Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

We study the system of root functions (SRF) of Hill operator Ly =−y + vy with a singular (complexvalued) potential v ∈ H−1 per and the SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L2- potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in Lp-spaces and other rearrangement invariant function spaces. © 2012 Elsevier Inc. All rights reserved.