Analysis of the Radiation Loss: Asymptotics Beyond all Orders

Abstract

Kath and Kriegsmann recently studied a model in bent fibre-optic tunnelling (see [6]). An interesting singular perturbation problem on the half axis:

$${ \in ^2}y'' + Q(x;\lambda )y = 0,x \in (0, + \infty )$$

((1.1))

(1.1) arises. Here 0 < ∈ ≪ 1 is a parameter and λ is an eigenvalue. One boundary condition associated with equation (1.1) is

$$ \in y'(0) + hy(0) = 0$$

((1.2))

(1.2) for some h > 0. The other boundary condition is imposed at ϰ = +∞. Several authors have computed a desired quantity ImQ(0;λ), which is called the radiation loss, for several special cases. The radiation loss problem is an nonlinear eigenvalue problem. In this paper, we try to have a general discussion for a variety of functions Q.