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Scattering of Dirac particles by mabnetic fields and spectral theory

磁场和谱理论对狄拉克粒子的散射

基本信息

批准号：
15540206
负责人：
TAMURA Hideo
金额：
$ 2.43万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540206/
关键词：
scattering by magnetic fields Dirac operator Pauli operator Aharonov-Bohm effect resonance at zero energy Aharonov-Bohm effect scattering by magnetic field Dirac operators Pauli operators resonance at zero energy Dirac operator Feynm

项目摘要

The subject of this research project is the spectral theory for Dirac operators with magnetic fields in two dimensions, and the special emphasis is placed on the study about the relation between scattering of Dirac particles by magnetic fields and resonance at zero energy of Schrodinger operators. The motion of massless Dirac particles is governed by the operator D(A,V)=σ・ (-i∇-A)+V acting on [L^2(R^2)]^2, where A(x):R^2→R^2 is a magnetic potential, V(x):R^2→R is a electronic potential and σ=(σ_1,σ_2) is a vector with 2 × 2 Pauli matrices as components. The square of Dirac Operator D(A,0) without scalar potential V becomes the diagonal operator with Schrodinger operators H_±=(-i∇-A)^2±b as diagonal components (Pauli operator), where b=∇×A:R^2→R denotes the magnetic field associated with vector potential A. The both operators H_± 【greater than or equal】 0 are nonnegative, but they have a different spectral structure at zero energy. If, for example, b∈C^∞_0(R^2) is compactly supported an … More d it has the noninteger flux α defined by α=∫b(x)dx/2π 【not a member of】 Z, then equation H_u=0 has a bounded solution (resonance) not in L^2, while H_+u=0 does not have such a solution. Thus the zero energy resonance of Schrodinger operators appears in the spectral theory for Dirac operators in a quite natural way. The present project deals with the following two subjects closely related to zero energy resonance : (1)resolvent convergence in norm to Dirac operators with solenoidal magnetic fields (point-like fields) : (2)scattreing by electromagnetic fields with small support. These subjects have both been discussed in physical articles when electromagnetic fields are spherically symmetric, and the results obtained are based on a calculation using the Bessel functions. The main achievement is that we have made clear the role of zero energy resonance hidden behind the explicit calculation from a mathematical point of view by eliminating the assumption of spherical symmetry. The scattering of Dirac particles by electromegnetic fields with small support appears in the model of cosmic string as an important problem of mathematical physics. The application to it has been also studied. Less

本研究课题的研究内容是二维磁场中Dirac算子的谱理论，重点研究Dirac粒子与磁场的散射与Schrodinger算子的零能共振之间的关系。无质量狄拉克粒子的运动由作用于[L^2（R^2）]^2的算符D（A，V）=σ·（-i <$-A）+V控制，其中A（x）：R^2→R^2是磁势，V（x）：R^2→R是电子势，σ=（σ_1，σ_2）是以2 × 2泡利矩阵为分量的向量。没有标势V的狄拉克算子D（A，0）的平方成为以薛定谔算子H ±=（-i <$-A）^2±B为对角分量的对角算子（泡利算子），其中B=<$x ×A：R^2→R表示与矢势A有关的磁场。这两个算子H ± 0都是非负的，但它们在零能量处具有不同的谱结构。例如，如果B∈C^∞_0（R^2）是紧支撑的， ...更多信息 d它有非整数通量α，定义为α= α B（x）dx/2π [不是] Z的成员，则方程H_u=0有不在L^2中的有界解（共振），而H_+u=0没有这样的解.这样，Schrodinger算子的零能共振就以一种十分自然的方式出现在Dirac算子的谱理论中。本文研究了与零能共振密切相关的两个问题：（1）具有螺线管磁场（点状场）的Dirac算子的预解式依范数收敛问题;（2）小支撑电磁场的散射问题。当电磁场是球对称时，这些问题都在物理学文章中讨论过，得到的结果是基于使用贝塞尔函数的计算。主要成果是通过消除球对称假设，从数学角度明确了隐藏在显式计算背后的零能共振的作用。狄拉克粒子与小支撑电磁场的散射是宇宙弦模型中的一个重要数学物理问题。并对其应用进行了研究。少