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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.

开始，我们与Shin-ichi Shimada研究了点相互作用的Schrodinger算子。也就是说，我们研究的薛定谔算子的磁点相互作用的原点被称为Aharonov-Bohm效应哈密顿量。由于定义在紧支撑光滑函数空间上的这个算子本质上不是自伴的，w决定了这个算子的所有自伴扩张及其在原点的边界条件。我们还证明了可以计算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。得到了波算符、散射矩阵和散射振幅的表达式，并给出了本征函数的展开式。这使得Aharonov和Bohm所处理的算子在波函数的原点的行为的特征。我们还研究了与Shin-ichi Doi的积分态密度的唯一性。我们证明了在相对一般的假设下，在磁场存在的情况下，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或边界条件的选择无关。得到了用全空间算符定义的积分态密度与用有限区域算符定义的积分态密度重合的一个充分条件。这使得通过计算有限区域算子的谱来计算整个空间算子的谱。