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Magnetic Transport Along Translationally Invariant Obstacles

A thesis by Michal Grňo, supervised by prof. Pavel Exner.

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Abstract

We discuss the spectra of magnetic Schrödinger operators of the form ${(-\mathrm i\nabla + \vec{A}(x))^2 + V(x)}$ on ${L^2(\Omega)}$, where $\Omega$ is either $\mathbb R^2$, a half-plane, a strip, or a thin layer in $\mathbb R^3$. This class of operators includes the Landau Hamiltonian, as well as the Iwatsuka Hamiltonian and other translationally invariant perturbations of it. We provide a comprehensive list of the known results regarding these operators, and inspect two Hamiltonians that have not yet been studied: the Landau Hamiltonian with a $\delta$-interaction supported on a line and the Landau Hamiltonian in a half-plane with a Robin boundary condition. We prove that the spectra of these two Hamiltonians are purely absolute continuous and that the former has gaps between adjacent Landau levels.

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A bachelor's thesis on Quantum Mechanics

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