1992 Fiscal Year Final Research Report Summary
Microlocal Analysis of Differential Equations
| Project/Area Number |
03452007
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | University of Tokyo |
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Principal Investigator |
KOMATSU Hikosaburo Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (40011473)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAHIGASHI Yasuyuki Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (90214684)
TSUTSUMI Yoshio Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (10180027)
KATAOKA Kiyoomi Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (60107688)
SUNADA Toshikazu Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (20022741)
KOTANI Shinichi Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (10025463)
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| Project Period (FY) |
1991 – 1992
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| Keywords | Microlocal analysis / Differential equations / Semigroups of operators / Schrodinger operators / Second microlocal analysis / zakharov equations / Classification of subfactors |
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| Research Abstract |
There are two ways of microlocal analysis, one by M. Sato et al. employs the theory of several complex variables and the cohomologies with coefficients in sheaves, and the other by L. Hormander et al. multiplication by cut-off functions and Fourier transforms. Komatsu established in between a third method of microlocal analysis employing Poisson integrals and their analytic continuations. This has the advantage of carrying out microlocalanalysis for various classes of generalized functions, including the Gevrey classes, between Sato's hyperfunctions and Schwartz' distributions at the same time.
Komatsu extended, moreover, the theory of Laplace transforms of hyperfunctions to the case where hyperfunctions have values in a Banach space, and applied it in order to extend the Hille-Yosida theory of semigroups of linear operators to the case where semigroups are various classes of generalized functions.
Kotani and Sunada investigated the spectra of Laplace operators and Schrodinger operators acting on the functions on Riemannian manifolds. In particular, Kotani gave a probabilistic proof to an estimate of the supremum of spectra in terms of curvatures. Sunada gave a sufficient condition for the spectrum has the band structure as a property of the C^*-group algebra of the discrete group acting on the manifold.
Kataoka compared and distinguished many theories called the second microlocal analysis, and showed the importance of choozing a suitable theory in applying the second microlocal analysis to differential equations.
Tsutsumi investigated the solvability of the initial value problem for the Zakharov equations describing the strong disturbance of Langmuin waves in plasmas.
Kawahigashi gave rigorous formulations and their proofs to the so-called Ocneanu theory for the classification of subfactors in operator algebras for the first time. On this established foundation there will be fruitful applications of the theory.
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