Equiconvergence property for spectral expansions related to perturbations of the operator −u′′(−x) with initial data


Leonid V Kritskov, Abdizhahan M Sarsenbi




Uniform equiconvergence of spectral expansions related to the second-order differential opera- tors with involution: −u′′(−x) and −u′′(−x) + q(x)u(x) with the initial data u(−1) = 0, u′(−1) = 0 is obtained. Starting with the spectral analysis of the unperturbed operator, the estimates of the Green’s functions are established and then applied via the contour integrating approach to the spectral expansions. As a corollary, it is proved that the root functions of the perturbed operator form the basis in L2(−1, 1) for any complex-valued coefficient q(x) ∈ L2(−1, 1)