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

We consider an equation y (x) = q(x)y(x), x ∈ R, (1) under the following assumptions on q: 0 q ∈ Lloc 1 (R), x −∞ q(t) dt >0, ∞ x q(t) dt >0 forallx ∈ R. (2) Let v (respectively u) be a positive non-decreasing (respectively non-increasing) solution of (1) such that v (x)u(x)− u (x)v(x) = 1, x∈ R. These properties determine u and v up to mutually inverse positive constant factors, and the function ρ(x) = u(x)v(x), x ∈ R, is uniquely determined by q. In the present paper, we obtain an asymptotic formula for computing ρ(x) as |x|→∞. As an application, under conditions (2), we study the behavior at infinity of solution of the Riccati equation z (x) +z(x)2 = q(x), x ∈ R. © 2006 Elsevier Inc. All rights reserved.