Quantum Hamiltonians with magnetic fields: effective dynamics and transport properties

Abstract

This thesis deals with both non-relativistic and relativistic quantum Hamiltonians with mag netic elds constrained to a hypersurface or its tubular neighbourhood. In the non-relativistic case, the magnetic Laplacian will be considered, whereas in the relativistic situation we will be concerned with the Dirac operator with magnetic eld. Firstly, we will look for an e ective operator for the magnetic Laplacian on a very thin neighbourhood of the hypersurface. We will show that, as the width of the neighbourhood tends to zero, the limit operator is the magnetic Laplace-Beltrami operator on the hypersurface with an additional scalar potential, which may be expressed in terms of the principal curvatures of the hypersurface. If the hypersurface is embed ded into R 3 then the e ective magnetic eld is given as the projection of the ambient magnetic eld to the normal direction. Next, we will focus on the two-dimensional magnetic Laplacian. It has been proved for a variety of translationally invariant magnetic elds that the spectrum of this operator is purely absolutely continuous. We will show that this still may be true even after adding a translationally invariant electrostatic perturbation. Moreover, we will prove the absolute continuity of the Laplacian with constant magnetic eld on neighbourhoods of certain curved translationally invariant two-dimensional hypersurfaces. Recall that the absolutely con tinuous spectrum is typically associated with transport properties of a model. Besides, for many two-dimensional models with a translationally invariant magnetic barrier, there exists a lower bound on currents along the barrier. We will show that these currents may be carried by states that disperse slowly or not at all and we will nd several su cient conditions for existence of such states in the relativistic case. If the vector potential that is associated with the magnetic barrier is of very thin but high pro le, it seems reasonable to work formally with the simple layer distribution instead of the true, possibly complicated, potential. Moreover, due to the symme try with respect to translations, it is possible to consider one-dimensional Dirac operators after employing the partial Fourier transform. Therefore, it makes sense to be concerned with the one-dimensional relativistic point interaction and its approximations by more realistic regular potentials. We will provide an approximation result in the norm resolvent sense for any type of the point interaction. Finally, we will introduce rigorously as a self-adjoint operator the two dimensional Dirac operator with potential that is proportional to the simple layer distribution supported on a closed curve and, similarly as in the one-dimensional setting, we will solve the problem of approximations by regular potentials. We will investigate in more detail the case when the singular potential may be associated with a vector potential, i.e., we will introduce a sort of magnetic δ-shell interaction.