Isotropic contact forces in arbitrary representation: heterogeneous few-body problems and low dimensions

TL;DR

This paper develops a formalism for modeling short-range contact forces in arbitrary spatial dimensions, applying it to few-body problems, dimensional reduction, and Efimov physics, with new integral equations and universal results.

Contribution

It introduces a generalized Bethe-Peierls approach in arbitrary dimensions, deriving integral equations and conditions for few-body systems, including Efimov states and low-dimensional reductions.

Findings

Derived integral equations for few-body contact interactions.

Established a nodal condition to prevent Thomas collapse in Efimov physics.

Computed the 2D '3+1' bosonic ground state as a function of mass ratio.

Abstract

The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for three-body systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov Ter-Martirosian equation and the Thomas collapse is avoided. Four-body bound state eigenequations are derived and the 2D '3+1' bosonic ground state is computed as a function of the mass ratio.