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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 3, Pages 29–64 (Mi izv4113)  

This article is cited in 8 scientific papers (total in 8 papers)

On the spectrum of a two-dimensional periodic operator with a small localized perturbation

D. I. Borisov

Bashkir State Pedagogical University

Abstract: We consider a two-dimensional periodic self-adjoint second-order differential operator on the plane with a small localized perturbation. The perturbation is given by an arbitrary (not necessarily symmetric) operator. It is localized in the sense that it acts on a pair of weighted Sobolev spaces and sends functions of sufficiently rapid growth to functions of sufficiently rapid decay. By studying the spectrum of the perturbed operator, we establish that the essential spectrum is stable, the residual spectrum is absent, and the set of isolated eigenvalues is discrete. We obtain necessary and sufficient conditions for the existence of new eigenvalues arising from the ends of lacunae in the essential spectrum. In the case when such eigenvalues exist, we construct the first terms of asymptotic expansions of these eigenvalues and the corresponding eigenfunctions.

Keywords: non-selfadjoint operator, perturbation, zone spectrum, eigenvalue, asymptotics.


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English version:
Izvestiya: Mathematics, 2011, 75:3, 471–505

Bibliographic databases:

UDC: 517.984
MSC: 35C20, 35J10
Received: 03.05.2009
Revised: 15.03.2010

Citation: D. I. Borisov, “On the spectrum of a two-dimensional periodic operator with a small localized perturbation”, Izv. RAN. Ser. Mat., 75:3 (2011), 29–64; Izv. Math., 75:3 (2011), 471–505

Citation in format AMSBIB
\by D.~I.~Borisov
\paper On the spectrum of a~two-dimensional periodic operator with a~small localized perturbation
\jour Izv. RAN. Ser. Mat.
\yr 2011
\vol 75
\issue 3
\pages 29--64
\jour Izv. Math.
\yr 2011
\vol 75
\issue 3
\pages 471--505

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    This publication is cited in the following articles:
    1. Golovina A.M., “On the resolvent of elliptic operators with distant perturbations in the space”, Russ. J. Math. Phys., 19:2 (2012), 182–192  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. M. Golovina, “On the spectrum of elliptic operators with distant perturbation in the space”, St. Petersburg Math. J., 25:5 (2014), 735–754  mathnet  crossref  mathscinet  zmath  isi  elib
    3. D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883  mathnet  crossref  crossref  mathscinet  isi  elib
    4. 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
    5. 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
    6. 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
    7. D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{PT}$-symmetric operator in a thin layer”, Sb. Math., 208:2 (2017), 173–199  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Borisov D.I., Dmitriev S.V., “On the Spectral Stability of Kinks in 2D Klein-Gordon Model with Parity-Time-Symmetric Perturbation”, Stud. Appl. Math., 138:3 (2017), 317–342  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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