2012 Fiscal Year Final Research Report
Analysis of wave propagation phenomena in the magnetic fields and inverse scattering problems
| Project/Area Number |
22540204
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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 | Tokyo Metropolitan University |
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Principal Investigator |
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| Project Period (FY) |
2010 – 2012
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| Keywords | 磁場中の波動伝播 / Schro"dinger 作用素 / 平滑化効果 / 散乱理論 / グラフ上の散乱逆問題 |
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| Research Abstract |
In this project we are interested in various wave propagation phenomena of Acoustic, Klein-Gordon, Schro”dinger equations. Main part of the study is the asymptotics of the waves propagation phenomena in the magnetic fields. We further study scattering direct and inverse problems which are important in applied physics. In order to explain the results more specifically, we list here the abstracts of the papers (1), (2) and (3).
(1) We survey some basic problems of Schro”dinger, Klein-Gordon and wave equations in the frame fork of general scattering theory. The following topics are treated under suitable decay and/or smallness conditions on the perturbationterms. Growth estimates of generalized eigenfunctions, Resolvent estimates, Scattering direct and inverse problems, Smoothing properties and Strichartz estimates. Due to our formulation of the weighted energy method, some topics are naturally extended to time-dependent and/or non-selfadjoint perturbations.
(2) We treat an inverse scattering problem on a graph with an infinite ray and a loop joined at one point. Our problem amounts to the reconstruction of potential on the basis of scattering data of operator.
(3) In this article we survey some basic resulta for the magnetic Schro”dinger operator with external potential which has a strong singularity. The following topics are treated under suitable decay conditions on the magnetic field and external potential: Selfadjointness of the operator, Growth estimates of generalized eigenfunctions, Principle og limiting absorption, Uniform resolvent estimates, and Smoothing properties for corresponding evolution equations.
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[Remarks] 1. Spectral representations and scattering for Schro"dinger operators on star graphs, 2011.10.12, 筑波大学解析セミナー
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[Remarks] 2. レゾルベントの低エネルギー評価と消散型方程式方程式の解の挙動, 2011.8.3, 三大学偏微分方程式セミナー
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[Remarks] 3. スターグラフ上の Schro"dinger 作用素のスペクトル表現と散乱問題,東京大学応用解析セミナー
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[Remarks] 4. Magnetic Schro"dinger 作用素に対する Resolvent の一様評価と関連する発展方程式の平滑化効果, 2011.6.30, 名古屋大学解析セミナー
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[Remarks] 5. Resolvent estimates and smoothing effects, 2010.11.18, 首都大学東京数理解析セミナー
