Abstract
In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation
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
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
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Sjöstrand, J. (2019). Quasi-Modes and Spectral Instability in One Dimension. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_4
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DOI: https://doi.org/10.1007/978-3-030-10819-9_4
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