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

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We study the system of root functions (SRF) of Hill operator Ly = -y′ ′ +vy with a singular potential v ∈ H⁻¹per and SRF of 1D Dirac operator Ly = i pmatrix 1 & 0 0 & -1 pmatrix dy/dx + vy with matrix L²-potential v=pmatrix 0 & P Q & 0 pmatrix, 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.