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2013 | 11 | 12 | 2234-2256
Tytuł artykułu

Asymptotic analysis of non-self-adjoint Hill operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
12
Strony
2234-2256
Opis fizyczny
Daty
wydano
2013-12-01
online
2013-10-08
Twórcy
autor
Bibliografia
  • [1] Dernek N., Veliev O.A., On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. Math., 2005, 145(1), 113–123 http://dx.doi.org/10.1007/BF02786687
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  • [3] Djakov P., Mityagin B.S., Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Dokl. Math., 2011, 83(1), 5–7 http://dx.doi.org/10.1134/S1064562411010017
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  • [8] Gesztesy F., Tkachenko V., When is a non-self-adjoint Hill operator a spectral operator of scalar type?, C. R. Math. Acad. Sci. Paris, 2006, 343(4), 239–242 http://dx.doi.org/10.1016/j.crma.2006.06.014
  • [9] Gesztesy F., Tkachenko V., A criterion for Hill operators to be spectral operators of scalar type, J. Anal. Math., 2009, 107, 287–353 http://dx.doi.org/10.1007/s11854-009-0012-5
  • [10] Gesztesy F., Tkachenko V., A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations, 2012, 253(2), 400–437 http://dx.doi.org/10.1016/j.jde.2012.04.002
  • [11] Kerimov N.B., Mamedov Kh.R., On the Riesz basis property of root functions of some regular boundary value problems, Math. Notes, 1998, 64(3–4), 483–487 http://dx.doi.org/10.1007/BF02314629
  • [12] Kessel’man G.M., On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vysš. Učebn. Zaved. Matematika, 1964, 2(39), 82–93 (in Russian)
  • [13] Makin A.S, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math., 2006, 73(1), 71–76 http://dx.doi.org/10.1134/S1064562406010194
  • [14] Maksudov F.G., Veliev O.A., Spectral analysis of differential operators with periodic matrix coefficients, Differential Equations, 1989, 25(3), 271–277
  • [15] Marchenko V.A., Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel, 1986 http://dx.doi.org/10.1007/978-3-0348-5485-6
  • [16] McGarvey D., Operators commuting with translations by one. Part I. Representation theorems, J. Math. Anal. Appl., 1962, 4(3), 366–410 http://dx.doi.org/10.1016/0022-247X(62)90038-0
  • [17] McGarvey D.C., Operators commuting with translations by one. Part II. Differential operators with periodic coefficients in L p(−∞,∞), J. Math. Anal. Appl., 1965, 11, 564–596 http://dx.doi.org/10.1016/0022-247X(65)90105-8
  • [18] McGarvey D.C., Operators commuting with translations by one. Part III. Perturbation results for periodic differential operators, J. Math. Anal. Appl., 1965, 12(2), 187–234 http://dx.doi.org/10.1016/0022-247X(65)90033-8
  • [19] Mikhailov V.P., On Riesz bases in L 2(0, 1), Dokl. Akad. Nauk USSR, 1962, 114(5), 981–984 (in Russian)
  • [20] Shkalikov A.A., On the basis problem of the eigenfunctions of an ordinary differential operator, Russian Math. Surveys, 1979, 34(5), 249–250 http://dx.doi.org/10.1070/RM1979v034n05ABEH003901
  • [21] Shkalikov A.A., Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, 37(6), 12–21 (in Russian)
  • [22] Shkalikov A.A., Veliev O.A., On the Riesz basis property of eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 2009, 85(5-6), 647–660 http://dx.doi.org/10.1134/S0001434609050058
  • [23] Titchmarsh E.C., Eigenfunction Expansions Associated with Second-Order Differential Equations. II, Clarendon Press, Oxford, 1958
  • [24] Tkachenko V.A., Spectral analysis of the one-dimensional Schrödinger operator with a periodic complex-valued potential, Soviet Math. Dokl., 1964, 5, 413–415
  • [25] Veliev O.A., The one-dimensional Schrödinger operator with a periodic complex-valued potential, Soviet Math. Dokl., 1980, 21, 291–295
  • [26] Veliev O.A., The spectrum and spectral singularities of differential operators with periodic complex-valued coefficients, Differentsial’nye Uravneniya, 1983, 19(8), 1316–1324 (in Russian)
  • [27] Veliev O.A., Spectral expansion of nonselfadjoint differential operators with periodic coefficients, Differentsial’nye Uravneniya, 1986, 22(12), 2052–2059 (in Russian)
  • [28] Veliev O.A., Spectral expansion for a nonselfadjoint periodic differential operator, Russ. J. Math. Phys., 2006, 13(1), 101–110 http://dx.doi.org/10.1134/S1061920806010109
  • [29] Veliev O.A., Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Bound. Value Probl., 2008, #628973
  • [30] Veliev O.A., On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math., 2010, 176, 195–208 http://dx.doi.org/10.1007/s11856-010-0025-x
  • [31] Veliev O.A., On the basis property of the root functions of differential operators with matrix coefficients, Cent. Eur. J. Math., 2011, 9(3), 657–672 http://dx.doi.org/10.2478/s11533-011-0015-1
  • [32] Veliev O.A., Toppamuk Duman M., The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl., 2002, 265(1), 76–90 http://dx.doi.org/10.1006/jmaa.2001.7693
Typ dokumentu
Bibliografia
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Identyfikator YADDA
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