RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb. (N.S.), 1974, Volume 94(136), Number 1(5), Pages 89–113 (Mi msb3634)

The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case

V. V. Kucherenko

Abstract: This paper considers the action of the operator $a(x_1-ih\frac\partial{\partial x})u\overset{\mathrm{def}}=\int a(x,h\xi)\times\exp i(x\xi)\widetilde u(\xi) d\xi$ on functions of the form $\exp(\frac{iS}h)\varphi(x)=u(x)$, where $\varphi\in C_0^\infty(\mathbf R^n)$ and $S\in C^\infty(\mathbf R^n)$. In particular, when $S(x,h)=S(x)$, $\operatorname{im}S(x)\geqslant0$, one has
$$a(x_1-ih\frac\partial{\partial x})\exp(-\frac{iS}h)\varphi=\exp(\frac{iS}h)\sum_{j=0}^N h^jL_j\varphi+O(h^{N+1}).$$
It is proved that for $\operatorname{im}S\not\equiv0$ the differential operators $L_j$ can be obtained from the analogous differential operators for $\operatorname{im}S\equiv0$ by means of “almost analytic extension” with respect to the arguments $S',S",…,S^{(k)}$.
Bibliography: 12 titles.

Full text: PDF file (2086 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1974, 23:1, 85–109

Bibliographic databases:

UDC: 517.43
MSC: Primary 35S05, 47G05; Secondary 35J10

Citation: V. V. Kucherenko, “The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case”, Mat. Sb. (N.S.), 94(136):1(5) (1974), 89–113; Math. USSR-Sb., 23:1 (1974), 85–109

Citation in format AMSBIB
\Bibitem{Kuc74} \by V.~V.~Kucherenko \paper The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case \jour Mat. Sb. (N.S.) \yr 1974 \vol 94(136) \issue 1(5) \pages 89--113 \mathnet{http://mi.mathnet.ru/msb3634} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=343104} \zmath{https://zbmath.org/?q=an:0293.35061} \transl \jour Math. USSR-Sb. \yr 1974 \vol 23 \issue 1 \pages 85--109 \crossref{https://doi.org/10.1070/SM1974v023n01ABEH002174} 

• http://mi.mathnet.ru/eng/msb3634
• http://mi.mathnet.ru/eng/msb/v136/i1/p89

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Kucherenko, “Asymptotic solutions of equations with complex characteristics”, Math. USSR-Sb., 24:2 (1974), 159–207
2. M. V. Karasev, V. E. Nazaikinskii, “On the quantization of rapidly oscillating symbols”, Math. USSR-Sb., 34:6 (1978), 737–764
3. M. V. Karasev, V. P. Maslov, “Asymptotic and geometric quantization”, Russian Math. Surveys, 39:6 (1984), 133–205
•  Number of views: This page: 306 Full text: 115 References: 31