2006 Fiscal Year Final Research Report Summary
Semiclassical Analysis of Schroedinger equations
| Project/Area Number |
17540141
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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 |
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Principal Investigator |
FUJIIE Setsuro University of Hyogo, Graduate School of Material Science, Associate Professor, 大学院物質理学研究科, 助教授 (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)
DOI Shin-ichi Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (00243006)
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| Project Period (FY) |
2005 – 2006
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| Keywords | semi-classical analysis / WKB method / microlocal analysis / resonance / Schroedinger equation / hyperbolic fixed point / almost analytic extension / propagation of siugularity |
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| Research Abstract |
The main researches during this period are the fallowings.
First, in collaboration with J.-F. Bony, T. Ramond and M. Zerzeri, I considered a Hamiltonian with a hyperbolic fixed point and the corresponding incoming and outgoing stable manifolds. We showed, under a generic assumption, that the microlocal solution of the corresponding Schroedinger equation on the outgoing stable manifold (output data) is uniquely determined by that on the incoming stable manifold (input data). Moreover, we succeeded in describing the output data in terms of the input data as Fourier integral operator, whose phase and amplitude are explicitely given by geometrical quantities. These results are written in the preprint "Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point.
Second, in collaboration with A. L. Benbernou and A. Martinez, I considered the asymptotic expansion of the width of shape resonances created by a "well in an island". About 20 years ago, Helffer and Sjostrand showed for analytic potentials that it has a classical expansion with an exponentially small prefactor whose rate is given by the Agmon distance from the well to the sea. We conjectured the same result for only smooth potentials.
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Research Products
(9 results)
