We study conditions of discreteness of spectrum of the operator defined by . 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

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... 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. MSC: Primary 34K08; 34K10; secondary 34K12