Microdifferential operator

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In mathematics, a microdifferential operator is a linear operator on a cotangent bundle (phase space) that generalizes a differential operator and appears in the framework of microlocal analysis as well as in the Kyoto school of algebraic analysis.

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The notion was originally introduced by L. Boutet de Monvel and P. Krée [1] as well as by M. Sato, T. Kawai and M. Kashiwara. [2] There is also an approach due to J. Sjöstrand. [3]

Definition

We first define the sheaf of formal microdifferential operators on the cotangent bundle of an open subset . [4] A section of that sheaf over an open subset is a formal series: for some integer m,

where each is a holomorphic function on that is homogeneous of degree in the second variable.

The sheaf of microdifferential operators on is then the subsheaf of consisting of those secctions satisfying the growh condition on the negative terms; namely, for each compact subset , there exists an such that

[5]

See also

Reference

Notes

  1. L. Boutet De Monvel, Louis & P. Krée
  2. M. Sato, T. Kawai & M. Kashiwara
  3. Sjöstrand
  4. Schapira 1985 , Ch. I., § 1.2.
  5. Schapira 1985 , Ch. I., § 1.3.

Works

Further reading