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 (complex-valued) potential v is an element of H-per(-1). and the SRF of 1D Dirac operator Ly = i((1)(0) (0)(-1))dy/dx + vy with matrix L-2-potential v = ((0)(Q) (P)(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 L-p-spaces and other rearrangement invariant function spaces. (C) 2012 Elsevier Inc. All rights reserved.