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.
Citation:
D. I. Borisov, “On the spectrum of a two-dimensional periodic operator with a small localized perturbation”, Izv. Math., 75:3 (2011), 471–505
\Bibitem{Bor11}
\by D.~I.~Borisov
\paper On the spectrum of a~two-dimensional periodic operator with a~small localized perturbation
\jour Izv. Math.
\yr 2011
\vol 75
\issue 3
\pages 471--505
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Linking options:
https://www.mathnet.ru/eng/im4113
https://doi.org/10.1070/IM2011v075n03ABEH002541
https://www.mathnet.ru/eng/im/v75/i3/p29
This publication is cited in the following 18 articles:
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., 146:4 (2021), 834–880
Borisov I D., Zezyulin D.A., “Bifurcations of Essential Spectra Generated By a Small Non-Hermitian Hole. i. Meromorphic Continuations”, Russ. J. Math. Phys., 28:4 (2021), 416–433
Yang Ch., Sun H., “Essential Spectra of Singular Hamiltonian Differential Operators of Arbitrary Order Under a Class of Perturbations”, Stud. Appl. Math., 147:1 (2021), 209–229
Drouot A., “The Bulk-Edge Correspondence For Continuous Dislocated Systems”, Ann. Inst. Fourier, 71:3 (2021), 1185–1239
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
Drouot A. Fefferman C.L. Weinstein M.I., “Defect Modes For Dislocated Periodic Media”, Commun. Math. Phys., 377:3 (2020), 1637–1680
Lu J., Watson A.B., Weinstein M.I., “Dirac Operators and Domain Walls”, SIAM J. Math. Anal., 52:2 (2020), 1115–1145
D. I. Borisov, A. M. Golovina, A. I. Mukhametrakhimova, “Analytic Continuation of Resolvents of Elliptic Operators in a Multidimensional Cylinder”, J Math Sci, 250:2 (2020), 260
Alexis Drouot, “Characterization of edge states in perturbed honeycomb structures”, Pure Appl. Analysis, 1:3 (2019), 385
D. I. Borisov, “Perturbations of the Continuous Spectrum of a Certain Nonlinear Two-Dimensional Operator Sheaf”, J. Math. Sci. (N. Y.), 252:2 (2021), 135–146
D. I. Borisov, M. Znojil, “On eigenvalues of a PT-symmetric operator in a thin layer”, Sb. Math., 208:2 (2017), 173–199
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
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
D.I. Borisov, “The Emergence of Eigenvalues of a PT-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
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
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
A. M. Golovina, “On the spectrum of elliptic operators with distant perturbation in the space”, St. Petersburg Math. J., 25:5 (2014), 735–754
Golovina A.M., “On the resolvent of elliptic operators with distant perturbations in the space”, Russ. J. Math. Phys., 19:2 (2012), 182–192
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