Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application

本项目致力于研究磁场薛定谔算子的谱和散射理论。重点讨论了二维大间距点状场磁散射中Aharonov-Bohm效应的数学研究。研究了以下三个主题。(1)本文分析了二维磁场中散射振幅在低能时的渐近行为，并讨论了小支撑磁场散射与散射的关系。所得结果在很大程度上依赖于磁场的通量和零能量处的共振空间。(2)二维点状磁场薛定谔算符本质上不是自伴的。它的亏指数为（2，2），每个自伴扩张都是一个微分算子，在原点处有一些边界条件。研究了当磁场支集收缩时，通过范数预解式收敛到点状磁场下的薛定谔算子的边界条件。(3)本文研究了两个大间距点状磁场散射中的Aharonov-Bohm效应。分析了当两个光场中心之间的距离趋于无穷大时，散射振幅的渐近行为。所得到的结果很大程度上取决于场的通量和入射方向和最终方向。

The present project has been devoted to the study on the spectral and scattering theory for the Schrodinger operators with magnnetic fields. The special emphasis is placed on the mathematical study on the Aharonov-Bohm effect in magnetic scattering by point-like fields at large separation in two dimensions. The following three subjects has been studied.(1) The asymptotic behavior at low energy of scattering amplitudes has been analysed for magnetic scattering in two dimensional fields, and the relation to scattering by magnetic fields with small support has been also discussed. The results obtained heavily depend on the flux of magnetic field and on the resonance space at zero energy.(2) The Schrodinger operator with point-like magnetic field in two dimensions is known to be not essentially self-adjoint. It has the deficiency indices (2,2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We have studied which boundary condition is realized through the norm resolvent convergence to Schrodinger operator with point-like magnetic field when the support of magnetic fields shrinks.(3) We have studied the Aharonov-Bohm effect in the scattering by two point-like magnetic fields at large separation. The asymptotic behavior of scattering amplitude has been analyzed when the distance between the centers of two fields goes to infinity. The obtained result heavily depends on the fluxes of fields and on incident and final directions.