On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

By Артем Маркович Савчук, A. Savchuk, Андрей Андреевич Шкаликов, A. Shkalikov. 2006

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We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = −y″ + q(x)y with potentials from the Sobolev space W2θ−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 s2k(q) = λk1/2(q) − k, s2k−1(q) = μk1/2(q) − k − 1/2, where {λk}1∞ and {μk}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 t2θ such that the mapping F:W2θ−1→t2θ defined by the equality F(q) = {sn}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 W2θ−1 and t2θ, 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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