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

2008/11/14 by Андрей Андреевич Шкаликов, Shkalikov, A. A., O. A. Veliev +1 · 1 citation
Materials Science · Mathematics · #34Bxx #34Lxx #FOS: Mathematics #Graph theory and applications #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0811.2337

openalex publication_date 2008/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper deals with the Sturm-Liouville operator Ly=-y^′′+q(x)y, x∈\lbrack0,1], generated in the space L₂=L₂[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. \textslLet q belong to Sobolev space W₁^p[0,1] with some integer p≥0 and satisfy the conditions q^(k)(0)=q^(k)(1)=0 for 0≤ k≤ s-1, where s≤ p. \textslLet the functions Q and S be defined by the equalities Q(x)=∫₀^xq(t) dt, S(x)=Q²(x) and let q_n%, Q_n,S_n be the Fourier coefficients of q,Q,S with respect to the trigonometric system \e^2πinx\_-∞^∞. Assume that the sequence q_2n-S_2n+2Q₀Q_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 L₂[0,1] (provided that the eigenfunctions are normalized) if and only if the condition q_2n-S_2n+Q₀Q_2n\asymp q_-2n-S_-2n+2Q₀Q_-2n, ngt;1, holds.

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