| Project/Area Number |
15540149
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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 | Tohoku University |
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Principal Investigator |
FUJIIE Setsuro Tohoku University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (00238536)
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| Co-Investigator(Kenkyū-buntansha) |
CHIHARA Hiroyuki Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70273068)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Schr"odinger Equation / WKB method / Semicalssical analysis / Resonance / Monodromy operator / Schrodinger方程式 / モノドロミー作要素 / ホモクリニック軌道 / 超局所解析 |
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| Research Abstract |
Thanks to the Grant-In-Aid for Scientific Research, I did the following 4 researches :
1.Propagation of the microsupport at a hyperbolic fixed point (with J.-F.Bony, T.Ramond, M.Zerzeri)
2.Imaginary part of shape resonances created by a well in an island (with A.L.Benbernou, A.Martinez)
3.A conically crossing model for 2-dim 2-level Schr"odinger operators (with C.Lasser, L.Nedelec)
4.Theory of exact WKB method for first order systems (with L.Nedelec).
The first problem is about the propagation of the microsupport from the stable manifold to the unstable manifold associated with the hyperbolic fixed point. We solved this problem in both analytic and smooth categories. Bony has talked about this result in a-congress in Paris.
The second is an extension of the result by Helffer-Sj"ostrand in the case of analyhtic potential to the case of smmoth potential.
There appears a caustics from the boundary of the island. We succeeded to extend a WKB solution beyond the caustics by representing it in the form of Airy type intagral and extending the smooth phase and the amplitude by almost analytic extension to the complex plane.
The third is to obtain the quantization condition of resonances of the 2-dim 2-level Schr"odinger operator with conically crossing eigenpotentials. We reduced this operator to a 1-dim one and applied the exact WKB method. We have already written a paper about this result.
The last is a generalization of the method used in the previous research 3. It generalizes the theory of exact WKB method for single Schr"odinger equations to 2-level systems. I talked about this result in an international congress held in Kyoto and we are now preparing a paper.
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