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.
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
editWe 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
See also
editReference
editNotes
edit- ^ L. Boutet De Monvel, Louis & P. Krée
- ^ M. Sato, T. Kawai & M. Kashiwara
- ^ Sjöstrand
- ^ Schapira 1985, Ch. I., § 1.2.
- ^ Schapira 1985, Ch. I., § 1.3.
Works
edit- Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d'ordre infini, I, Ann. Inst. Fourier, Grenoble, 33–4 (1983), 227–250.
- Boutet De Monvel, Louis ; Krée, Paul, Pseudo-differential operators and Gevrey classes, Annales de l'Institut Fourier, Volume 17 (1967) no. 1, pp. 295-323
- M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, in: Lecture Notes in Math. 287, Springer, 1973, 265–529.
- Schapira, Pierre (1985). Microdifferential Systems in the Complex Domain. Grundlehren der mathematischen Wissenschaften. Vol. 269. Springer. doi:10.1007/978-3-642-61665-5. ISBN 978-3-642-64904-2.
- Sjöstrand, Johannes. Singularités analytiques microlocales, dans Singularités analytiques microlocales - équation de Schrödinger et propagation des singularités..., Astérisque, no. 95 (1982), pp. iii-166. https://www.numdam.org/item/AST_1982__95__R3_0/