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

By A. Shkalikov, O. Veliev. 2008

72 citations recorded by the source corpus.

AbstractThe 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4421a19659f05c4b38421bc799b7c5a9e3761fc5

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Citations in the corpus

A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schr\ (2011, background): “In [78] a specific system of root vectors corresponding to a generic potential in the space L(2)([0, π]; dx) was introduced, and a criterium for it to form a Riesz basis was”
Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators (2011, background): “, in the 2000’s, many authors [18, 26, 29, 34, 33, 35, 37, 38, 39, 40, 52] focused on the problem of convergence of eigenfunction (or more generally root function) decompositions in the case of regular but not strictly regular bc.”