| Project/Area Number |
25887038
|
|---|
| Research Category |
Grant-in-Aid for Research Activity Start-up
|
| Allocation Type | Single-year Grants |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Kobe University |
|---|
Principal Investigator |
KAJINO Naotaka 神戸大学, 理学(系)研究科(研究院), 助教 (90700352)
|
|---|
| Project Period (FY) |
2013-08-30 – 2015-03-31
|
| Project Status |
Completed (Fiscal Year 2014)
|
| Budget Amount *help |
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
|
|---|
| Keywords | フラクタル解析 / ディリクレ形式 / ラプラシアン / 固有値漸近挙動 / コンヌのトレース定理 / 測度論的リーマン構造 / アポロニウスの詰め込み / 熱核 / 国際研究者交流（ドイツ，イギリス） / 国際研究者交流（アメリカ） |
|---|
| Outline of Final Research Achievements |
In this research the author has studied asymptotic behavior of the distributions of the eigenvalues of Laplacians (the eigenfrequencies) on fractals and has proved the following assertions:
Connes' trace theorem, which characterizes the notion of volume as an operator-theoretic paraphrase of eigenvalue asymptotics, holds for a large class of Laplacians on fractals including the case of the measurable Riemannian structure on the Sierpinski gasket, and surface area can be also characterized in a similar manner.
Moreover, for the measurable Riemannian structure on a classical fractal called the Apollonian gasket, the author has proved several fundamental facts such as the self-adjointness of the associated Laplacian, the discreteness of the eigenvalue distribution of the Laplacian, and estimates of the smallest eigenvalue.
|
