Asymptotic analysis of non-self-adjoint Hill operators

Oktay Veliev

Open Mathematics (2013)

  • Volume: 11, Issue: 12, page 2234-2256
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Oktay Veliev. "Asymptotic analysis of non-self-adjoint Hill operators." Open Mathematics 11.12 (2013): 2234-2256. <http://eudml.org/doc/269430>.

@article{OktayVeliev2013,
abstract = {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.},
author = {Oktay Veliev},
journal = {Open Mathematics},
keywords = {Asymptotic formulas; Hill operator; Spectral singularities; Spectral operator; asymptotic formulas; spectral singularities; spectral operator},
language = {eng},
number = {12},
pages = {2234-2256},
title = {Asymptotic analysis of non-self-adjoint Hill operators},
url = {http://eudml.org/doc/269430},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Oktay Veliev
TI - Asymptotic analysis of non-self-adjoint Hill operators
JO - Open Mathematics
PY - 2013
VL - 11
IS - 12
SP - 2234
EP - 2256
AB - 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.
LA - eng
KW - Asymptotic formulas; Hill operator; Spectral singularities; Spectral operator; asymptotic formulas; spectral singularities; spectral operator
UR - http://eudml.org/doc/269430
ER -

References

top
  1. [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 Zbl1073.34094
  2. [2] Djakov P., Mityagin B.S., Instability zones of periodic one-dimensional periodic Schrödinger and Dirac operators, Russian Math. Surveys, 2006, 61(4), 663–776 http://dx.doi.org/10.1070/RM2006v061n04ABEH004343 
  3. [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 Zbl1242.34148
  4. [4] Dunford N., Schwartz J.T., Linear Operators, III, Wiley Classics Lib., John Wiley & Sons, New York, 1988 
  5. [5] Eastham M.S.P., The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973 Zbl0287.34016
  6. [6] Gasymov M.G., Spectral analysis of a class of second-order nonselfadjoint differential operators, Funktsional. Anal. i Prilozhen., 1980, 14(1), 14–19 (in Russian) 
  7. [7] Gel’fand I.M., Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.), 1950, 73, 1117–1120 (in Russian) 
  8. [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 Zbl1108.34064
  9. [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 Zbl1193.47037
  10. [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 Zbl1251.34100
  11. [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. [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. [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 Zbl1155.34365
  14. [14] Maksudov F.G., Veliev O.A., Spectral analysis of differential operators with periodic matrix coefficients, Differential Equations, 1989, 25(3), 271–277 
  15. [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. [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. [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. [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. [19] Mikhailov V.P., On Riesz bases in L 2(0, 1), Dokl. Akad. Nauk USSR, 1962, 114(5), 981–984 (in Russian) 
  20. [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 Zbl0471.34014
  21. [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. [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 Zbl1190.34111
  23. [23] Titchmarsh E.C., Eigenfunction Expansions Associated with Second-Order Differential Equations. II, Clarendon Press, Oxford, 1958 Zbl0097.27601
  24. [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 Zbl0188.46103
  25. [25] Veliev O.A., The one-dimensional Schrödinger operator with a periodic complex-valued potential, Soviet Math. Dokl., 1980, 21, 291–295 Zbl0449.34015
  26. [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. [27] Veliev O.A., Spectral expansion of nonselfadjoint differential operators with periodic coefficients, Differentsial’nye Uravneniya, 1986, 22(12), 2052–2059 (in Russian) Zbl0647.34019
  28. [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 Zbl1130.34060
  29. [29] Veliev O.A., Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Bound. Value Probl., 2008, #628973 Zbl1158.34051
  30. [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 Zbl1204.34117
  31. [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 Zbl1245.34083
  32. [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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.