Spectral almost-Riemannian geometry and the magnetic Aharonov-Bohm effect

Yasaka, Kenzo (2020) Spectral almost-Riemannian geometry and the magnetic Aharonov-Bohm effect. Bachelor's Thesis, Mathematics.

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Abstract

In this thesis, we study the magnetostatic Aharonov-Bohm effect when charged particles are constrained to move on several spaces. To do so, we find explicit descriptions of the spectrum and eigenfunctions of a generalized Laplace-Beltrami operator which admits a vector potential A (in general, a one-form) on several two-dimensional almost-Riemannian manifolds (a generalization of Riemannian manifolds). We study three examples, namely the punctured plane and the unit cylinder both with Euclidean metric (these are in fact Riemannian manifolds) and finally the Grushin cylinder. We find in the case of the Grushin cylinder that the spectrum is extremely sensitive to changes in the magnetic flux, in contrast to the Euclidean cases. We also discuss several modifications of this effect, including the addition of relativistic effects (from a quantum field theoretical perspective) and the addition of magnetic monopoles.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Seri, M. and Biondini, S.
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	14 Jul 2020 12:36
Last Modified:	14 Jul 2020 12:36
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/22690

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