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Derivation of the Maxwell-Schrödinger equations from the Pauli-Fierz Hamiltonian. (English) Zbl 1455.35210

The Pauli-Fierz Hamiltonian describes a quantum system of identical, non-relativistic particles which are coupled to a quantized electromagnetic field. The authors study the time evolution of this system in a mean-field limit, where the number of particles \(N\) becomes large and the coupling to the radiation field is scaled by \(N^{-1/2}\). At time zero, it is assumed that there is a Bose-Einstein condensate. As \(N\to\infty\), the authors show that the time evolution preserves the condensate and that it can be approximated by the Maxwell-Schrödinger system. In order to obtain these results, it is assumed that the repulsive interaction potential is positive, real, and even, such that the Pauli-Fierz Hamiltonian is self-adjoint on its domain. It is also assumed that there is enough regularity on solutions of the Maxwell-Schrödinger system.

MSC:

35Q40 PDEs in connection with quantum mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
78A35 Motion of charged particles

Citations:

Zbl 1242.81150
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References:

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