Asymptotic behavior of the spectra of second-order non-self-adjoint systems of differential operators

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 8–16, April, 1992.     

Wave operators for non-self-adjoint operators and the functional model

Explicit formulas for the wave operators satisfying certain conditions are obtained in terms of the functional model of Nagy-Foias.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 129–135, 1977.The author is grateful to B. S. Pavlov for his attention to this work.     

Dichotomy of the spectrum of non-self-adjoint operators

Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 24–30, May–June, 1991.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Trace formulas for non-self-adjoint operators with discrete spectrum

We prove a trace formula, which is an abstract analog of the well-known Gel'fand-Levitan formula.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 228–236, 1988.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

On localization of the spectrum of non-self-adjoint differential operators

In this paper, we consider the problem on localization of the spectrum of non-self-adjoint differential operators on unbounded domains with power coefficients. To find the location of spectrum points in the complex plane, we use isospectral deformations of differential operators and the properties of families of closed operators analytic in the Kato sense. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 11–19, 2006.     

The Krein string and characteristic functions of non-self-adjoint operators

The operator generated by the Krein string is investigated in the framework of the extension theory of symmetric operators. A simple proof of the complete non-self-adjointness of the operator is proposed. The scattering function of the string is obtained with the help of the Derkach-Malamud formula for characteristic functions of almost solvable extensions. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 115–134.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.