Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations (Q996886)

The goal of this paper is to study some asymptotic properties of solution of the equation \[ y''(x)=q(x)y(x), \quad x\in\mathbb R \] under the assumptions \[ 0\leq q\in L_1^{\text{loc}}(\mathbb R),\int_{-\infty}^x q(t)\,dt>0, \quad \int_x^\infty q(t)\,dt>0 \] for all \(x\in{\mathbb{R}}\). As an application, the behavior at infinity of a solution of the Riccati equation \[ z'(x)+z(x)^2=q(x),\qquad x\in\mathbb R, \] is studied.