Asymptotics for equations with singularities in the characteristics
Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 315-347.

Voir la notice de l'article provenant de la source Math-Net.Ru

The operator method of constructing the asymptotics for degenerate equations is considered in this article. Machinery for asymptotic representations of Poisson algebras is constructed, which permits explicit expressions for the parametrices of such equations to be derived. Bibliography: 52 titles.
@article{IM2_1982_19_2_a6,
     author = {V. P. Maslov and V. E. Nazaikinskii},
     title = {Asymptotics for equations with singularities in the characteristics},
     journal = {Izvestiya. Mathematics },
     pages = {315--347},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1982},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a6/}
}
TY  - JOUR
AU  - V. P. Maslov
AU  - V. E. Nazaikinskii
TI  - Asymptotics for equations with singularities in the characteristics
JO  - Izvestiya. Mathematics 
PY  - 1982
SP  - 315
EP  - 347
VL  - 19
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a6/
LA  - en
ID  - IM2_1982_19_2_a6
ER  - 
%0 Journal Article
%A V. P. Maslov
%A V. E. Nazaikinskii
%T Asymptotics for equations with singularities in the characteristics
%J Izvestiya. Mathematics 
%D 1982
%P 315-347
%V 19
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a6/
%G en
%F IM2_1982_19_2_a6
V. P. Maslov; V. E. Nazaikinskii. Asymptotics for equations with singularities in the characteristics. Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 315-347. http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a6/

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[2] Arnold V. I., “O kharakteristicheskom klasse, vkhodyaschem v usloviya kvantovaniya”, Funktsionalnyi analiz i ego prilozheniya, 1967, no. 1, 1–14 | MR

[3] Berezin F. A., Metod vtorichnogo kvantovaniya, Nauka, M., 1965 | MR

[4] Berezin F. A., “Kvantovanie”, Izvestiya AN SSSR. Ser. matemat., 38:5 (1974), 1116–1175 | MR | Zbl

[5] Godeman R., Algebraicheskaya topologiya i teoriya puchkov, IL, M., 1961

[6] Godbiion K., Differentsialnaya geometriya i analiticheskaya mekhanika, Mir, M., 1973

[7] Grushin V. V., “Ob odnom klasse gipoellipticheskikh operatorov”, Matemat. sbornik, 83(125):3 (1970), 456–473 | Zbl

[8] Danford N., Shvarts Dzh. T., Lineinye operatory (Spektralnaya teoriya. Samosopryazhennye operatory v gilbertovom prostranstve), Mir, M., 1966

[9] Ivrii V. Ya., “O volnovykh frontakh reshenii nekotorykh psevdodifferentsialnykh uravnenii”, Funktsionalnyi analiz i ego prilozheniya, 10:2 (1976), 71–72 | MR | Zbl

[10] Karasev M. V., Maslov V. P., “Algebry s obschimi perestanovochnymi sootnosheniyami i ikh prilozheniya. II: Operatornye unitarno-nelineinye uravneniya”, Sovremennye problemy matematiki. Itogi nauki i tekhniki, 13, VINITI AN SSSR, 1979, 145–267 | MR

[11] Karasev M. V., Maslov V. P., “Globalnye asimptoticheskie operatory regulyarnogo predstavleniya”, Doklady AN SSSR, 257:1 (1981), 33–37 | MR

[12] Karasev M. V., Nazaikinskii V. E., “O kvantovanii bystroostsilliruyuschikh simvolov”, Matematicheskii sbornik, 106(148):2(6) (1978), 183–213 | MR | Zbl

[13] Kirillov A. A., “Unitarnye predstavleniya nilpotentnykh grupp Li”, Uspekhi matematicheskikh nauk, 17:4 (1962), 57–101 | MR

[14] Kirillov A. A., Elementy teorii predstavlenii, Nauka, M., 1972 | MR

[15] Kostant B., “Kvantovanie i unitarnye predstavleniya”, Uspekhi matematicheskikh nauk, 28:1 (1973), 163–225 | MR | Zbl

[16] Krein S. G., Shikhvatov A. M., “Lineinye differentsialnye uravneniya na gruppe Li”, Funktsionalnyi analiz i ego prilozheniya, 4:1 (1970), 62–61 | MR

[17] Kucherenko V. V., “Asimptoticheskie resheniya sistemy $A(x,-ih\partial/\partial x)u =0$ pri $h\to0$ v sluchae kharakteristik peremennoi kratnosti”, Izvestiya AN SSSR. Ser. matemat., 38:3 (1974), 625–662 | Zbl

[18] Kucherenko V. V., “Parametriks dlya uravnenii s vyrozhdayuschimsya simvolom”, Doklady AN SSSR, 229:4 (1976), 797–800 | MR | Zbl

[19] Kucherenko V. V., Osipov Yu. V., “Asimptoticheskie resheniya giperbolicheskogo uravneniya s kharakteristikami peremennoi kratnosti”, Doklady AN SSSR, 249:2 (1979), 289–293 | MR | Zbl

[20] Levitan B. M., Teoriya operatorov obobschennogo sdviga, Nauka, M., 1973 | MR | Zbl

