Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Djakov, Plamen; Mityagin, Boris

doi:10.1007/s00208-010-0612-5

Cited by 38 publications

(40 citation statements)

References 21 publications

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“…It extends and slightly generalizes [18, Theorem 1] (or [17,Theorem 2]) in the case of Hill operators, and [15,Theorem 12] in the case of Dirac operators.…”

Section: 2mentioning

confidence: 75%

“…They played a crucial role in proving estimates for and inequalities between γ n , δ n , β ± n and t n (z) := β − n (v; z) / β + n (v; z) , (6) see [8,Lemma 49 and Proposition 66]. Moreover, it turns out that there is an essential relation between the Riesz basis property of the system of root functions and the ratio functionals t n (v, z) which made possible to give criteria for existence of (Riesz) bases consisting of root functions not only for Hill operators but for Dirac operators as well (see, for example, [18,Theorem 1] or [17,Theorem 2] for Hill, or [15,Theorem 12] for Dirac operators). These criteria are quite general and applicable to wide classes of potentials.…”

Section: 2mentioning

confidence: 98%

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

Djakov

Mityagin

2012

Journal of Functional Analysis

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We study the system of root functions (SRF) of Hill operator Ly = −y + vy with a singular (complexvalued) potential v ∈ H −1 per and the SRF of 1D Dirac operator Ly = i 1 0Q 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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“…It extends and slightly generalizes [18, Theorem 1] (or [17,Theorem 2]) in the case of Hill operators, and [15,Theorem 12] in the case of Dirac operators.…”

Section: 2mentioning

confidence: 75%

Section: 2mentioning

confidence: 98%

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

Djakov

Mityagin

2012

Journal of Functional Analysis

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“…In connection with Corollary 1.3 and Proposition 1.4 we mention the papers [49,50,21,39] and [6][7][8][9][10][11][12][13], that appeared during the last decade. Basically they are devoted to Riesz basis property of EAF for BVP with strictly regular (and just regular) BC for 2 × 2 Dirac systems.…”

Section: It Is Clear That T a (C D) = T −A (D C)mentioning

confidence: 99%

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

Malamud

Оридорога

2012

Journal of Functional Analysis

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The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2 × 2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.

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“…However we did not proved yet the completeness of the corresponding basis. An obstacle for that can be found in the works [9], [10], [11], [12]. The Darboux-Crum transformations play only technical role here.…”

Section: Appendixmentioning

confidence: 99%

Singular soliton operators and indefinite metrics

Grinevich

Novikov

2013

Bull Braz Math Soc, New Series

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Abstract. We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebro-geometrical potentials (i.e. "singular finite-gap" and "singular solitons") satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functions with singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2 . Here g is the genus of spectral curve.

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Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Cited by 38 publications

References 21 publications

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

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

Singular soliton operators and indefinite metrics

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