Geodesic
The spectrum of a self-adjoint differential
Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1063-1093 Cet article a éte moissonné depuis la source Math-Net.Ru

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The asymptotic behaviour of the spectrum of a self-adjoint second-order differential operator on the axis is investigated. The coefficients of this operator depend on rapid and slow variables and are periodic in the rapid variable. The period of oscillations in the rapid variable is a small parameter. The dependence of the coefficients on the rapid variable is localized, and they stop depending on it at infinity. Asymptotic expansions for the eigenvalues and the eigenfunctions of the operator in question are constructed. It is shown that, apart from eigenvalues convergent to eigenvalues of the homogenized operator as the small parameter converges to zero, the perturbed operator can also have an eigenvalue convergent to the boundary of the continuous spectrum. Necessary and sufficient conditions for the existence of such an eigenvalue are obtained. Bibliography: 22 titles. 
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D. I. Borisov; R. R. Gadyl'shin. The spectrum of a self-adjoint differential. Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1063-1093. http://geodesic.mathdoc.fr/item/SM_2007_198_8_a0/

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., 5, North-Holland, Amsterdam–New York–Oxford, 1978 | MR | Zbl

[2] E. Sanches-Palensiya, Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 ; E. Sánchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–Heidelberg–New York, 1980 | MR | MR | Zbl

[3] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, Nauka, M., 1984 ; N. S. Bakhvalov, G. Panasenko, Homogenisation: Averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Math. Appl. (Soviet Ser.), 36, Kluwer Acad. Publ., Dordrecht, 1989 | MR | Zbl | MR | Zbl

[4] O. A. Oleinik, G. A. Iosifyan, A. S. Shamaev, Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo MGU, M., 1990 ; O. A. Oleǐnik, A. S. Shamaev, G. A. Yosifian, Mathematical problems in elasticity and homogenization, Stud. Math. Appl., 26, North-Holland, Amsterdam, 1992 | MR | Zbl | MR | Zbl

[5] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Nauka, M., 1993 ; V. V. Zhikov, S. M. Kozlov, O. A. Olejnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994 | MR | Zbl | MR | Zbl

[6] A. L. Pyatnitskii, G. A. Chechkin, A. S. Shamaev, Usrednenie. Metody i nekotorye prilozheniya, Tamara Rozhkovskaya, Novosibirsk, 2007 | MR

[7] P. Kuchment, “The mathematics of photonic crystals”, Mathematical modeling in optical science, Proceedings of a minisymposium on optics at SIAM's annual meeting (Stanford Univ., Palo Alto, CA, 1997), Frontiers Appl. Math., 22, SIAM, Philadelphia, PA, 2001, 207–272 | MR | Zbl

[8] V. V. Zhikov, “O lakunakh v spektre nekotorykh divergentnykh ellipticheskikh operatorov s periodicheskimi koeffitsientami”, Algebra i analiz, 16:5 (2004), 34–58 ; V. V. Zhikov, “On spectrum gaps of some divergent elliptic operators with periodic coefficients”, St. Petersburg Math. J., 16:5 (2005), 773–790 | MR | Zbl | DOI

[9] V. S. Buslaev, “Kvaziklassicheskoe priblizhenie dlya uravnenii s periodicheskimi koeffitsientami”, UMN, 42:6 (1987), 77–98 | MR | Zbl

[10] M. Marx, “On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator”, Asymptot. Anal., 48:4 (2006), 295–357 | MR | Zbl

[11] B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions”, Ann. Physics, 97:2 (1976), 279–288 | DOI | MR | Zbl

[12] M. Klaus, “On the bound state of Schrödinger operators in one dimension”, Ann. Physics, 108:2 (1977), 288–300 | DOI | MR | Zbl

[13] R. Blankenbecler, M. L. Goldberger, B. Simon, “The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians”, Ann. Physics, 108:1 (1977), 69–78 | DOI | MR

[14] M. Klaus, B. Simon, “Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case”, Ann. Physics, 130:2 (1980), 251–281 | DOI | MR | Zbl

[15] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh operatora Shredingera na osi”, TMF, 132:1 (2002), 97–104 | MR | Zbl

[16] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 ; T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin–Heidelberg–New York, 1966 | MR | Zbl | MR | Zbl

[17] I. M. Glazman, Pryamye metody spektralnogo kachestvennogo analiza singulyarnykh differentsialnykh operatorov, Fizmatgiz, M., 1963 ; I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Transl., Jerusalem; Oldbourne Press, London, 1965 | MR | Zbl | MR | Zbl

[18] V. V. Zhikov, “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

[19] D. I. Borisov, R. R. Gadylshin, “O spektre operatora Shredingera s bystro ostsilliruyuschim finitnym potentsialom”, TMF, 147:1 (2006), 58–63 | MR

[20] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki. I. Funktsionalnyi analiz, Mir, M., 1977 ; V. Reed, B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York–London, 1972 | MR | MR | Zbl

[21] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 ; V. P. Mikhailov, Partial differential equations, Mir, Moscow, 1978 | MR | Zbl | MR | Zbl

[22] D. I. Borisov, “O spektre operatora Shredingera, vozmuschennogo bystro ostsilliruyuschim potentsialom”, Problemy matematicheskogo analiza, 33 (2006), 13–78 | Zbl