On the Infrared Problem for the Dressed Non-Relativistic Electron in a Magnetic Field

Abstract

We consider a non-relativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. The system is translation invariant in the $x_3$-direction and we consider the reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the $x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that $H(P_3)$ has a ground state in the Fock representation if and only if $E^{\prime }(P_3)=0$, where $P_3\mapsto E^{\prime }(P_3)$ is the derivative of the map $P_3\mapsto E(P_3)=\inf \sigma (H(P_3))$. If $E^{\prime }(P_3)\ne 0$, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant.

References in corpus (1)

• Existence and non existence of a ground state for the massless Nelson model under binding condition