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Dirac system with potential lying in Besov spaces

Dirac system with potential lying in Besov spaces We study the spectral properties of the Dirac operator L P,U generated in the space (L 2[0, π])2 by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y 1, y 2) t , $$B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)$$ , and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ n } n∈Z of the operator L P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B 1,∞ θ , θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L P,U , and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Dirac system with potential lying in Besov spaces

Savchuk, A.
Differential Equations , Volume 52 (4) – May 20, 2016
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References (16)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Pleiades Publishing, Ltd.
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266116040042
Publisher site
See Article on Publisher Site

Abstract

We study the spectral properties of the Dirac operator L P,U generated in the space (L 2[0, π])2 by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y 1, y 2) t , $$B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)$$ , and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ n } n∈Z of the operator L P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B 1,∞ θ , θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L P,U , and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.

Journal

Differential Equations – Springer Journals

Published: May 20, 2016

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