Perturbative Methods for Modelling (M2)

Example 1, p. 486

We consider the Schrödinger equation  $\fs2 \epsilon^2\,y''=Q(x)\,y$ with $\fs2 Q(x)\ne 0$ for the $\fs2 x$ considered. Show that $\fs2 y(x) \sim c_1\,Q^{-1/4}(x)\,\exp\left({1\over \epsilon}\, \int_a^x\sqrt{Q(t)}\,dt\right)+ c_2\,Q^{-1/4}(x)\,\exp\left(-{1\over \epsilon}\,\int_a^x\sqrt{Q(t)}\,dt\right)$, provided $\fs2 \epsilon\,S_1\ll S_0$, $\fs2 \epsilon\,S_2\ll S_1$ and $\fs2 \epsilon\,S_2\ll 1$ for $\fs2 \epsilon\to 0$  with  $\fs2 S_0(x) = \int^x \sqrt{Q(t)}\,dt$,  $\fs2 S_1(x) = -{1\over 4}\, \ln Q(x)$, $\fs2 S_2(x)= \int^x\left[{Q''\over 8\, Q^{3/2}} - {5\,\left(Q'\right)^2\over 32\, Q^{5/2}}\right]\, dt$.