Studies on Spectral Structure of Shrodinger Operators with Electromagnetic Fields

电磁场薛定谔算子的谱结构研究

基本信息

批准号：
10640204
负责人：
IWATSUKA Akira
金额：
$ 2.11万
依托单位：
KYOTO INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

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.

首先，我们与 Shin-ichi Shimada 研究了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，其原点处具有磁点相互作用，称为阿哈罗诺夫-玻姆效应哈密顿量。由于定义在紧支持光滑函数空间上的该算子本质上不是自共轭的，因此 w 确定了该算子的所有自共拓及其在原点的边界条件。我们还表明，可以计算这些保留角动量的算子的渐近完整性、相移公式、本征函数展开式等。得到了波算子、散射矩阵和散射振幅的表达式，并给出了本征函数展开式。这使得能够根据波函数原点的行为来表征阿哈罗诺夫和玻姆处理的算子。我们还与土井信一一起研究了积分态密度的唯一性。我们表明，在存在磁场的相对一般的假设下，状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边界条件的选择无关。我们还得到了全空间算子定义的积分态密度与有限域算子定义的积分态密度一致的充分条件。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。