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On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spacesстатья
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 3 февраля 2015 г.
- Авторы: Savchuk A.M., Shkalikov A.A.
- Журнал: Mathematical Notes
- Том: 80
- Номер: 5
- Год издания: 2006
- Издательство: Pleiades Publishing, Ltd
- Местоположение издательства: Road Town, United Kingdom
- Первая страница: 814
- Последняя страница: 832
- DOI: 10.1007/s11006-006-0204-6
- Аннотация: We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = −y″ + q(x)y with potentials from the Sobolev space W 2 θ−1, θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s 2k (q) = λ k 1/2(q) − k, s 2k−1(q) = μ k 1/2(q) − k − 1/2, where {λ k } 1 ∞ and {μ k } 1 ∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t 2 θ such that the mapping F:W 2 θ−1→t 2 θ defined by the equality F(q) = {s n } 1 ∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W 2 θ−1 and t 2 θ, and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥τ ≤ C∥q∥θ−1, where the exact value of τ = τ(θ) > θ − 1 is given and the constant C depends only on the radius of the ball ∥q∥θ− ≤ R, but is independent of the function q varying in this ball.
- Добавил в систему: Савчук Артем Маркович