The variational theorem for the scattering length in low dimensions and its applications to universal systems

TL;DR

This paper extends the variational theorem for scattering length to low-dimensional systems, providing a new approach to analyze universal many-body bosonic systems with arbitrary interactions.

Contribution

It introduces an extension of the variational theorem to one and two dimensions and applies it to universal bosonic systems, including a generalized Tan adiabatic sweep theorem.

Findings

Extended variational theorem applicable to 1D and 2D systems.

Derived a generalized Tan adiabatic sweep theorem.

Calculated pair distribution functions at short distances.

Abstract

The variational theorem for the scattering length [Cherny and Shanenko, Phys. Rev. E 62, 1646 (2000)] is extended to one and two dimensions. It is shown that the arising singularities can be treated in terms of generalized functions. The variational theorem is applied to a universal many-body system of spinless bosons. The extended Tan adiabatic sweep theorem is obtained for interacting potentials of arbitrary shape with the variation of the one-particle dispersion. The pair distribution function is calculated at short distances by means of the variation of the potential. The suggested scheme is based on simple quantum mechanics; it is physically transparent and free from any divergence.