Electron in a Magnetic Field
Laurent Amour, J´ er´ emy Faupin, Benoˆ ıt Gr´ ebert, and Jean-Claude Guillot
Abstract. We consider a non-relativistic electron interacting with a classical
magnetic ﬁeld pointing along the x3 -axis and with a quantized electromagnetic
ﬁeld. The system is translation invariant in the x3 -direction and we consider
the reduced Hamiltonian H(P3 ) associated with the total momentum P3 along
the x3-axis. For a ﬁxed momentum P3 suﬃciently small, we prove that H(P3 )
has a ground state in the Fock representation if and only if E (P3) = 0, where
P3 → E (P3) is the derivative of the map P3 → E(P3 ) = inf σ(H(P3)). If
E (P3) = 0, we obtain the existence of a ground state in a non-Fock represen-
tation. This result holds for suﬃciently small values of the coupling constant.
In this paper we pursue the analysis of a model considered in [AGG1], de-
scribing a non-relativistic particle (an electron) interacting both with the quan-
tized electromagnetic ﬁeld and a classical magnetic ﬁeld pointing along the x3-axis.
An ultraviolet cutoﬀ is imposed in order to suppress the interaction between the
electron and the photons of energies bigger than a ﬁxed, arbitrary large parameter
Λ. The total system being invariant by translations in the x3-direction, it can be
seen (see [AGG1]) that the corresponding Hamiltonian admits a decomposition of
the form H
H(P3)dP3 with respect to the spectrum of the total momentum
along the x3-axis that we denote by P3. For any given P3 suﬃciently close to 0,
the existence of a ground state for H(P3) is proven in [AGG1] provided an in-
frared regularization is introduced (besides a smallness assumption on the coupling
parameter). Our aim is to address the question of the existence of a ground state
without requiring any infrared regularization.
The model considered here is closely related to similar non-relativistic QED
models of freely moving electrons, atoms or ions, that have been studied recently
(see [BCFS, FGS1, Hi, CF, Ch, HH, CFP, FP] for the case of one single
electron, and [AGG2, LMS, FGS2, HH, LMS2] for atoms or ions). In each of
these papers, the physical systems are translation invariant, in the sense that the
associated Hamiltonian H commutes with the operator of total momentum P . As
a consequence, H
H(P )dP , and one is led to study the spectrum of the ﬁber
Hamiltonian H(P ) for ﬁxed P ’s.
1991 Mathematics Subject Classiﬁcation. 81V10, 81Q10, 81Q15.
Volume 500, 2009