A Sturm-Liouville problem is an ordinary differential equation of the
form $$-(p(x)y'(x))'+(q(x)-\lambda w(x))y(x)=0\quad \text{für } -\infty
\le a0$. In general, the differential equation has two
linearly independent solutions. These solutions are not necessarily in the
Hilbert space $$L^2(a,b,w\,dx):=\bigg\{u:(a,b)\mapsto
\mathbb{C}:\int_{a}^{b}w|u|^2\,dx<\infty\bigg\}.$$ If $p$ and $p$ are
real-valued, then by well-known result of H. Weyl says either all solutions
of the eigenvalue problem for each $\lambda\in\mathbb{C}$ belong to
$L^2(a,b,w\,dx)$ or only one solution (and its multiple) belongs to
$L^2(a,b,w\,dx)$. If all solutions belong to $L^2(a,b,w\,dx)$ we are in the
limit-circle-case otherwise in the limit-point-case. In this thesis we
present some of the results of the paper ”Secondary conditions for linear
differential operator of the second order“ by A. R. Sims (Journal of
Mathematics and Mechanics, 6 (1957), 247-285). A. R. Sims extended the
limit-point, limit-circle classification of Weyl for complex-valued
coefficients with $p=w\equiv 1$. This classification consists of three
cases, one more than as in the classification of H. Weyl for real
coefficients $p$ and $q$. Furthermore we discuss the properties of the
Weyl-function $M$. We declare a solution operator $R_{\lambda}$ which turn
out to be a Hilbert-Schmidt operator in the cases where all the solutions
belong to $L^2[a,b)$. Then the spectrum consists only of isolated
eigenvalues of finite algebraic multiplicity, which belong to
$\mathbb{C}^-:=\{z\in \mathbb{C}:\mathrm{Im}\,z\le 0\}$.