Geodesic
On the spectrum of a~periodic operator with a~small localized perturbation
Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 659-688.

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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. 
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D. I. Borisov; R. R. Gadyl'shin. On the spectrum of a~periodic operator with a~small localized perturbation. Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 659-688. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a2/

[1] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Daniel Davey Co., Inc., New York, 1966 | MR | MR | Zbl | Zbl

[2] M. S. P. Eastham, The spectral theory of periodic differential equations, Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973 | Zbl

[3] F. S. Rofe-Beketov, “A test for the finiteness of the number of discrete levels introduced into the gaps of a continuous spectrum by perturbations of a periodic potential”, Sov. Math. Dokl., 5 (1964), 689–692 | MR | Zbl

[4] V. A. Zheludev, “O sobstvennykh znacheniyakh vozmuschennogo operatora Shrëdingera s periodicheskim potentsialom”, Problemy matem. fiziki, 2 (1967), 108–123 | Zbl

[5] V. A. Zheludev, “Perturbations of the spectrum of the Schrödinger operator with a complex periodic potential”, Spectral Theory, 3 (1969), 25–41 | Zbl

[6] F. Gesztesy, B. Simon, “A short proof of Zheludev's theorem”, Trans. Amer. Math. Soc., 335:1 (1993), 329–340 | DOI | MR | Zbl

[7] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, New York, 1966 | MR | MR | Zbl | Zbl

[8] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Dover Publications, Inc., New York, 1975 | MR | Zbl

[9] E. Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–New York, 1980 | MR | MR | Zbl

[10] R. R. Gadyl'shin, “Local perturbations of the Schrödinger operator on the axis”, Theoret. and Math. Phys., 132:1 (2002), 976–982 | DOI | MR | Zbl

[11] A. F. Filippov, Differential equations with discontinuous right-hand sides, Mathematics and Its Applications: Soviet Series, 18, Kluwer, Dordrecht, 1988 | MR | MR | Zbl