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Keywords:
spectrum; functional differential operator
spectrum; functional differential operator
Summary:
We study conditions of discreteness of spectrum of the functional-differential operator \[ \mathcal {L} u=-u''+p(x)u(x)+\int _{-\infty }^\infty (u(x)-u(s)) {\rm d}_s r(x,s) \] 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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