Study of asymptotic distribution of the Schroedinger operators with strong magnetic fields

强磁场下薛定谔算子渐近分布的研究

• 批准号: 15540168
• 负责人: IWATSUKA Akira
• 金额: $ 2.3万
• 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: 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-15540168/
• 关键词: Schroedinger operator; magnetic field; spectrum; asymptotic distribution; boundary condition; eigenvalue; 自己共役作用素

Iwasuka showed that the spectrum of the Schroedinger operators with Poisson random potential is the whole real line potential has some negative part.Mine obtained some detailed estimate of the number of the eigenvalues lying between the Landau levels when constant magnetic field is perturbed by multiple solenoidal (point) magnetic field, by using method of the perturbation of the canonical commutation relation.Doi studied the structure of the singularity of the solution to the Schroedinger equations with quadratic potential with perturbation by using the analysis of the asymptotic behavior of the Hamilton flow.Ito and Tamura proved the selfadjointness, the asymptotic completeness of the wave operator and the nonexistence of the eigenvalue for the Schrodinger operators with multiple delta like magnetic fields, and gave the leading term of the asymptotic behavior of their scattering amplitude when the distance between the centers of the magnetic fields tends to infinity.Tamura also proved the convergence in norm of the resolvent of the two dimensional Dirac operators when a delta like magnetic field is approximated by smooth fields. He also obtained an asymptotic formula for the kernel of the Schroedinger semigroup in the case of singular potentials by applying his recent result that the Trotter product formula converges in the sense of operator norm for unbounded operators.

Iwasuka 表明，泊松随机势的薛定谔算子的谱是整个实线势，有一些负部分。我得到了当恒定磁场受到多个螺线管扰动时位于朗道能级之间的特征值数量的一些详细估计（点）磁场，采用正则交换关系的微扰方法。Doi 研究了薛定谔方程组解的奇点结构： Ito 和 Tamura 证明了具有多个类 Delta 磁场的薛定谔算子的自共轭性、波算子的渐近完备性以及特征值的不存在性，并给出了当磁场中心之间的距离趋于无穷大时，它们的散射幅度渐近行为的首项。田村还证明了当类似 Delta 的磁场被平滑场近似时，二维狄拉克算子的求解。他还通过应用他最近的结果（Trotter 乘积公式在无界算子的算子范数意义上收敛），获得了奇异势情况下薛定谔半群核的渐近公式。

项目成果

Scattering of Dirac particles by electromagnetic fields with small support in two dimensions and effect from scalar potentials

二维小支撑电磁场对狄拉克粒子的散射以及标量势的影响

• 发表时间: 2004
• 期刊: Annales Henri Poincare 5・1
• 作者: Hiroshi T.ITO;Osanobu YAMADA;Chiaki TSUKAMOTO;Shin-ich DOI;Shin-ichi DOT;Hideo TAMURA
• 通讯作者: Hideo TAMURA

The Aharonov-Bohm solenoids in a constant magnetic field

恒定磁场中的阿哈罗诺夫-玻姆螺线管

• 发表时间: 2005
• 期刊: Annales Henri Poincare 6・1
• 作者: Kazunori ANDO;Akira IWATSUKA;Masahiro KAMINAGA;Fumihiko NAKANO;Fumio MAITANI;Takuya MINE
• 通讯作者: Takuya MINE

Hiroshi T.ITO: "Aharonov-Bohm effect in scattering by a chain of point-like magnetic fields"Asymptotic Analysis. 34. 199-240 (2003)

Hiroshi T.ITO：“点状磁场链散射中的阿哈罗诺夫-玻姆效应”渐近分析。

Dispersion of singularities of solutions for Schroedinger equations

薛定谔方程解奇点的色散

• 发表时间: 2004
• 期刊: Communications in Mathematical Physics 250・3
• 作者: Hiroshi T.ITO;Osanobu YAMADA;Chiaki TSUKAMOTO;Shin-ich DOI;Shin-ichi DOT
• 通讯作者: Shin-ichi DOT

Conformal slit mappings from periodic domains

来自周期域的共形狭缝映射

• 期刊: Kodai Mathematical Journal (掲載予定)
• 作者: Kazunori ANDO;Akira IWATSUKA;Masahiro KAMINAGA;Fumihiko NAKANO;Fumio MAITANI
• 通讯作者: Fumio MAITANI