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

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

2011-06-28

arxiv.org/abs/1106.5774v1

Mathematics

Spectral Theory

Scientific paper

We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-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 $L^p$-spaces and other rearrangement invariant function spaces.

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