Microlocal analysis

The micro- local analysis is a branch of mathematics that has developed in the 1960s and 1970s from the theory of partial differential equations and from the Fourier Analysis. The term micro- local analysis comes from joint work of Mikio Sato, Takahiro Kawai, and Masaki Kashiwara. It is in the physical domain of quantum mechanics or the Semiklassik important as the Heisenberg uncertainty principle can be systematically characterized with her.

Overview

The micro- local analysis has evolved in the 1960s and 1970s from the theory of linear partial differential equations out. Many basic idea of ​​micro- local analysis for example, come from Lars Hörmander, Louis Nirenberg and Viktor Pavlovich Maslov. These and others started the micro- local analysis with methods from Fourier analysis and from the theory of partial differential equations in the category build. The investigated objects were thus defined and investigated on smooth manifolds. In the field of partial differential equations, the distribution theory offers key techniques for solving these equations, so this theory also plays a fundamental role in the field of micro- local analysis. In the distribution theory of the notion of singular support has been introduced. This includes all the points in whose vicinity a selected distribution can not be created or represented by a smooth function. In the area of ​​micro- local analysis, this term has been generalized to the central object of the wave front set. This subset of the cotangent bundle contains information as both the place and the frequency of the singularities.

A little later they started the micro- local analysis on the category of analytic functions expand. In this context, introduced by Mikio Sato hyperfunctions are as a generalization of distributions important objects. Even under the conditions of the category of analytic functions, the wavefront amount ( slightly different than in the category) defined.

Important properties of the micro- local analysis

Distribution

The distribution theory is a separate theory for solving partial differential equations and is not directly part of the micro- local analysis. Decisive this theory by Laurent Schwartz was developed in the 1940s. He defined, for example, the Fourier transform of tempered distributions and proved to the core set of Schwartz. For the micro local analysis the distribution theory is of fundamental importance, because in the micro- local analysis seeks to distributional solutions of partial differential equations.

Pseudo- differential operator

A pseudo- differential operator is a generalization of the differential operator. It was developed from techniques of Fourier analysis to the solutions of certain partial differential equations. For example, let a linear partial differential operator with constant coefficients and is the Fourier transform and its inverse transform. Then you can the differential equation in

Shall transfer and due to the Differentationseigenschaften the Fourier transform

Differentationseigenschaft this together with the return transformation of the Fourier transform is an important technique of Fourier analysis to solve partial differential equations. In the micro- local analysis we consider integral operators, the representation of the

Have. Compared to Fourier analysis, the polynomial function has been replaced by a more general function, which depends on two variables, in the operator. Of course, in this context, the existence of the integrals must be secured, in this context, therefore, the concept of the oscillating integral was introduced and the function is a member of a class of symbol and is therefore called icon. In the micro- local analysis one is interested in, for example, for the behavior of operators in certain "small" environments. Pseudo-differential operators, for example, pseudo- locally, ie the application of a pseudo- differential operator on a distribution does not increase its singular support.

By Lars Hörmander both the symbol classes as well as the oscillatory integral were introduced. The pseudo- differential operator goes back to works by Joseph Kohn and Louis Nirenberg.

Wavefront amount

The wave front set is a central object of the micro- local analysis. It is a generalization of the concept of the singular of a distribution medium. In the category, the wavefront amount of a distribution in Euclidean space is defined as the complement in those points for which there are environments and such that for uniformly converges for all test functions and is suitable for all. As this definition takes into account only local aspects of the distribution, one can also define the wave front lot with a card similar to distributions on manifolds, there is a subset of the cotangent bundle. The projection of the wave-front amount to the variable again corresponds to the singular support of the observed distribution. Similarly, we also defined the analytic wave front set.

Fourier integral operator

Another object of the micro- local analysis is the Fourier integral operator. This is a generalization of the pseudo- differential operator. The term is replaced by the more general expression, which now is a phase function. In addition, shall these operators and apply with. The representation of a Fourier integral operator is therefore