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

We consider an equation(1)y^″ (x) = q (x) y (x), x ∈ R, under the following assumptions on q:(2)0 ≤ q ∈ L₁^loc (R), underover(∫, - ∞, x) q (t) d t > 0, underover(∫, x, ∞) q (t) d t > 0 for all  x ∈ R . Let v (respectively u) be a positive non-decreasing (respectively non-increasing) solution of (1) such thatv^′ (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 equationz^′ (x) + z (x)² = q (x), x ∈ R .