Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators
Djakov, Plamen Borissov and Mityagin, Boris (2012) Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators. Journal of Functional Analysis, 263 (8). pp. 2300-2332. ISSN 0022-1236
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Official URL: http://dx.doi.org/10.1016/j.jfa.2012.07.003
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.
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