Inverse scattering problems for Dirac equations

狄拉克方程的逆散射问题

• 批准号: 09640182
• 负责人: ITO Hiroshi
• 金额: $ 1.98万
• 依托单位: Ehime University (1998)Kyoto 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-09640182/
• 关键词: Dirac equation; scattering operator; scattering theory; inverse scattering theory; relativistic quantum mechanics; relativistic Schrodinger equation; 相対論的不変性

We have considered an inverse scattering problem for Dirac equations with a time-dependent electromagnetic field. In this work it has been shown that some part of the electromagnetic field can be reconstructed from the scattering operator. Moreover, we have shown that the field can be completely reconstructed if the field is time-independent or satisfies some exponential decay condition. Our assumptions and results are independent of a choice of inertial frames, which is important for the relativistic theory. We have also shown that a similar result holds for a relativistic Schrodinger equation. The study for Schrodinger and Pauli equations is important for that of Dirac equations. Iwatsuka has studied the asymptotic distribution of eigenvalues for Pauli operators with Hideo Tamura. Okaji has shown, with De Carli, that the strong unique continuation property holds for Dirac operators with scalar potentials. Shimada has investigated Schrodinger equations with a potential supported in a line. Igari has studied the propagation of the singularities of a solution for some Cauchy problem in a complex domain. Sadamatsu has obtained a necessary condition for the well-posedness for initial value problems for some degenerate parabolic equations.

我们考虑了具有随时间变化的电磁场的狄拉克方程的逆散射问题。在这项工作中，已经表明电磁场的某些部分可以通过散射算子来重建。此外，我们已经证明，如果场与时间无关或满足某些指数衰减条件，则可以完全重建场。我们的假设和结果与惯性系的选择无关，这对于相对论理论很重要。我们还表明，类似的结果也适用于相对论薛定谔方程。薛定谔方程和泡利方程的研究对于狄拉克方程的研究具有重要意义。 Iwatsuka 与 Hideo Tamura 一起研究了泡利算子特征值的渐近分布。奥卡吉和德卡利一起证明，对于具有标量势的狄拉克算子来说，强大的独特连续性是成立的。岛田研究了直线支持势的薛定谔方程。 Igari 研究了复杂域中某些柯西问题解的奇点传播。 Sadamatsu 得到了一些简并抛物方程初值问题适定性的必要条件。

项目成果

Akira Iwatsuka: "asymptotic distribution of eigenvalves for Pauli operators with non constant magnetic fields" Duke Mathematical Journal. 93・3. 535-574 (1998)

Akira Iwatsuka：“具有非恒定磁场的泡利算子的特征值的渐近分布”杜克数学杂志 93・3（1998）。

島田伸一: "A solvable model for line interaction" 京都大学数理解析研究所講究録. 994. 168-183 (1997)

Shinichi Shimada：“线相互作用的可解模型”京都大学数学科学研究所 Kokyuroku 994. 168-183 (1997)。

Hiroshi Ito: "Inverse scattering problems for Dirac operators with time-dependent electromagnetic potentials(in Japanese)" RIMS Kokyuroku. 1047. 26-35 (1998)

Hiroshi Ito：“具有与时间相关的电磁势的狄拉克算子的逆散射问题（日语）”RIMS Kokyuroku。

Hiroshi Ito: "An inversescattering problem for Dirac equations with time-dependent electromagnetic potentials" Pub.RIMS.,Kyoto Univ.34・. 355-381 (1998)

Hiroshi Ito：“具有时间相关电磁势的狄拉克方程的逆散射问题”Pub.RIMS.，Kyoto Univ.34・355-381（1998）。

Akira Iwatsuka: "Asymptotic distribution of eigenvalues for Pauli Operators with nonconstant magnetic fields" Duke Mathematical Journal. 93-3. 535-574 (1998)

Akira Iwatsuka：“具有非常量磁场的泡利算子特征值的渐近分布”杜克数学杂志。