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Ground state properties in the quasiclassical regime

Michele Correggi, Marco Falconi and Marco Olivieri

Vol. 16 (2023), No. 8, 1745–1798
Abstract

We study the ground state energy and ground states of systems coupling nonrelativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasiclassical 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 nonrelativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to that of a nonlinear quasiclassical 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 quasiclassical energy.

Keywords
quasiclassical limit, interaction of matter and light, semiclassical analysis
Mathematical Subject Classification
Primary: 81Q20, 81T10
Secondary: 81Q05, 81Q10, 81S30, 81V10
Milestones
Received: 13 November 2020
Revised: 2 November 2021
Accepted: 14 February 2022
Published: 16 October 2023
Authors
Michele Correggi
Dipartimento di Matematica
Politecnico di Milano
Milan
Italy
Marco Falconi
Dipartimento di Matematica
Politecnico di Milano
Milan
Italy
Marco Olivieri
Fakultät für Mathematik
Karlsruhe Institut für Technologie
Karlsruhe
Germany
Department of Mathematics
Aarhus Universitet
Aarhus
Denmark

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