Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application

磁场薛定谔算子的谱理论及其应用

• 批准号: 11440056
• 负责人: TAMURA Hideo
• 金额: $ 4.8万
• 依托单位: Okayama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440056/
• 关键词: magnetic Schrodinger operator; scattering at low energy; resolvent convergence; point interaction; scattering by magnetic field; Aharonov-Bohm effect; スペクトル理論; 磁場散乱; アハロノフ・ボウム効果; 散乱振幅

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.

本课题主要研究具有磁场的薛定谔算符的谱和散射理论。重点研究了二维大间距点状磁场散射中的阿哈罗诺夫-玻姆效应的数学研究。研究了以下三个主题。(1)分析了二维磁场中散射幅值的低能渐近行为，并讨论了小支撑磁场与散射的关系。得到的结果很大程度上依赖于磁场的通量和零能量下的共振空间。(2)已知二维点状磁场的薛定谔算子本质上不是自伴随的。它具有缺陷指数（2,2），并且每个自伴随扩展都被实现为在原点处具有一定边界条件的微分算子。研究了当磁场支撑收缩时，通过范数解析收敛到具有点状磁场的薛定谔算子来实现哪些边界条件。(3)研究了两点状磁场大间距散射中的Aharonov-Bohm效应。分析了两场中心距离趋近于无穷大时散射振幅的渐近特性。所得结果在很大程度上取决于场的通量以及入射方向和最终方向。

项目成果

H.Tamura: "Norm resolvent convergence to magnetic Schrodinger operators with point interactions"Rev.Math.Phys.. (in press).

H.Tamura：“范数解析收敛到具有点相互作用的磁薛定谔算子”Rev.Math.Phys..（正在出版）。

田村英男: "Magnetic scattering at low energy in two dimensions"Nagoya Math.J.. 155. 95-151 (1999)

田村秀夫：“二维低能量磁散射”Nagoya Math.J.. 155. 95-151 (1999)

H.Tamura: "Magnetic scattering at low energy in two dimensions"Nagoya Math.J.. 155. 95-151 (1999)

H.Tamura：“二维低能量磁散射”Nagoya Math.J.. 155. 95-151 (1999)

M.Hirokawa: "Canonical quantization on a doubly connected space and the Aharanov-Bohm effect"J.Func.Anal.. 174. 322-363 (2000)

M.Hirokawa：“双重连通空间的正则量子化和阿哈拉诺夫-玻姆效应”J.Func.Anal.. 174. 322-363 (2000)

田村英男: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Ann.H.Poincare. (発表予定).

Hideo Tamura：“大间距点状磁场散射中的阿哈罗诺夫-玻姆效应”Ann.H.Poincare（即将介绍）。