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
$$ Ly=-y''+q(x)y, \qquad x\in[0,1], $$
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 $p\ge0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\le k\le s-1$, where $s\le p$. Let the functions $Q$ and $S$ be defined by the equalities
$$ Q(x)=\int_0^xq(t)\,dt,\qquad S(x)=Q^2(x) $$
and let $q_n$, $Q_n$, and $S_n$ be the Fourier coefficients of $q$, $Q$, and $S$ with respect to the trigonometric system $\{e^{2\pi inx}\}_{-\infty}^\infty$. Assume that the sequence $q_{2n}-S_{2n}+2Q_0Q_{2n}$ 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
$$ q_{2n}-S_{2n}+2Q_0Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_0Q_{-2n},\qquad n>1, $$
holds.