  We introduce a novel approach for dealing with eigenvalue problems of
Sturm-Liouville operators generated by the differential expression
\begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right]
)^{\prime }+sp\left[ y^{\prime }+sy\right] +qy\right) \end{equation*} which is
based on norm resolvent convergence of classical Sturm-Liouville operators.
This enables us to describe the continuous dependence of the $n$-th eigenvalue
on the space of self-adjoint boundary conditions and the coefficients of the
differential equation after giving the inequalities among the eigenvalues.
Moreover, oscillation properties of the eigenfunctions are also characterized.
In particular, our main results can be applied to solve a class of
Sturm-Liouville problems with transmission conditions.
