Perturbative Methods for Modelling (M2)

Section 10.4, p. 504

We consider the Schrödinger equation $\fs2 \epsilon^2\, y''=Q(x)\,y$ with boundary condition $\fs2 y(+\infty)=0$ such that $\fs2 Q(x)>0$ for $\fs2 x>0$, $\fs2 Q(x)$ for $\fs2 x$ and $\fs2 Q(x)\sim a\,x$ for $\fs2 x\to0$ with $\fs2 a>0$. We assume that $\fs2 Q(x)\ll x^{-2}$ for $\fs2 x\to \pm \infty$. Show the approximations for $\fs2 \epsilon\to 0$:

$\fs2 y_{III}(x) = 2\, C\, [-Q(x)]^{-1/4}\,\sin\left[{1\over \epsilon}\int_x^0 \sqrt{-Q(t)}\,dt +{\pi/4}\right]$ for $\fs2 x$ and $\fs2 (-x)\gg \epsilon^{2/3}$

$\fs2 y_{II}(x) = 2\, C\, \sqrt{\pi}\, (a\, \epsilon)^{-1/6}\,{\rm Ai}\left(\epsilon^{-2/3}\,a^{1/3}\,x\right)$ for  $\fs2 |x|\ll 1$

$\fs2 y_{I}(x)=C\, [Q(x)]^{-1/4}\,\exp\left[{1\over \epsilon}\int_0^x \sqrt{Q(t)}\,dt \right]$ for $\fs2 x>0$  and   $\fs2 x\gg \epsilon^{2/3}$

where $\fs2 C$is an arbitrary constant. Draw schematically the solution.