Micro-local analysis

The micro-local analysis is a branch of mathematics that in the 1960s and 1970s from the theory of partial differential equations and from the Fourier analysis has developed. The term microlocal analysis comes from the joint work of Mikio Satō , Takahiro Kawai and Masaki Kashiwara . It is important in the physical area of quantum mechanics or semiclassics , as it can be used to systematically characterize Heisenberg's uncertainty relation .

overview

Microlocal analysis developed in the 1960s and 1970s from the theory of linear partial differential equations. Many of the fundamental ideas of micro-local analysis come from, for example, Lars Hörmander , Louis Nirenberg and Wiktor Pawlowitsch Maslow . These and others began to build microlocal analysis using methods from Fourier analysis and the theory of partial differential equations in the category . The examined objects were thus defined and examined on smooth manifolds . In the area of partial differential equations, distribution theory offers important techniques for solving these equations, which is why this theory also plays a fundamental role in the area of microlocal analysis. The concept of the singular carrier was introduced in distribution theory. This includes all points in the vicinity of which a selected distribution cannot be generated or represented by a smooth function . In the field of micro-local analysis, this term was generalized to the central object of the wavefront set . This subset of the cotangential bundle contains both the location and the frequency of the singularities as information. ${\ displaystyle C ^ {\ infty}}$

A little later, microlocal analysis began to be expanded to include the category of analytical functions . In this context, the hyperfunctions introduced by Mikio Satō as a generalization of the distributions are important objects. The wavefront set was also defined under the conditions of the category of analytical functions (somewhat differently than in the category). ${\ displaystyle C ^ {\ infty}}$

Important objects of micro-local analysis

distribution

Distribution theory is an independent theory for solving partial differential equations and is not a direct part of micro-local analysis. This theory was largely developed by Laurent Schwartz in the 1940s. For example, he defined the Fourier transform for tempered distributions and proved Schwartz's core theorem . Distribution theory is of fundamental importance for micro-local analysis, because in micro-local analysis one seeks 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 solve certain partial differential equations. For example, let be a linear partial differential operator with constant coefficients and be the Fourier transform and its inverse transform. Then we can use the differential equation in ${\ displaystyle D}$ ${\ displaystyle {\ mathcal {F}}}$ ${\ displaystyle {\ mathcal {F}} ^ {- 1}}$ ${\ displaystyle you = f}$

{\ displaystyle {\ mathcal {F}} ^ {- 1} ({\ mathcal {F}} (you)) = f}

and due to the differentiation properties of the Fourier transform applies

{\ displaystyle {\ mathcal {F}} ^ {- 1} ({\ mathcal {F}} (you)) (y) = {\ mathcal {F}} ^ {- 1} (i ^ {| \ alpha |} \ xi ^ {\ alpha} {\ mathcal {F}} (u)) = {\ mathcal {F}} ^ {- 1} \ left (i ^ {| \ alpha |} \ xi ^ {\ alpha } \ int _ {\ mathbb {R} ^ {n}} e ^ {- ix \ xi} u (x) \ mathrm {d} x \ right) = \ int _ {\ mathbb {R} ^ {n} } \ int _ {\ mathbb {R} ^ {n}} e ^ {i (yx) \ xi} i ^ {| \ alpha |} \ xi ^ {\ alpha} u (x) \ mathrm {d} x \, \ mathrm {d} \ xi \ ,.}

This differentiation property together with the back and forth transformation of the Fourier transform is an important technique of Fourier analysis for solving partial differential equations. In micro-local analysis one considers integral operators , which represent the representation

{\ displaystyle Au (x) = \ int _ {\ mathbb {R} ^ {n}} \ int _ {\ mathbb {R} ^ {n}} e ^ {i (xy) \ xi} a (x, \ xi) u (y) \ mathrm {d} y \, \ mathrm {d} \ xi}

to have. Compared to Fourier analysis, the polynomial function in the operator has been replaced by a more general function that depends on two variables. Of course, the existence of the integrals must also be ensured in this context, in this context the term oscillating integral was introduced and the function is an element of a symbol class and is therefore also called a symbol. In micro-local analysis, for example, one is interested in the behavior of operators in certain “small” environments. For example, pseudo differential operators are pseudo-local, that is, applying a pseudo differential operator to a distribution does not increase its singular carrier. ${\ displaystyle i ^ {| \ alpha |} \ xi ^ {\ alpha}}$ ${\ displaystyle a}$

Lars Hörmander introduced both the symbol classes and the oscillating integral. The pseudo differential operator goes back to the work of Joseph Kohn and Louis Nirenberg.

