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 finitelymany non-relativistic charged particles interactingwith 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 systemto the infimum over all possible classical field configurations of the ground state energy of the effective Schrödinger operator.