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 $$Ly=-y^{\prime \prime }+q(x)y,x\in [0,1],$$ generated in the space $L_2=L_2[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator $L$ are proved. One of the main results is the following. \textsl{Let $q$ belong to Sobolev space $W_1^p[0,1]$ with some integer $p\ge 0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\le k\le s-1$, where s}$\le p.$ \textsl{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 $q_n$ be the Fourier coefficients of $q,Q,S$ with respect to the trigonometric system $\{e^{2\pi inx}{\}}_{-\infty }^{\infty }$. 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 eigen 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 $$q_{2n}-S_{2n}+Q_0{Q}_{2n}\asymp q_{-2n}-S_{-2n}+2Q_0{Q}_{-2n},n>1,$$ holds.

16 pages