Wavefront set

The amount of wavefronts is a central object of micro-local analysis. It is a generalization of the concept of the singular carrier of a distribution. In the -category the wavefront set of a distribution in Euclidean space is defined as the complement in those points for which neighborhoods of and of exist such that for in converges uniformly for all test functions and for all . Since this definition only takes into account local aspects of the distribution, the wavefront set can also be defined by means of maps analogously to distributions on manifolds , where it is a subset of the cotangent bundle. The projection of the wavefront set onto the variable again corresponds to the singular carrier of the distribution under consideration. The analytical wavefront set is also defined in a similar way. ${\ displaystyle C ^ {\ infty}}$ ${\ displaystyle {\ rm {WF}} (u)}$ ${\ displaystyle u}$ ${\ displaystyle \ mathbb {R} ^ {n} \ times (\ mathbb {R} ^ {n} \ backslash 0)}$ ${\ displaystyle (x_ {0}, \ xi _ {0}) \ in \ mathbb {R} ^ {n} \ times (\ mathbb {R} ^ {n} \ backslash 0)}$ ${\ displaystyle U}$ ${\ displaystyle x_ {0}}$ ${\ displaystyle V}$ ${\ displaystyle \ xi _ {0}}$ ${\ displaystyle u (\ phi \ exp (-itx \ cdot \ xi)) = {\ mathcal {O}} (t ^ {- N})}$ ${\ displaystyle t \ to \ infty}$ ${\ displaystyle \ xi}$ ${\ displaystyle \ phi \ in C_ {c} ^ {\ infty} (U)}$ ${\ displaystyle N> 0}$ ${\ displaystyle x}$

Fourier integral operator

Another object of 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 term , where is now a phase function . In addition, for these operators and with may apply. The representation of a Fourier integral operator is thus ${\ displaystyle e ^ {- i (xy) \ xi}}$ ${\ displaystyle e ^ {- i \ phi (x, y, \ xi)}}$ ${\ displaystyle \ phi}$ ${\ displaystyle x, y \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle \ xi \ in \ mathbb {R} ^ {N}}$ ${\ displaystyle n \ neq N}$

{\ displaystyle Bu (x) = \ int _ {\ mathbb {R} ^ {n}} \ int _ {\ mathbb {R} ^ {N}} e ^ {i \ phi (x, y, \ xi) } a (x, \ xi) u (x) \ mathrm {d} \ xi \, \ mathrm {d} y \ ,.}

literature

A. Kaneko: Microlocale Analysis . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
MA Shubin: Wave front . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Alain Grigis, Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 .

Individual evidence

^ Johannes Sjöstrand: Microlocal Analysis . In: Jean-Paul Pier (Ed.): Development of mathematics 1950-2000 . Birkhäuser, Basel / Boston / Berlin 2000, ISBN 3-7643-6280-4 , p. 970 .
^ Johannes Sjöstrand: Microlocal Analysis . In: Jean-Paul Pier (Ed.): Development of mathematics 1950-2000 . Birkhäuser, Basel / Boston / Berlin 2000, ISBN 3-7643-6280-4 , p. 967 .
↑ Microlocal Analysis . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
↑ Alain Grigis & Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 1.
↑ Alain Grigis, Johannes Sjöstrand: Micro Local analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 18.
↑ Alain Grigis, Johannes Sjöstrand: Micro Local analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 40.
↑ Wave front crowd . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .

[1] Johannes Sjöstrand: Microlocal Analysis . In: Jean-Paul Pier (Ed.): Development of mathematics 1950-2000 . Birkhäuser, Basel / Boston / Berlin 2000, ISBN 3-7643-6280-4 , p. 970 .

[2] Johannes Sjöstrand: Microlocal Analysis . In: Jean-Paul Pier (Ed.): Development of mathematics 1950-2000 . Birkhäuser, Basel / Boston / Berlin 2000, ISBN 3-7643-6280-4 , p. 967 .

[3] Microlocal Analysis . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .

[4] Alain Grigis & Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 1.

[5] Alain Grigis, Johannes Sjöstrand: Micro Local analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 18.

[6] Alain Grigis, Johannes Sjöstrand: Micro Local analysis for differential operators: an introduction. Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 40.

[7] Wave front crowd . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .