Bounds for eigenvalues of singular differential operators

CMMSE: Maximal regularity and two-sided estimates of the approximation numbers of the nonlinear Sturm-Liouville equation solutions with rapidly oscillating coefficients in $L_{2} (R)$

“…It follows that all the conditions of Theorem 4 of [11] are fulfilled. Consequently, 𝑞 (𝑥, 𝑦) 𝑦 (𝑥) , 𝑦 ′′ ∈ 𝐿 2 (𝑅).…”

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L2(ℝ)

“…Hence, according to Theorem 3 in [11], both functions 𝑦 ′′ and q(x, 𝑦)𝑦 belong to L 2 (R). Theorem 1 is proved.…”

“…(A) to find out the conditions on the potential function q(x, 𝑦) which provide 𝑦 ′′ ∈ L 2 (R), when 𝑦(x) is a solution of the nonlinear equation L𝑦 = 𝑓 ∈ L 2 (R). We note that the linear case is well studied and reviews are available in previous studies [7][8][9][10][11][12]. It is known that eigenvalues 𝜆 n (n = 1, 2, ...) of the self-adjoint positive completely continuous operator A in the Hilbert space H are numbered according to their decreasing magnitude and observing their multiplicities have the following approximative properties (a) 𝜆 n = min š∈l n ‖A − K‖, where l n is the set of all finite-dimensional operators with dimension no greater than n, K is a linear n-dimensional operator; (b) 𝜆 n → 0, when n → ∞, wherein the faster convergence to zero, the operator A better approximated by finite rank operators.…”

Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function

“…However, there is not much information on the distribution of the spectrum. We mention works [18,19,21] where authors study the asymptotic behaviour of the eigenvalues of the Schrodinger operators where the regularity of the potentials is relaxed.…”

“…Our main tool will be the Otelbaev function q * µ + associated to the positive part µ + of µ. The Otelbaev function first appeared in works by its name giver, Kazakh mathematician Mukhtarbai O. Otelbaev, in the 1970s, when he successfully studied spectral properties of operators − d 2 dx 2 + q, where q was some lower bounded function [18,19]. The main point of using the Otelbaev function (which Otelbaev himself simply denoted as q * ) was that the method essentially does not need any smoothness assumptions on the potential q.…”