Analysis on Schroedinger operators with delta-like magnetic fields on Riemannian manifolds

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The Aharonov-Bohm effect is known as an observable quantum effect by the magnetic vector potential, which is introduced in the classical electrodynamics only as a computational tool. In this subject, we study the effect of the Aharonov-Bohm magnetic fields (delta-like magnetic fields) on the spectrum of the Schroedinger operators on Riemannian manifolds. Especially, we consider the Schroedinger operators on the hyperbolic plane with a constant magnetic field plus the Aharonov-Bohm magnetic fields placed periodically on a hyperbolic lattice, and study the threshold value of the magnetic fluxes for the existence of the infinitely degenerated Landau levels. Moreover, we consider the Schroedinger operators on the Euclidean plane with two quantized Aharonov-Bohm magnetic fields, and give an explicit form of the eigenfunctions in terms of the Mathieu functions.