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∈L1loc(R),∫−∞xq(t)dt>0,∫x∞q(t)dt>0for 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.