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

摘要

We consider an equation y"(x)=q(x)y(x), x is an element of R, under the following assumptions on q: [GRAPHICS] 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 is an element of R. These properties determine u and v up to mutually inverse positive constant factors, and the function rho(x) = u(x)v(x), x is an element of R, is uniquely determined by q. In the present paper, we obtain an asymptotic formula for computing rho(x) as x --> infinity. 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 is an element of R. (c) 2006 Elsevier Inc. All rights reserved.