On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems

Abstract

The paper deals with the Sturm-Liouville operator $$ Ly = - y'' + q(x)y, x \in [0,1], $$ generated in the space L2 = L2[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 spaceW1p [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 $$ Q(x) = \int_0^x {q(t)dt, S(x) = Q^2 (x)} $$ and let qn, Qn, and Sn be the Fourier coefficients of q, Q, and S with respect to the trigonometric system \( \{ e^{2\pi inx} \} _{ - \infty }^\infty \). Assume that the sequence q2n − S2n + 2Q0Q2n 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 L2[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n} - s_{2n} + 2Q_0 Q_{2n} \asymp q_{ - 2n} - s_{2n} + 2Q_0 Q_{ - 2n} , n > 1, $$ holds.