Quasi-Modes and Spectral Instability in One Dimension

Abstract

In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation

$$\displaystyle P(x,hD_x)u=\sum _{k=0}^m b_k(x)(hD_x)^ku=0, $$

on an interval α < x < β, where we assume that the coefficients b _k ∈ C ^∞(]α, β[). Here h ∈ ]0, h ₀] is a small parameter and we wish to solve (above equation) up to any power of h. We look for u in the form

$$\displaystyle u(x;h)=a(x;h)e^{i\phi (x)/h}, $$

where ϕ ∈ C ^∞(]α, β[) is independent of h. The exponential factor describes the oscillations of u, and when ϕ is complex valued it also describes the exponential growth or decay; a(x;h) is the amplitude and should be of the form

$$\displaystyle a(x;h)\sim \sum _{\nu =0}^\infty a_\nu (x)h^\nu \mbox{ in }C^\infty (]\alpha ,\beta [). $$