WKB approximation

Definition

Let $I$ be an open interval, let $Q \colon I \to \R$ be smooth with $Q(x) \neq 0$ for all $x \in I$, and let $\varepsilon > 0$ be a small parameter. Consider the linear ordinary differential equation $$\varepsilon^2 y''(x) = Q(x) \, y(x), \quad x \in I.$$ The WKB ansatz is $y = \exp\bigl(\frac{1}{\varepsilon} \sum_{n \ge 0} \varepsilon^n S_n(x)\bigr)$. Inserting it and collecting powers of $\varepsilon$ gives $S_0'^2 = Q$ at order $\varepsilon^0$ and $2 S_0' S_1' + S_0'' = 0$ at order $\varepsilon^1$. The WKB approximation, or Liouville–Green approximation, keeps these two orders. It gives the two functions $$y_\pm(x) = Q(x)^{-1/4} \exp\Bigl(\pm \frac{1}{\varepsilon} \int_{x_0}^{x} \sqrt{Q(s)} \, ds\Bigr),$$ where $x_0 \in I$ and the roots are taken with a fixed branch. On compact subintervals of $I$, the equation has solutions $y = y_\pm \, (1 + O(\varepsilon))$ as $\varepsilon \to 0$.