Geodesic
Asymptotics of the Spectrum and Quantum Averages near the Boundaries of Spectral Clusters for Perturbed Two-Dimensional Oscillators
Matematičeskie zametki, Tome 92 (2012) no. 4, pp. 583-596.

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The eigenvalue problem for the perturbed resonant oscillator is considered. A method for constructing asymptotic solutions near the boundaries of spectral clusters using a new integral representation is proposed. The problem of calculating the averaged values of differential operators on solutions near the cluster boundaries is studied.
Keywords: spectral cluster, resonance, operator averaging method, coherent transform, WKB-approximation, turning point. 
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A. V. Pereskokov. Asymptotics of the Spectrum and Quantum Averages near the Boundaries of Spectral Clusters for Perturbed Two-Dimensional Oscillators. Matematičeskie zametki, Tome 92 (2012) no. 4, pp. 583-596. http://geodesic.mathdoc.fr/item/MZM_2012_92_4_a8/

[1] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v difraktsii korotkikh voln. Metod etalonnykh zadach, Nauka, M., 1972 | MR | Zbl

[2] S. Yu. Dobrokhotov, V. P. Maslov, “Nekotorye prilozheniya teorii kompleksnogo rostka k uravneniyam s malym parametrom”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat., 5, VINITI, M., 1975, 141–211 | MR | Zbl

[3] V. P. Maslov, Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nelineinyi analiz i ego prilozheniya, Nauka, M., 1977 | MR | Zbl

[4] M. V. Karasev, V. P. Maslov, “Asimptoticheskoe i geometricheskoe kvantovanie”, UMN, 39:6 (1984), 115–173 | MR | Zbl

[5] M. V. Karasev, “Birkhoff resonances and quantum ray method”, Proc. Intern. Seminar “Days of Diffraction – 2004”, St. Petersburg and Steklov Math. Institute, St. Petersburg, 2004, 114–126 | DOI

[6] M. V. Karasev, “Noncommutative algebras, nanostructures, and quantum dynamics generated by resonances”, Quantum Algebras and Poisson Geometry in Mathematical Physics, 216, Amer. Math. Soc. Trans. Ser. 2, Providence, RI, 2005, 1–17 ; Adv. Stud. Contemp. Math. (Kyungshang), 11:1 (2005), 33–56 ; Russ. J. Math. Phys., 13:2 (2006), 131–150 ; arXiv: math/0412542 | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[7] A. Weinstein, “Asymptotics of eigenvalue clusters for the Laplacian plus a potential”, Duke Math. J., 44:4 (1977), 883–892 | DOI | MR | Zbl

[8] A. Weinstein, “Eigenvalues of the Laplacian Plus a Potential”, Proceedings of the International Congress of Mathematicians, Helsinki, 1978, 803–805

[9] M. V. Karasev, E. M. Novikova, “Predstavlenie tochnykh i kvaziklassicheskikh sobstvennykh funktsii cherez kogerentnye sostoyaniya. Atom vodoroda v magnitnom pole”, TMF, 108:3 (1996), 339–387 | MR | Zbl

[10] M. V. Karasëv, E. M. Novikova, “Algebra s kvadratichnymi kommutatsionnymi sootnosheniyami dlya aksialno-vozmuschennogo polya Kulona–Diraka”, TMF, 141:3 (2004), 424–454 | MR | Zbl

[11] M. V. Karasëv, E. M. Novikova, “Algebra s polinomialnymi kommutatsionnymi sootnosheniyami dlya effekta Zeemana v pole Kulona–Diraka”, TMF, 142:1 (2005), 127–147 | MR | Zbl

[12] M. V. Karasëv, E. M. Novikova, “Algebra s polinomialnymi kommutatsionnymi sootnosheniyami dlya effekta Zeemana–Shtarka v atome vodoroda”, TMF, 142:3 (2005), 530–555 | MR | Zbl

[13] J. Schwinger, On angular momentum, U.S. Atomic Energy Commission, Report NYO-3071, 1952; Quantum Theory of Angular Momentum, A Collection of Reprints and Original Papers, eds. L. C. Biedenham, H. van Dam, Academic Press, New York, 1965, 229–279 | MR

[14] M. Karasev, E. Novikova, “Non-Lie permutation relations, coherent states, and quantum embedding”, Coherent Transform, Quantization, and Poisson Geometry, 187, Amer. Math. Soc. Transl. Ser. 2, Providence, RI, 1998, 1–202 | MR | Zbl

[15] V. V. Golubev, Lektsii po analiticheskoi teorii differentsialnykh uravnenii, GITTL, M.–L., 1950 | MR | Zbl

[16] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1974 | MR | Zbl

[17] M. V. Fedoryuk, Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1983 | MR | Zbl

[18] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987 | MR | Zbl

[19] Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, eds. M. Abramovits, I. Stigan, Nauka, M., 1979 | MR | Zbl

[20] M. V. Fedoryuk, Asimptotika: integraly i ryady, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1987 | MR | Zbl