Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application

本课题致力于研究具有磁场的薛定谔算符的谱和散射理论。重点对二维大间隔类点场磁散射中的Aharonov-Bohm效应进行了数学研究。主要研究了以下三个方面的内容：(1)分析了二维场中磁散射在低能量散射幅度下的渐近行为，并讨论了与小支撑磁场散射的关系。所得结果在很大程度上依赖于零能量下的磁场通量和共振空间。(2)二维类点磁场薛定谔算符本质上不是自伴的。它具有亏度指数(2，2)，并且每个自伴扩张都被实现为一个在原点具有一定边界条件的微分算子。研究了当磁场支承收缩时，通过范数预解收敛到点状磁场薛定谔算子的边界条件。(3)研究了大间隔两个点状磁场散射中的Aharonov-Bohm效应。分析了当两场中心之间的距离为无穷大时，散射幅度的渐近行为。得到的结果在很大程度上依赖于场的通量、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.