Studies on Spectral Structure of Shrodinger Operators with Electromagnetic Fields

首先，我们研究了具有点相互作用的Shin-ichi Shimada薛定谔算子。也就是说，我们研究了在原点具有磁点相互作用的薛定谔算符，称为Aharonov-Bohm效应哈密顿量。由于定义在紧支撑光滑函数空间上的算子本质上不是自伴的，因此我们确定了该算子的所有自伴扩张及其在原点的边界条件。我们还证明了这些保持角动量的算符的渐近完备性、相移公式、本征函数展开式等。我们得到了波算符、散射矩阵和散射幅度的表达式，并给出了本征函数展开公式。这使得Aharonov和Bohm所处理的算符能够用波函数的原点处的行为来刻画。我们还研究了具有Shin-ichi Doi的积分态密度的唯一性。我们证明了在磁场存在的情况下，在相对一般的假设下，积分态密度是唯一定义的，与将区域扩展到整个空间的方式无关，也与边界条件的选择无关。我们还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密度重合的一个充分条件。这使得能够通过计算有限区域算子的谱来计算整个空间算子的谱。

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.