Spectral analysis for a class of integral-difference operators: known facts, new results, and open problems.

Melnikov, Yuri B.

Displaying similar documents to “Spectral analysis for a class of integral-difference operators: known facts, new results, and open problems.”

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The paper is concerned with conditions guaranteeing that a bounded operator in a reflexive Banach space is a scalar type spectral operator. The cases where the spectrum of the operator lies on the real axis and on the unit circle are studied separately.

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We study conditions of discreteness of spectrum of the functional-differential operator $ℒ u = - u^{''} + p (x) u (x) + \int_{- \infty}^{\infty} (u (x) - u (s)) d_{s} r (x, s)$ on $(- \infty, \infty)$ . In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

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On the localization of the spectrum for quasi-selfadjoint extensions of a Carleman operator

S. M. Bahri (2012)

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In the present work, using a formula describing all scalar spectral functions of a Carleman operator $A$ of defect indices $(1, 1)$ in the Hilbert space $L^{2} (X, μ)$ that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator $A$ .