Magnetic Schrödinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models

Abstract

We study the quasi-classical limit of the Pauli-Fierz model: the system is composed of finitely many non-relativistic charged particles interacting with a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schrödinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic system to the infimum over all possible classical field configurations of the ground state energy of the effective Schrödinger operator.

26 pages, pdfLatex. Final version to appear in J. Spectr. Theory

References in corpus (5)

• Classical limit of the Nelson model with cut off
• Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit
• Concentration of cylindrical Wigner measures
• Quantum mean field asymptotics and multiscale analysis
• Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

Cited by in corpus (7)

• Ground State Properties in the Quasi-Classical Regime
• Quasi-Classical Dynamics
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• Microscopic Derivation of Time-dependent Point Interactions
• Quasi-classical Limit of a Spin Coupled to a Reservoir
• Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields
• Differential Equations of Quantum Mechanics