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

This article is cited in 3 scientific papers (total in 3 papers)

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.

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

Bibliographic databases:

UDC: 517.43
MSC: Primary 35S05, 47G05; Secondary 35J10
Received: 07.06.1973

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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath
    2. M. V. Karasev, V. E. Nazaikinskii, “On the quantization of rapidly oscillating symbols”, Math. USSR-Sb., 34:6 (1978), 737–764  mathnet  crossref  mathscinet  zmath
    3. M. V. Karasev, V. P. Maslov, “Asymptotic and geometric quantization”, Russian Math. Surveys, 39:6 (1984), 133–205  mathnet  crossref  mathscinet  zmath  adsnasa  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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