Quantum Hamiltonians with magnetic fields: effective dynamics and transport properties
Typ dokumentu
habilitační prácehabilitation thesis
Autor
Tušek, Matěj
Metadata
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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.
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