Ground State Properties in the Quasi-Classical Regime

TL;DR

This paper investigates the ground state energy of quantum particles coupled with Bose fields in the quasi-classical limit, showing convergence to a nonlinear functional and characterizing the ground states.

Contribution

It establishes the convergence of the microscopic ground state energy to a quasi-classical nonlinear functional and describes the limiting behavior of ground states.

Findings

Ground state energy converges to a nonlinear quasi-classical functional.

Ground states concentrate on minimizers of the quasi-classical energy.

The results connect microscopic models with effective classical descriptions.

Abstract

We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations,and thus behaves in a semiclassical way, while the non-relativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to the one of a nonlinear quasi-classical functional depending on both the particles' wave function and the classical configuration of the field. Equivalently, this energy can be interpreted as the lowest energy of a Pekar-like functional with an effective nonlinear interaction for the particles only. If the particles are confined, the ground state of the microscopic system converges as well, to a probability measure concentrated on the set of minimizers of the quasi-classical energy.