This master thesis discusses second-order differential expressions of
$\tau[y]=\frac{1}{w}[-(py')'+qy]$. The functions $p^{-1}$, $q$ and $w$ are
locally integrable on the half-open, not necessarily bounded, interval
$[a,b)$. The corresponding Eigenvalue problem $\tau[y]=\lambda y$ is called
Sturm-Liouville-Eigenvalue-problem. This differential equations has two
linearly independent solutions, but they are not necessarily in the
Hilbertspace $$L^2(a,b,w\, dx):= \left{u:(a,b)\rightarrow
\mathbb{C}:\int_a^bw|u|^2\, dx <\infty \right}$$. A famous result from H.
Weyl says, that either all solutions of the eigenvalue problem belong to
$L^2(a,b,w\, dx)$ for all $\lambda$ or only one solution (and its multiple)
belongs to $L^2(a,b,w\, dx)$. The first case is named limit-circle-case and
the second one limit-point-case. We present the results of the paper "On
the spectrum of second-order differential operators with complex
coefficients" of B.M. Brown, D.K.R. McCormack, W.D. Evans und M. Plum
(Proc. R. Soc. Lond. 455 (1999), 1235-1257). They discuss the
Sturm-Liouville-Eigenvalue-problem with complex coefficients $p$ and $q$.
It is possible to copy partial the results of H.Weyl also in the
complex-valued case. The characterization yields to two limit-circle-cases
and one limit-point-case. Furthermore we discuss the Weyl-function $m$ and
an operator realization of $\tau$. Finally we determine sets, where the
spectrum of the operator consists only of eigenvalues of finite algebraic
multiplicity.