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

Abstract

We study the system of root functions (SRF) of Hill operator Ly=−y″+vy with a singular (complex-valued) potential v∈Hper−1 and the SRF of 1D Dirac operator Ly=i(100−1)dydx+vy with matrix L2-potential v=(0PQ0), 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.