Asymptotic Behavior of the Density of the Spectral Measure of the Singular Sturm–Liouville Operator

Abstract

holds. Every function α(λ) of this kind is called a spectral function (measure) of the operator L. An upper bound as λ → +∞ for the spectral function α(λ), sinα = 0, in the case of a locally integrable potential q(x) was obtained for the first time in the paper [2], and an asymptotic formula as λ → +∞ was later established in the paper [3]: α(λ) ∼ (2/π)λ sin−2 α if sinα = 0, and α(λ) ∼ (2/3)π−1λ3/2 if sinα = 0. The papers [4–9] deal with the spectral function α(λ) of the Sturm–Liouville problem. The most comprehensive results concerning the asymptotics of the spectral function of the Laplace operator in an arbitrary domain G ⊂ R were obtained in [10, Chap. 1]. The asymptotic behavior of the spectral function for higher-order ordinary differential operators on the entire line and on the half-line was studied in detail in [11]. The asymptotics of α(λ) as λ → +∞ was refined mainly under two kinds of conditions imposed on the potential q : I. q ∈ L(0,+∞); II. there exists an x0 > 0 such that q(x) = 0 for x ≥ x0, ∫ +∞ x0 ( q′|q|−5/2 + |q′′| |q|−3/2 ) dx < ∞, and ∫ +∞ x0 |q|−1/2dx = ∞.