Geodesic
The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case
Sbornik. Mathematics, Tome 23 (1974) no. 1, pp. 85-109 
V. V. Kucherenko. The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case. Sbornik. Mathematics, Tome 23 (1974) no. 1, pp. 85-109. http://geodesic.mathdoc.fr/item/SM_1974_23_1_a4/
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This paper considers the action of the operator $a\bigl(x_1-ih\frac\partial{\partial x}\bigr)u\overset{\mathrm{def}}=\int a(x,h\xi)\times\exp i(x\xi)\widetilde u(\xi)\,d\xi$ 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 $$ a\biggl(x_1-ih\frac\partial{\partial x}\biggr)\exp\biggl(-\frac{iS}h\biggr)\varphi=\exp\biggl(\frac{iS}h\biggr)\sum_{j=0}^N h^jL_j\varphi+O(h^{N+1}). $$ 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 $S',S'',\dots,S^{(k)}$. Bibliography: 12 titles.

[1] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, izd-vo MGU, Moskva, 1965

[2] V. P. Maslov, M. V. Fedoryuk, “Kanonicheskii operator (veschestvennyi sluchai)”, Sovremennye problemy matematiki, 1, VINITI, Moskva, 1973, 85–167

[3] V. S. Buslaev, “Proizvodyaschii integral i kanonicheskii operator Maslova v metode VKB”, Funkts. analiz, 3:3 (1969), 17–31 | MR | Zbl

[4] V. V. Kucherenko, “Kvaziklassicheskaya asimptotika funktsii tochechnogo istochnika dlya statsionarnogo uravneniya Shredingera”, Teor. i matem. fiz., 1:3 (1969), 384–406 | MR

[5] L. Khërmander, “Psevdodifferentsialnye operatory”, Psevdodifferentsialnye operatory, izd-vo «Mir», Moskva, 1967, 63–87 | MR

[6] V. Maslov, “The caracteristics of pseudo-differential operators and differences schemes”, Actes Congr. int. math. 1970, Paris, 1971, 755–769 | MR | Zbl

[7] V. V. Kucherenko, “Kanonicheskii operator Maslova na rostke kompleksnogo pochti analiticheskogo mnogoobraziya”, DAN SSSR, 213:6 (1973), 1251–1254 | MR | Zbl

[8] V. P. Maslov, B. Yu. Sternin, “Kanonicheskii operator (kompleksnyi sluchai)”, Sovremennye problemy matematiki, I, VINITI, Moskva, 1973, 169–195

[9] V. V. Kucherenko, “Uravnenie Gamiltona-Yakobi v kompleksnoi neanaliticheskoi situatsii”, DAN SSSR, 213:5 (1973), 1021–1024 | Zbl

[10] V. P. Maslov, Operatornye metody, izd-vo «Nauka», Moskva, 1973 | MR

[11] L. Khërmander, “Integralnye operatory Fure”, Matematika, 16:1 (1972), 17–61 | Zbl

[12] A. O. Gelfond, Ischislenie konechnykh raznostei, izd-vo «Nauka», Moskva, 1967 | MR