Basis properties of eigenfunctions of the differential operator −u″(−x) + q(x)u(x) with Cauchy data

Uniform equiconvergence of spectral expansions associated with two second-order differential operators with involution −u″(−x) + q(x)u(x) and the Cauchy data u(−1) = 0, u′(−1) = 0 is obtained. The proof uses the Cauchy integral method and the Green’s function asymptotics of the considered operator. As a corollary, it is proved that the root functions of this operator form the basis in L 2(−1, 1) for any continuous complex-valued coefficient q(x).