Random Schrödinger operators with magnetic vortices

Abstract

We study magnetic Schrödinger operators with Aharonov-Bohm type potentials. These Hamiltonians describe the interaction of a particle in the two-dimensional plane with one or more magnetic flux tubes or vortices perpendicular to the plane. First, we consider the question of self-adjointness of the single vortex Aharonov-Bohm Hamiltonian, which is known to have a four-parameter family of self-adjoint extensions. We examine whether any of the self-adjoint extensions of this operator may be obtained as the limit of the Pauli Hamiltonians for a finite-width flux tube. We show that two one-parameter subfamilies may be obtained in this way, with convergence in the norm resolvent sense to the corresponding self-adjoint extension. The same applies when we approximate by smooth flux tubes. We then study this operator using the Feynman-Kac-Ito path integral theorem for magnetic Schrödinger operators. We show that this theorem can be applied to operators with a singular Aharonov-Bohm type vector potential, and apply it to studying the density of states of the operator with one or more magnetic vortices. We then consider two models of random magnetic operators. First, we consider a model where infinitely thin magnetic flux tubes with random flux are distributed on a lattice. We find that the almost sure spectrum of this operator is the positive half line and prove the existence of the integrated density of states. Secondly, we exploit the work on smooth approximations to study the case where the potential consists of a Poisson distribution of these smooth flux tubes with cutoffs. We prove that the almost sure spectrum is not bounded below. We also obtain bounds for the asymptotic behaviour of the density of states at the bottom of the spectrum.