We study conditions of discreteness of spectrum of the functional-differential operator Lu = −u ′′ + p(x)u(x) + ∫ ∞ −∞ (u(x) − u(s)) dsr(x, s) on (−∞,∞). 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.
We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations u ′′(x) +∑ i pi(x)u ′ (hi(x)) +∑ i qi(x)u(gi(x)) = 0 without the delay conditions hi(x), gi(x) ≤ x, i = 1, 2, . . ., and u ′′(x) + ∫ ∞ 0 u ′ (s)dsr1(x, s) + ∫ ∞ 0 u(s)dsr0(x, s) = 0.