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On discreteness of spectrum of a functional differential operator. (English) Zbl 1340.34231
Summary: We study conditions for the discreteness of the 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 Schrödinger operator. In this paper, we consider a symmetric operator with real spectrum. Conditions for the discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain a simple condition for the discreteness of the spectrum.
34K08 Spectral theory of functional-differential operators
34K06 Linear functional-differential equations
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
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