2007 Fiscal Year Final Research Report Summary
Analysis of properties of the solutions to Scrodinger equations via canonical transforms
| Project/Area Number |
18540176
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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 | Osaka University |
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Principal Investigator |
SUGIMOTO Mitsuru Osaka University, Dept. Math, Asso. Prof (60196756)
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| Co-Investigator(Kenkyū-buntansha) |
HOSHIRO Toshihiko Univ. Hyogo, Dept. Math, Prof (40211544)
DOI Shin-ichi Osaka Univ, Dept. Math, Prof (00243006)
HAYASHI Nakao Osaka Univ, Dept. Math, Prof (30173016)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Fourier Integral operator / Scrodinger equation / smoothing effect / dispersive equation / non-linear problem / time-space estimate / modulation space / time-frequency analysis |
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| Research Abstract |
This research is an attempt to apply the method of canonical transform to various problems of Scrodinger or more general equations.
In the first year, we first established the theory of Fourier integral operators which is a main tool to realize our idea. To put it concretely, we gave the symbolic calculus for the composition of Fourier integral operators and pseudo- differential operators with their phase and amplitude functions in wider classes. By using this result, we got be able to discuss the mapping property of them in various kinds of function spaces like Sobolev or Bosov spaces.
In the second year, we developed some method to show the time-space estimates for dispersive equations. We found the fact that the comparison of some quantity of symbols implies the comparison of time-space estimates. We named it "comparison principle", and by using it, we developed some method which deduce the estimates in one or two space dimension to some simple model "estimate". Furthermore, if we combine the method of canonical transforms and the comparison principle, we can deduce time-space estimates for almost every dispersive equation to just a trivial estimate. We also investigated fundamental properties of modulation spaces, and apply them to prove some mapping properties of pseudo-differential equations.
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