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We study conditions of discreteness of spectrum of the operator defined by Lu=−1ρ(x)u″(x)−∫0∞u(s)dsr(x,s), x∈[0,∞). The operator has two singularities at the ends of the interval (0,∞). The second question is positivity of solutions of the equation Lu=f under boundary conditions u(0)=0, u′(∞)=0. The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators.

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis