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

Abstract

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = −y″ + q(x)y with potentials from the Sobolev space W ^θ−1₂ , 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 ^1/2_k (q) − k, s _2k−1(q) = μ ^1/2_k (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 ^θ₂ such that the mapping F:W ^θ−1₂ →t ^θ₂ 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 ^θ−1₂ and t ^θ₂ , 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.