Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Garkavenko G. V., Uskova N. B. Spectral Analysis of a Class of Difference Operators with Growing Potential. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 395-402. DOI: 10.18500/1816-9791-2016-16-4-395-402, EDN: XHPYGT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 178)

Spectral Analysis of a Class of Difference Operators with Growing Potential

Garkavenko Galina Valerievna, Voronezh State Technical University
Uskova Natal’ya Borisovna, Voronezh State Technical University

The similar operator method is used for the spectral analysis of the closed difference operator of the form (A x)(n) = x(n + 1) + x(n − 1) − 2x(n) + a(n)x(n), n ∈ Z under consideration in the Hilbert space l2(Z) of bilateral sequences of complex numbers, with a growing potential a : Z → C. The asymptotic estimates of eigenvalue, eigenvectors, spectral estimation of equiconvergence applications for the test operator and the operator of multiplication by a sequence a : Z → C. For the study of the operator, it is represented in the form of A − B, where (Ax)(n) = a(n)x(n), n ∈ Z, x ∈ l2(Z) with the natural domain. This operator is normal with known spectral properties and acts as the unperturbed operator in the method of similar operators. The bounded operator (Bx)(n) = −x(n + 1) − x(n − 1) + 2x(n), n ∈ Z, x ∈ l2(Z), acts as the perturbation. 

  1. Musilimov B., Otelbaev M. Estimation of the least eigenvalues for the matrix class corresponding to the Sturm-Liouville difference equation. U.S.S.R. Comput. Math. Math. Phys., 1981, vol. 21, iss. 6, pp. 68–73. DOI: https://doi.org/10.1016/0041-5553(81)90151-8.
  2. Baskakov A. G. Method of abstract harmonic analysis in the theory of perturbation of linear operators. Siberian Math. J., 1983, vol. 24, no. 1, pp. 17– 32 (in Russian).
  3. Baskakov A. G. A theorem on splitting an operator, and some related questions in the analytic theory of perturbations. Math. USSRIzv., 1987, vol. 28, iss. 3, pp. 421–444. DOI: https://doi.org/10.1070/IM1987v028n03ABEH000891.
  4. Baskakov A. G. Spectral analysis of perturbed nonquasianalytic and spectral operators. Russian Acad. Sci. Izv. Math., 1995, vol. 45, iss. 1, pp. 1– 31. DOI: https://doi.org/10.1070/IM1995v045n01ABEH001621.
  5. Baskakov A. G., Derbushev A. V., Shcherbakov A. O. The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials. Izv. Math., 2011, vol. 75, iss. 3, pp. 445–469. DOI: https://doi.org/10.1070/IM2011v075n03ABEH002540.
  6. Uskova N. B. On spectral properties of Shturm – Liouville operator with matrix potential. Ufa Math. J., 2015, vol. 7, iss. 3, pp. 84–94. DOI: https://doi.org/10.13108/2015-7-3-84.
  7. Polyakov D. M. Spectral analiysis of a forth-order nonsefaioint operator with nonsmoth coefficients. Siberian Math. J., 2015, vol. 56, iss. 1, pp. 138– 154. DOI: https://doi.org/10.1134/S0037446615010140.
  8. Baskakov A. G. Estimates for the Green’s function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relation. Sb. Math., 2015, vol. 206, no. 8, pp. 1049–1086. DOI: https://doi.org/10.1070/SM2015v206n08ABEH004489.
  9. Garkavenko G. V. On diagonalization of certian classes of linear operator. Russian Math. (Iz. VUZ), 1994, vol. 38, iss. 11, pp. 11–16.
  10. Uskova N. B. On the method of similar operators in Banach algebras. Russian Math. (Iz. VUZ), 2005, vol. 49, iss. 3, pp. 75–81.
  11. Uskova N. B. On the spectral properties of a second-order differential operator with a matrix potential. Differential Equations, 2016, vol. 52, no. 5, pp. 557–567.
  12. Danford N., Schwartz J. T. Linear Operators. Pt. III : Spectral Operators. New York, Interscience Publ., 1971. 689 p. (Russ. ed.: Danford N., Schwartz J. T. Lineinye operatory : v 3 t. T. 3 : Spektral’nye operatory. Moscow, Mir, 1974. 664 p.)