| Project/Area Number |
15540178
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | University of Hyogo (2004-2006)
Himeji Institute of Technology (2003) |
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Principal Investigator |
UMEDA Tomio University of Hyogo, Graduate School of Material Science, Professor, 大学院物質理学研究科, 教授 (20160319)
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| Co-Investigator(Kenkyū-buntansha) |
IWASAKI Chisato University of Hyogo, Graduate School of Material Science, Professor, 大学院物質理学研究科, 教授 (30028261)
HOSHIRO Toshihiko University of Hyogo, Graduate School of Material Science, Professor, 大学院物質理学研究科, 教授 (40211544)
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| Project Period (FY) |
2003 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | relativistic Sshrodinger operators / genreralized eigenfunctions / pseudo-differential operators |
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| Research Abstract |
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
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