Asymptotics on the Diagonal of the Green Function of a Sturm-Liouville Operator and its Applications

Abstract

A Asymptotics on the diagonal of the Green function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm-Liouville operator are obtained.

AN ASYMPTOTIC MAJORANT FOR SOLUTIONS OF STURM–LIOUVILLE EQUATIONS IN $L_P(\mathbb{R})$

“…This paper continues the authors' work in [2,3,5]. We consider the equation By a solution of equation (1.1), we mean any function y such that y, y ∈ AC loc (R) and equality (1.1) hold almost everywhere in R. We also assume that (1.1) is correctly solvable in L p (R).…”

“…Minor technical changes in the proof allow one to keep the results of [3] when condition (2.20) is replaced by conditions (1.2) and (2.1). These changes arise when the inequality d 0 1, which follows from (2.20) (see [3]), is replaced with a more general inequality, (2.5) (see Lemma 2.4).…”

“…These changes arise when the inequality d 0 1, which follows from (2.20) (see [3]), is replaced with a more general inequality, (2.5) (see Lemma 2.4).…”

Davies–Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations

“…(1.1). In addition, they are applied, for example, in the spectral theory of the Sturm-Liouville operator and in the theory of the Riccati equation (see [3,4]). Therefore, their further development may be useful for Eq.…”

“…To state it, we give Definition 1.2, but first note that throughout the sequel we denote by c absolute positive constants which are not essential for exposition and which may differ even within a single chain of computations. [3].) Suppose that condition (1.2) holds.…”