Spectral and Scattering Theory and its Application

光谱与散射理论及其应用

• 批准号: 09640151
• 负责人: TAMURA Hideo
• 金额: $ 1.79万
• 依托单位: Okayama University (1998)Ibaraki University (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640151/
• 关键词: exponential product formula; Schrodinger semigroup; Pauli operator; asymptotic distribution of eigenvalues; scattering by magnetic fields; scattering amplitudes; レゾルベント; 固有値分布; 非一様磁場

The present project has been devoted to the study on the following three subjects related to the spectral and scattering theory for Schr_dinger operators.(1) For exponential product formula(Lie-Trotter-Kato product formula), the convergence in operator norm has been proved and the error estimate has been also established. The 9btained results have been applied to Schr_dinger semi-groups or propagators with singular or time-dependent potentials.(2) The unperturbed Pauli operator without electric potentials has zero eigenvalue with infinite multiplicities as its bottom of essential spectrum. When the operators are perturbed by potentials falling off at infinity, the asymptotic distribution of discrete eigenvalues near the origin has been studied. The special emphasis is placed on the case that Pauli operators do not necessarily have constant magnetic fields.(3) The asymptotic behavior at low energy of scattering amplitudes has been analysed for scattering by two dimensional magnetic fields and the relation to scattering by magnetic fields with small support has been also discussed.

本文主要研究了薛定谔算子的谱和散射理论中的三个问题：(1)对于指数乘积公式(Lie-Trotter-Kato乘积公式)，证明了算子范数收敛，并建立了误差估计。所得结果已应用于具有奇异势或含时变势的Schr_dinger半群或传播子。(2)无电势的未受扰动的Pauli算子的本征值为零且有无穷多个本征值作为其本质谱的底。当算子被无穷大处的势扰动时，研究了离散本征值在原点附近的渐近分布。特别强调了Pauli算符不一定具有恒定磁场的情况。(3)分析了二维磁场散射在低能散射幅度下的渐近行为，并讨论了与小支撑磁场散射的关系。

项目成果

H.Tamura: "Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields" Duke Math. J.93. 535-574 (1998)

H.Tamura：“具有非恒定磁场的泡利算子特征值的渐近分布”杜克数学。

H.Tamura: "Euor bounds on exponeutial product farmulas for Sohiodinger operators" J.Math.Soc.Japan. 50. 359-377 (1998)

H.Tamura：“Euor 对 Sohiodinger 算子的指数乘积公式的限制”J.Math.Soc.Japan。

H.Tamura: "Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with spherically symmetric magnetic fields" Tsukuba J. Math.22. 281-303 (1998)

H.Tamura：“具有球对称磁场的二维泡利算子负特征值的渐近分布”Tsukuba J. Math.22。

H.Tamura: "Error bounds on exponential product formulas for Sahiodinger operators" J.Math.Soc.Japan. 50. 359-377 (1998)

H.Tamura：“Sahiodinger 运算符的指数乘积公式的错误界限”J.Math.Soc.Japan。

H.Tamura and A.Iwatsuka: "Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields" Duke Math.J. 93. 535-574 (1998)

H.Tamura 和 A.Iwatsuka：“具有非恒定磁场的泡利算子特征值的渐近分布”Duke Math.J。