François Bentosela

ANOMALOUS ELECTRON TRAPPING BY LOCALIZED MAGNETIC FIELDS

We consider an electron with an anomalous magnetic moment g > 2 confined to a plane and interacting with a non-zero magnetic field B perpendicular to the plane. We show that if B has compact support and the magnetic flux in natural units is , the corresponding Pauli Hamiltonian has at least bound states, without making any assumptions about the field profile. Furthermore, in the zero-flux case there is a pair of bound states with opposite spin orientations. Using a Birman-Schwinger technique, we extend the last claim to a weak rotationally symmetric field with , thus correcting a recent result. Finally, we show that under mild regularity assumptions existence of the bound states can also be proved for non-symmetric fields with tails.

Absolute Continuity in Periodic Thin Tubes and Strongly Coupled Leaky Wires

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive δ interaction supported by a periodic curve in Rd = 2, 3.

Electron trapping by a current vortex

We investigate an electron in the plane interacting with the magnetic field due to an electric current forming a localized rotationally symmetric vortex. We show that independently of the vortex profile an electron with spin antiparallel to the magnetic field can be trapped if the vortex current is strong enough. In addition, the electron scattering on the vortex exhibits resonances for any spin orientation. On the other hand, in distinction to models with a localized flux tube this situation exhibits no bound states for weak vortices.

The dynamics of one-dimensional Bloch electrons in constant electric fields

We study the dynamics of a one-dimensional Bloch electron subjected to a constant electric field. The periodic potential is supposed to be less singular than the δ-like potential (Dirac comb). We give a rigorous proof of Ao’s result that for a large class of initial conditions (high momentum regime) there is no localization in momentum space. The proof is based on the mathematical substantiation of the two simplifying assumptions made in physical literature: the transitions between far away bands can be neglected and the transitions at the quasicrossing can be described by Landau–Zener-type formulas. Using the connection between the above model and the driven quantum ring (DQR) shown by Avron and Nemirovski, our results imply the increase of energy for weakly singular such DQR and appropiate initial conditions.

Anomalous electron trapping by magnetic flux tubes and electric current vortices

We consider an electron with an anomalous magnetic moment, g > > 2, confined to a plane and interacting with a nonhomogeneous magnetic field B, and investigate the corresponding Pauli Hamiltonian. We prove a lower bound on the number of bound states for the case when B is of a compact support and the related flux is N + ∈, ∈∈(0, 1]. In particular, there are at least N + 1 bound states if B does not change sign. We also consider the situation where the magnetic field is due to a localized rotationally symmetric electric current vortex in the plane. In this case the flux is zero; there is a pair of bound states for a weak coupling, and higher orbital-momentum “spin-down” states appearing as the current strength increases.

On the guided states of 3D biperiodic Schrodinger operators

We consider the Laplacian operator H 0: = − Δ perturbed by a non-positive potential V, which is periodic in two directions, and decays in the third one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H: = H 0 + V in the periodic variables. If V is sufficiently small and decreases fast enough in the third direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay in the third direction faster than any rational function without real pole.