Microlocal Analysis of Differential Equations

点击翻译按钮获取中文摘要

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.