On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems

The paper deals with the Sturm-Liouville operator L-y = -y '' + q(x)y, x is an element of [0,1], generated in the space L-2 = L-2[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev space W-1(P)[0,1] for some integer p >= 0 and satisfy the conditions q((k))(0) = q((k))(1) = 0 for 0 <= k <= s - 1, where s <= p. Let the functions Q and S be defined by the equalities Q(x) = integral(x)(0) q(t) dt, S(x) = Q(2)(x) and let q(n), Q(n), and S-n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system {e(2 pi inx)}(-infinity)(infinity). Assume that the sequence q(2n) - S-2n + 2Q(0)Q(2n) decreases not faster than the powers n(-s-2). Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L-2[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q(2n) - S-2n + 2Q(0)Q(2n) asymptotic to q(-2n) - S-2n + 2Q(0)Q(-2n), n > 1, holds.