Microlocal Analysis of Differential Equations

微局部分析有两种方法，一种是M. Sato等人利用多复变理论和层中系数的上同调，L. Hormander等人，通过截止函数和傅立叶变换乘法。小松之间建立了第三种方法的microlocal分析采用泊松积分及其解析延拓。这具有的优点进行microlocalanalysis各类广义函数，包括Gevrey类，之间佐藤的hyperfunctions和Schwartz'distribution在同一时间。小松扩展，此外，理论的拉普拉斯变换的hyperfunctions的情况下，hyperfunctions有值的Banach空间，并将其应用于扩展希勒-Kotani和Sunada研究了拉普拉斯算子和Schrodinger算子的谱作用在黎曼流形上的函数上的算子。特别是，Kotani给出了一个概率证明的估计上确界的频谱曲率。作为离散群的C^*-群代数作用于流形的性质，Sunada给出了谱具有带结构的充分条件，Kataoka比较和区别了许多被称为第二微局部分析的理论，Tsutsumi研究了Zakharov的初值问题的可解性，并证明了在将第二微局部分析应用于微分方程时，建立适当的理论的重要性。Kawahigashi首次对算子代数中的子因子分类的Ocneanu理论给出了严格的公式和证明。在这一基础上，理论的应用将会取得丰硕的成果。

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.