Studies on Spectral Structure of Shrodinger Operators with Electromagnetic Fields

首先，我们研究了与shinichi Shimada点相互作用的薛定谔算子。也就是说，我们研究了薛定谔算子在原点处具有磁点相互作用，称为Aharonov-Bohm效应哈密顿量。由于定义在紧支持光滑函数空间上的算子本质上不是自伴随的，所以w确定了该算子的所有自伴随扩展及其在原点处的边界条件。我们还证明了可以计算出这些算子的渐近完备性、相移公式、特征函数展开等，并且这些算子保持了角动量。得到了波算符、散射矩阵和散射振幅的表达式，并给出了本征函数展开公式。这使我们能够根据波函数原点处的行为来描述由Aharonov和Bohm处理的算子。我们还与Doi Shin-ichi一起研究了状态积分密度的唯一性。我们证明了态的积分密度是唯一定义的，与将区域扩展到整个空间的方式或在磁场存在下相对一般假设的边界条件的选择无关。得到了用全空间算符定义的态的积分密度与用有限区域算符定义的态的积分密度重合的充分条件。这使得通过计算有限域算子的谱来计算整个空间算子的谱成为可能。

To begin with, we studied the Schrodinger operators with point interactions with Shin-ichi Shimada. Namely we studied the Schrodinger operator with a magnetic point interaction at the origin known as Aharonov-Bohm effect Hamiltonian. Since this operator defined on the space of compactly supported smooth functions is not essentially selfadjoint, w determined all the self adjoint extension of this operator and their boundary conditions at the origin. We also showed that one can calculate the asymptotic completeness, the phase shift formula, the eigenfunction expansions etc. for these operators which preserves the angular momentum. We obtained expressions of the wave operators, scattering matrices and the scattering amplitudes, and showed the eigenfunction expansion formula as well. This enabled to characterize the operator treated by Aharonov and Bohm in terms of the behavior at the origin of the wavefunctions.We also studied on the uniqueness of the integrated density of states with Shin-ichi Doi. We showed that the integrated density of states is uniquely defined independently of the way to extend the regions to the whole space or the choisce of the boundary conditions under relatively general assumptions in the presence of the magnetic fields. Also we obtained a sufficient conditions for the coincidence of the integrated density of states defined in terms of the whole space operator and that defined by using the finite region operators. This enabled to compute the spectrum of the whole space operators by calculating those of the finite region operators.