be generalized to the two-component fermions with interactions between opposite spins.