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

Aharonov-Bohm效应被称为磁矢势的一种可观测的量子效应，在经典电动力学中只是作为一种计算工具引入的。在本课题中，我们研究了Aharonov-Bohm磁场(类三角磁场)对黎曼流形上薛定谔算子谱的影响。特别地，我们考虑了具有恒定磁场的双曲平面上的薛定谔算符加上周期放置在双曲晶格上的Aharonov-Bohm磁场，并研究了存在无限简并Landau能级时磁通量的阈值。此外，我们还考虑了具有两个量子化Aharonov-Bohm磁场的欧几里德平面上的薛定谔算子，给出了本征值函数的马修函数的显式形式。

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.