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

By Plamen Djakov, Boris Mityagin. 2011

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ab3c58cf807b64fa2c85c5bf73a35590b258471

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Citations in the corpus

• On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators (2015, methodology): “Mityagin [12] for q ∈W−1,2[0, π] established by different methods a criterion for the system of root functions to contain a Riesz basis.”
• On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations (2012, methodology): “Mityagin [11], the authors established a criterion for eigenfunctions and associated functions to form a Riesz basis for periodic (resp.”