  In the previous papers \cite{HLWZ,KWZ}, the jump phenomena of the $j$-th
eigenvalue were completely characterized for Sturm-Liouville problems. In this
paper, we show that the jump number of these eigenvalue branches is exactly the
Maslov index for the path of corresponding boundary conditions. Then we
determine the sharp range of the $j$-th eigenvalue on each layer of the space
of boundary conditions. Finally, we prove that the graph of monodromy matrix
tends to the Dirichlet boundary condition as the spectral parameter goes to
$-\infty$.
