In this thesis we study ordinary linear differential equations
$f''=(p+q)f$. By means of WKB-approximation the asymptotic behaviour of
solutions is investigated. The differential equation is defined on a simply
connected domain with analytic functions $p$ and $q$. One aim is to
construct bounds for the WKB-approximation. This is done by solving a
Volterra integral equation. For this type of integral equation we proof a
theorem about existence of solutions and their growth behaviour. As an
application we use the WKB-method for polynomial coefficients $p$ and $q$.
This situation is currently intensive studied in Theoretical Physics (so
called $\PT$ quantum mechanics). We obtain solutions which exponentially
decay or exponentially diverge in a certain area (Stokes wedges and Stokes
lines) of the complex plane.