Geodesic
New formulas for Maslov's canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics
Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 355-386 Cet article a éte moissonné depuis la source Math-Net.Ru

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We suggest a new representation of Maslov's canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds. We present the results in the two-dimensional case and illustrate them with examples.
Keywords: semiclassical asymptotics, focal point, caustic, integral representation, Bessel function, Schrödinger equation, wave beam. 
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     title = {New formulas for {Maslov's} canonical operator in a~neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics},
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S. Yu. Dobrokhotov; G. N. Makrakis; V. E. Nazaikinskii; T. Ya. Tudorovskii. New formulas for Maslov's canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics. Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 355-386. http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a0/

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

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

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

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

[5] V. P. Maslov, V. E. Nazaikinskii, “Algebry s obschimi perestanovochnymi sootnosheniyami i ikh prilozheniya. I. Psevdodifferentsialnye uravneniya s rastuschimi koeffitsientami”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat., 13, VINITI, M., 1979, 5–144 | DOI | MR | Zbl

[6] V. V. Belov, S. Yu. Dobrokhotov, TMF, 92:2 (1992), 215–254 | DOI | MR

[7] S. Yu. Dobrokhotov, B. Tirotstsi, A. I. Shafarevich, Matem. zametki, 82:5 (2007), 792–796 | DOI | DOI | MR | Zbl

[8] S. Dobrokhotov, A. Shafarevich, B. Tirozzi, Russ. J. Math. Phys., 15:2 (2008), 192–221 | DOI | MR | Zbl

[9] L. Hörmander, Acta Math., 127:1–2 (1971), 79–183 | DOI | MR

[10] S. Yu. Dobrokhotov, G. Makrakis, V. E. Nazaikinskii, Fourier integrals and a new representation of Maslov's canonical operator near caustics, arXiv: 1307.2292 | MR

[11] V. I. Arnold, Osobennosti kaustik i volnovykh frontov, Fazis, M., 1996 | MR

[12] M. V. Berry, S. Klein, Proc. Nat. Acad. Sci. U.S.A., 93:6 (1996), 2614–2619 | DOI | MR | Zbl

[13] J. J. Stamns, B. Spjelkavik, Optica, 30:9 (1983), 1331–1358 | DOI | MR

[14] Yu. A. Kravtsov, Yu. I. Orlov, Geometricheskaya optika neodnorodnykh sred, Nauka, M., 1980 | MR

[15] B. R. Vainberg, Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, MGU, M., 1982 | MR

[16] V. V. Kucherenko, TMF, 1:3 (1969), 384–406 | DOI | MR

[17] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1989 | MR

[18] S. Yu. Dobrokhotov, M. Rulo, Matem. zametki, 87:3 (2010), 458–463 ; S. Yu. Dobrokhotov, M. Rouleux, Asymptotic Anal., 74:1–2 (2011), 33–73 | DOI | DOI | MR | Zbl | DOI | MR | Zbl

[19] M. V. Fedoryuk, Asimptotika: integraly i ryady, Nauka, M., 1987 | MR

[20] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii, v. 1, Nauka, M., 1982 | MR