Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

Abstract

For the system of root functions of an operator defined by the differential operation −u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L2(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L2 and W2−1, respectively, and may have singularities at the endpoints of G such that q(x) = qR(x) +q′S(x) and the functions qS(x), p(x), q 2 S (x)w(x), p2(x)w(x), and qR(x)w(x) are integrable on the whole interval G, where w(x) = x(1 − x).