 WKB approximation for a slowly varying oscillator

ID: wkb-approximation-for-a-slowly-varying-oscillator

For

ϵ^{2} y_{XX} + k^{2} (X) y = 0

with nonzero slowly varying

k

y \sim k (X)^{- 1/2} [C_{+} e^{i ϵ^{- 1} \int^{X} k (s) d s} + C_{-} e^{- i ϵ^{- 1} \int^{X} k (s) d s}] .

It fails near turning points and where

ϵ ∣ k^{'} ∣/∣ k ∣^{2}

is not small.

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