Quasi-Modes and Spectral Instability in One Dimension

  • Johannes Sjöstrand
Part of the Pseudo-Differential Operators book series (PDO, volume 14)


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 bk ∈ C(]α, β[). Here h ∈ ]0, h0] 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
$$\displaystyle a(x;h)\sim \sum _{\nu =0}^\infty a_\nu (x)h^\nu \mbox{ in }C^\infty (]\alpha ,\beta [). $$


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Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

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