[21] Maslov V. P., Operatornye metody, Nauka, M., 1973 | MR

[22] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965

[23] Maslov V. P., Nazaikinskii V. E., “Algebry s obschimi perestanovochnymi sootnosheniyami i ikh prilozheniya. I: Psevdodifferentsialnye uravneniya s rastuschimi koeffitsientami”, Sovremennye problemy matematiki. Itogi nauki i tekhniki, 13, VINITI AN SSSR, 1979, 5–144

[24] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[25] Mischenko A. S., Sternin B. Yu., Shatalov V. E., Kanonicheskii operator Maslova. Kompleksnaya teoriya, MIEM, M., 1974

[26] Mischenko A. S., Sternin B. Yu., Shatalov V. E., Lagranzhevy mnogoobraziya i metod kanonicheskogo operatora, Nauka, M., 1978 | MR

[27] Nazaikinskii V. E., Oshmyan V. G., Sternin B. Yu., Shatalov V. E., “Integralnye operatory Fure i kanonicheskii operator”, Uspekhi matematicheskikh nauk, 36:2 (1981), 81–140 | MR | Zbl

[28] Nelson E., “Analiticheskie vektory”, Matematika, 6:3 (1962), 89–131

[29] Obukhov M. A., “K voprosu o geostroficheskom vetre”, Izvestiya AN SSSR, ser. geograficheskaya i geofizicheskaya, 13:4 (1949), 295–306

[30] Oleinik O. A., “O lineinykh uravneniyakh vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Matematicheskii sbornik, 69:1 (1966), 111–140 | MR | Zbl

[31] Oleinik O. A., Radkevich E. V., “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Matematicheskii analiz. Itogi nauki i tekhniki, VINITI AN SSSR, M., 1969

[32] Petkov V. M., Ivrii V. Ya., “Neobkhodimye usloviya korrektnosti zadachi Koshi dlya nestrogo giperbolicheskikh uravnenii”, Uspekhi matematicheskikh nauk, 29:5 (1974), 3–40 | MR

[33] Radkevich E. V., “Gipoellipticheskie operatory s kratnymi kharakteristikami”, Matematicheskii sbornik, 79 (1969), 193–216 | Zbl

[34] Feigin V. I., “Dve algebry psevdodifferentsialnykh operatorov v $\mathbf{R}^n$ i nekotorye prilozheniya”, Trudy moskovskogo matematicheskogo obschestva, 36, 1977, 155–194 | MR

[35] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR

[36] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978 | MR

[37] Alinhac S., “Parametrix et propagation des singularitèe pour un problém de Cauchy a multiplicité variable”, Asterisque, no. 34,35, 1976, 3–26 | MR | Zbl

[38] Anderson R. F. V., “The Weyl Functional Calculus”, Journal of Functional Analysis, 4:2 (1969), 240–267 | DOI | MR | Zbl

[39] Cook C. H., Fischer H. R., “Uniform convergence structures”, Mathematische Annalen, 173 (1967), 290–306 | DOI | MR | Zbl

[40] Duistermaat A., Hörmander L., “Fourier integral operators, II”, Acta Mathematica, 128:3,4 (1972) | MR | Zbl

[41] Folland G. B., “Subelliptic estimates and function spaces on nilpotent Lie groups”, Arkiv für Mathematik, 13 (1975), 161–207 | DOI | MR | Zbl

[42] Hellfer B., “Construction de parametrixes pour des operateurs pseudo-differentiels caracteristiques sur la reunion de deux cones lisses”, Comptes Rendus Acad. Sci., 283:4 (1976), A193–A195 | MR

[43] Hörmander L., “Hypoelliptic second order differential equations”, Acta Mathematica, 119 (1967), 147–171 | DOI | MR | Zbl

[44] Hörmander L., “Fourier integral operators”, Acta Mathematica, 127:1,2 (1971) | MR

[45] Leray J., “Analyse lagrangienne et mecanique quantique”, Séminaire sur les Équations aux Dérivées Partielles (1976–1977), I, Collège de France, Paris, 1977, Exp. No. 1, 303 | MR

[46] Maslov V. P., Metodes operatorios, Mir publishers, M., 1981 | MR

[47] Maslov V. P., “The characteristics of pseudo-differential operators and difference schemes”, Actes Congres Internationale Mathematique, v. 2 (Nice), 1970, 755–769 | MR

[48] Melin A., Sjostrand J., “Fourier integral operators with complex-valued phase functions”, Lecture Notes, 459, Springer, 1976, 120–223 | MR

[49] Nosmas J.-C., “Paramétrix du problème de Cauchy pour une classe d'opérateurs à caractéristiques de multiplicité variable”, C. R. Acad. Sci. Paris Sér. A-B, 285:16 (1977), A1065–A1068 | MR

[50] Rotschild L. P., Stein E. M., “Hypoelliptic differential operators and nilpotent groups”, Acta Mathematica, 137:3,4 (1976), 247–320 | DOI | MR

[51] Treves F., Introduction to pseudodifferential and Fourier integral operators, I–II, Plenum Press, N.Y., 1980 | MR | Zbl

[52] Weinstein A., “Symplectic manifolds and their Lagrangian submanifolds”, Advances in Math., 6 (1971), 329–346 | DOI | MR | Zbl