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Correggi, Michele; Falconi, Marco; Olivieri, Marco

Ground state properties in the quasiclassical regime. (English) Zbl 1538.81023

Anal. PDE 16, No. 8, 1745-1798 (2023).
Summary: 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.

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MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81T10 Model quantum field theories
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics

Keywords:

quasiclassical limit; interaction of matter and light; semiclassical analysis
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