Sergey Labovskiy, Statistics and Informatics, Moscow State University of Economics, Nezhinskaya st. 7, 119501 Moscow, Russia, e-mail: labovski@gmail.com; Mário Frengue Getimane, Instituto Superior de Transportes e Comunicações, Prolong. da Av. Kim Il Sung (IFT/TDM) - Edificio D1 Caixa Postal, 2088 Maputo, Mozambique, e-mail: mgetimane@isutc.transcom.co.mz
Abstract: We study conditions of discreteness of spectrum of the functional-differential operator
\mathcal{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 Schrodinger 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.
Keywords: spectrum; functional differential operator
Classification (MSC 2010): 34K06, 34L05
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