Generalized eigenfunctions of relativistic Schroedinger operators, pseudo-differential operators and their related topics

相对论薛定谔算子、伪微分算子的广义本征函数及其相关主题

• 批准号: 15540178
• 负责人: UMEDA Tomio
• 金额: $ 1.98万
• 依托单位: University of Hyogo (2004-2006)Himeji Institute of Technology (2003)
• 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 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540178/
• 关键词: relativistic Sshrodinger operators; genreralized eigenfunctions; pseudo-differential operators

It has been shown that the resolvent kernel of the free relativistic Schroedinger operator consists of three parts, each of which is different from each other in nature: the fisrt term is the Riesz potential with strong singularity, the second one the same as the resolvent kernel of the negative Laplacian, and the third one an ingral kernel with tame property. A new inequality for the Riese potential has been proposed. The inequality is an essential tool in the research of the current project. It has been proven that the generalized eigenfintions of relativistic Schroedinger operators are obtained through the limiting absorption principle. In the process of proving this fact, the action of the square root of the negative Laplacian on distributions is examined, and established the fact that the action is meaningful for a sufficiently large class of distributions in all spatial dimensions. Also, it is shown that the differences between generalized eigenfunctions and the corresponding pla … More ne waves satisfy the Sommerfeld radiation conditions.It has been found that the spatial dimension plays a dominant role in the analysis of asymptotic behaviors at infinity of generalized eigenfunctions of relativisitic Schroedinger operators. It has been revealed that whether the spatial dimension is odd or even is decisive from the technical point of view. Motivated by this discovery, we computed the resolvent kernel of the free relativistic Schroedinger operator in the two dimensional case, and compered it with the one in the three dimensional case. It was found that the two dimensional resolvent kernel is very complicated: it consists of not only the Riesz potential but also the Bessel function, Neumann function and Struve function.Based upon the facts described above, we have proved that generalized eigenfunctions of relativistic Schroedinger operators in two dimension are bounded functions. This fact, together with the fact shown above, enables us to show that generalized eigenfunctions in the two dimension are asymptotically equal to the sum of the corresponding plane waves and spherical waves. Also, These two facts enable us to establish the completeness of the system of the generalized eigenfunctions in the two dimension, namely, the system spans the absolutely continuous subspace for the relativistic Schroedinger operator. Less

本文证明了自由相对论Schroedinger算子的预解核由三部分组成，每一部分性质不同：第一项是具有强奇异性的Riesz势，第二项与负Laplacian算子的预解核相同，第三项是具有驯服性质的奇异核。提出了Riese势的一个新的不等式。该不等式是本课题研究的重要工具。证明了相对论性薛定谔算子的广义本征值是由极限吸收原理得到的。在证明这一事实的过程中，负拉普拉斯算子的平方根对分布的作用被检查，并建立了这样一个事实，即该作用对所有空间维度中足够大的一类分布是有意义的。此外，本文还指出，广义本征函数与相应的平面的差异， ...更多信息 ne波满足Sommerfeld辐射条件.在分析相对论Schroedinger算子的广义本征函数在无穷远处的渐近行为时，空间维数起着主导作用.从技术角度来看，空间维度是奇数还是偶数是决定性的。基于这一发现，我们计算了二维情形下自由相对论Schroedinger算子的预解核，并与三维情形下的预解核进行了比较。发现二维预解核是非常复杂的，它不仅由Riesz势组成，而且由Bessel函数、Neumann函数和Struve函数组成，在此基础上，我们证明了二维相对论Schroedinger算子的广义本征函数是有界函数.这一事实，加上上面所示的事实，使我们能够证明，在二维广义本征函数渐近等于相应的平面波和球面波的总和。这两个事实也使我们能够建立二维广义本征函数系的完备性，即该系统是相对论性Schroedinger算子的绝对连续子空间。少

项目成果

Representation formulas of the solutions to the Cauchy problems for first order systems

一阶系统柯西问题解的表示公式

• 发表时间: 2007
• 期刊: Osaka J. Math 44, no.1
• 作者: M. Tajiri;T. Umeda
• 通讯作者: T. Umeda

Toshihiko Hoshiro: "Global Smoothing properties of dispersive equations with constant coefficients"Proceedings of 3rd International ISAAC Congress. 971-976 (2003)

Toshihiko Hoshiro：“具有常数系数的色散方程的全局平滑特性”第三届国际 ISAAC 大会论文集。

保城 寿彦: "ウェーブレット展開の無条件収束性について"京都大学数理解析研究所講究録. (発表予定). (2004)

Toshihiko Hojo：“论小波展开的无条件收敛”京都大学数学科学研究所 Kokyuroku（待发表）。

Grusin operator and heat kernel on nilpotent Lie groups

幂零李群上的 Grusin 算子和热核

• 发表时间: 2006
• 期刊: 京都大学数理解析研究所講究録 1502
• 作者: Masaki Tajiri;Tomio Umeda;Chisato Iwasaki;Tomio Umeda;Kenro Furutani
• 通讯作者: Kenro Furutani

Decay and regularity for dispersive equations with constant coeficients

常数系数色散方程的衰变和正则性

• 发表时间: 2004
• 期刊: Journal D' Analyse Mathematique 91
• 作者: Kenro Furutani;Chisato Iwasaki;Chisato Iwasaki;Chisato Iwasaki;Chisato Iwasaki;Toshihiko Hoshiro;Toshihiko Hoshiro
• 通讯作者: Toshihiko Hoshiro