Method of similar operators in research of spectral properties of difference operators with growthing potential
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 673-689.

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We study a difference operator with a general growing potential. We obtain asymptotic formulas for the eigenvalues, eigenvectors, and the spectral projections of such operators. We also obtain estimates for the equiconvergence of spectral decompositions. For an invertible difference operator, we estimate off-diagonal decay of the matrix of the inverse operator and the Fourier coefficients of the eigenvectors.
Keywords: difference operator, similar operator method, eigenvalue, eigenvector, spectral projector.
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G. V. Garkavenko; N. B. Uskova. Method of similar operators in research of spectral properties of difference operators with growthing potential. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 673-689. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a118/

[1] B. Musilimov, M. Otelbaev, “Estimation of the least eigenvalues to the matrix class corresponding to the Sturm–Liouville difference operators”, USSR Computational Mathematics and Mathematical Physics, 21:1 (1981), 68–83 | DOI | MR

[2] M. Otelbaev, “Coercive estimates for the solution of the difference equation”, Proceeding of the Steklov Institute of Mathematics, 181 (1989), 265–274 | MR | Zbl

[3] A.G. Kostyuchenko, I.S. Sargsyan, Distribution of eigenvalues (self-adjoint ordinary differential operators), Nauka, M., 1970 (in Russian) | MR

[4] M.A. Naimark, Linear differential operators, v. II, New York, 1968 | MR | Zbl

[5] A.G. Baskakov, “Estimates for the entries of inverse matrices and the spectral analysis of linear operators”, Izvestiya: Mathematics, 61:6 (1997), 1113–1135 | DOI | MR | Zbl

[6] A.G. Baskakov, I.A. Krishtal, “Memory estimation of inverse operators”, J. Funct. Anal., 267:8 (2014), 2551–2605 | DOI | MR | Zbl

[7] I.A. Blatov, “On algebras and applications of operators with pseudosparse matrices”, Siberian Math. Journal, 37:1 (1996), 32–52 | DOI | MR | Zbl

[8] I.Ts. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, RI, 1969 | MR

[9] A.G. Baskakov, “Methods of abstract harmonic analysis in the perturbation of linear operators”, Siberian Math. Journal, 24:1 (1983), 17–32 | DOI | MR

[10] A.G. Baskakov, “A theorem on splitting an operator, and some related questions in the analytic theory of perturbations”, Mathematics of the USSR-Izvestiya, 28:3 (1987), 421–444 | DOI | Zbl

[11] A.G. Baskakov, “Spectral analysis of perturbed nonquasianalytic and spectral operators”, Izvestiya: Mathematics, 45:1 (1995), 1–31 | DOI | MR | Zbl

[12] A.G. Baskakov, “Spectral analysis with respect to finite-dimensional perturbations of spectral operators”, Soviet Mathematics (Izvestiya VUZ. Matematika), 35:1 (1991), 1–11 | MR | Zbl

[13] A.G. Baskakov, A.V. Derbushev, A.O. Shcherbakov, “The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials”, Izvestiya: Mathematics, 75:3 (2011), 445–469 | DOI | MR | Zbl

[14] N.B. Uskova, “On a result of R. Terner”, Mathematical Notes, 76:6 (2004), 844–854 | DOI | MR | Zbl

[15] N.B. Uskova, “On the method of similar operators in Banach algebras”, Russian Mathematics (Izvestiya VUZ. Matematika), 49:3 (2005), 75–81 | MR | Zbl

[16] G.V. Garkavenko, “On diagonalization of certain classes of linear operators”, Russian Mathematics (Izvestiya VUZ. Matematika), 38:11 (1994), 11–16 | MR | Zbl

[17] A.G. Baskakov, “Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations”, Sbornik: Mathematics, 206:8 (2015), 1049–1086 | DOI | MR | Zbl

[18] N.B. Uskova, “On spectral properties of second order differential operator with matrix potential”, Differential Equations, 52:5 (2016), 557–567 | DOI | MR | Zbl

[19] N.B. Uskova, “On spectral properties of second order differential operator with matrix potential”, Ufa Math. Journal, 7:3 (2015), 88–99 | DOI | MR

[20] D.M. Polyakov, “Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients”, Siberian Mathematical Journal, 56:1 (2015), 138–154 | DOI | MR | Zbl

[21] A.G. Baskakov, D.M. Polyakov, “Spectral Properties of the Hill Operator”, Mathematical Notes, 99:4 (2016), 598–602 | DOI | MR | Zbl

[22] A.O. Shcherbakov, “Regularized Trace of the Dirac Operator”, Mathematical Notes, 98:1 (2015), 168–179 | DOI | MR | Zbl

[23] W. Rudin, Functional Analysis, McGraw-Hill book, 1973 | MR | Zbl

[24] N. Dunford, J.T. Schwartz, Linear operators. Spectral operators, v. III, Pure and Applied Mathematics, VII, Wiley-Interscience, New York, 1971 | MR | Zbl

[25] A.G. Baskakov, “Wiener's theorem and the asymptotic estimates of the elements of inverse matrices”, Functional Analysis and Its Applications, 24:3 (1990), 222–224 | DOI | MR | Zbl

[26] A.G. Baskakov, “Abstract harmonic analysis and asymptotic estimates of elements of inverse matrices”, Mathematical Notes, 52:2 (1992), 764–771 | DOI | MR | Zbl

[27] A.G. Baskakov, “Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis”, Siberian Mathematical Journal, 38:1 (1997), 10–22 | DOI | MR | Zbl

[28] J.-P. Kahane, Series de Fourier absolument convergence, New York, 1970 | MR

[29] M.A. Shubin, “Pseudodifference operators and their Green's function”, Mathematics of the USSR-Izvestiya, 26:3 (1986), 605–622 | DOI | Zbl

[30] A.G. Baskakov, Haromic analysis of linear operators, Voronezh University Press, Voronezh, 1987 (in Russian) | MR