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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 4, Pages 37–66 (Mi izv1146)  

This article is cited in 23 scientific papers (total in 23 papers)

On the spectrum of a periodic operator with a small localized perturbation

D. I. Borisov, R. R. Gadyl'shin

Bashkir State Pedagogical University

Abstract: The paper deals with the spectrum of a periodic self-adjoint differential operator on the real axis perturbed by a small localized non-self-adjoint operator. We show that the continuous spectrum does not depend on the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We study the problem of the existence of eigenvalues embedded in the continuous spectrum, obtain necessary and sufficient conditions for the existence of eigenvalues, construct asymptotic expansions of the eigenvalues and corresponding eigenfunctions and consider some examples.

DOI: https://doi.org/10.4213/im1146

Full text: PDF file (696 kB)
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English version:
Izvestiya: Mathematics, 2008, 72:4, 659–688

Bibliographic databases:

UDC: 517.984
MSC: 35C20, 35J05, 35J10, 35J25, 35B99, 35P25, 47A40, 76Q05
Received: 25.08.2006
Revised: 24.03.2008

Citation: D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of a periodic operator with a small localized perturbation”, Izv. RAN. Ser. Mat., 72:4 (2008), 37–66; Izv. Math., 72:4 (2008), 659–688

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Borisov I D., Zezyulin D.A., Znojil M., “Bifurcations of Thresholds in Essential Spectra of Elliptic Operators Under Localized Non-Hermitian Perturbations”, Stud. Appl. Math.  crossref  isi
    2. Bikmetov A., Gadyl'shin R., “On quantum waveguide with shrinking potential”, Russ. J. Math. Phys., 17:1 (2010), 19–25  crossref  mathscinet  zmath  isi  elib  scopus
    3. R. R. Gadyl'shin, I. Kh. Khusnullin, “Schrödinger operator on the axis with potentials depending on two parameters”, St. Petersburg Math. J., 22:6 (2011), 883–894  mathnet  crossref  mathscinet  zmath  isi
    4. D. I. Borisov, “On the spectrum of a two-dimensional periodic operator with a small localized perturbation”, Izv. Math., 75:3 (2011), 471–505  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. R. R. Gadyl'shin, I. Kh. Khusnullin, “Perturbation of a periodic operator by a narrow potential”, Theoret. and Math. Phys., 173:1 (2012), 1438–1444  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    6. Lescarret V., Schneider G., “Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide”, Z. Angew. Math. Phys., 63:3 (2012), 401–427  crossref  mathscinet  zmath  isi  scopus
    7. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    8. C. L. Fefferman, J. P. Lee-Thorp, M. I. Weinstein, “Topologically protected states in one-dimensional continuous systems and Dirac points”, Proceedings of the National Academy of Sciences, 111:24 (2014), 8759  crossref  mathscinet  zmath  scopus
    9. A. R. Bikmetov, V. F. Vil'danova, I. Kh. Khusnullin, “On perturbation of a Schrödinger operator on axis by narrow potentials”, Ufa Math. J., 7:4 (2015), 24–31  mathnet  crossref  isi  elib
    10. A. Yu. Trynin, “On some properties of sinc approximations of continuous functions on the interval”, Ufa Math. J., 7:4 (2015), 111–126  mathnet  crossref  isi  elib
    11. Duchene V., Vukicevic I., Weinstein M.I., “Oscillatory and Localized Perturbations of Periodic Structures and the Bifurcation of Defect Modes”, SIAM J. Math. Anal., 47:5 (2015), 3832–3883  crossref  mathscinet  zmath  isi  elib  scopus
    12. Duchene V., Vukicevic I., Weinstein M.I., “Homogenized Description of Defect Modes in Periodic Structures With Localized Defects”, Commun. Math. Sci., 13:3 (2015), 777–823  crossref  mathscinet  zmath  isi  elib  scopus
    13. V. V. Zhikov, S. E. Pastukhova, “Operator estimates in homogenization theory”, Russian Math. Surveys, 71:3 (2016), 417–511  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. Borisov D., Golovina A., Veselic I., “Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate”, Ann. Henri Poincare, 17:9 (2016), 2341–2377  crossref  mathscinet  zmath  isi  elib  scopus
    15. Fefferman C.L., Lee-Thorp J.P., Weinstein M.I., “Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures”, 2D Mater., 3:1 (2016), 014008  crossref  mathscinet  isi  elib  scopus
    16. Bikmetov A.R., Gadyl'shin R.R., “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18  crossref  mathscinet  zmath  isi  scopus
    17. A. R. Bikmetov, I. Kh. Khusnullin, “Perturbation of Hill operator by narrow potentials”, Russian Math. (Iz. VUZ), 61:7 (2017), 1–10  mathnet  crossref  isi
    18. Fefferman C.L., Lee-Thorp J.P., Weinstein M.I., “Topologically Protected States in One-Dimensional Systems”, Mem. Am. Math. Soc., 247:1173 (2017), 1+  crossref  mathscinet  isi  scopus
    19. I. Kh. Khusnullin, “Vozmuschenie volnovoda uzkim potentsialom”, Tr. IMM UrO RAN, 23, no. 2, 2017, 274–284  mathnet  crossref  elib
    20. D. I. Borisov, “Vozmuscheniya nepreryvnogo spektra odnogo nelineinogo dvumernogo operatornogo puchka”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 13–24  mathnet  mathscinet
    21. Borisov D., Cardone G., “Spectra of Operator Pencils With Small P & Xdcab;& X1D4Af;& Xdcaf;-Symmetric Periodic Perturbation”, ESAIM-Control OPtim. Calc. Var., 26 (2020), UNSP 21  crossref  isi
    22. Drouot A., Fefferman C.L., Weinstein M.I., “Defect Modes For Dislocated Periodic Media”, Commun. Math. Phys., 377:3 (2020), 1637–1680  crossref  mathscinet  isi
    23. Lu J., Watson A.B., Weinstein M.I., “Dirac Operators and Domain Walls”, SIAM J. Math. Anal., 52:2 (2020), 1115–1145  crossref  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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