  A survey of estimates on the number $N_-(\BM_{\a G})$ of negative eigenvalues
(bound states) of the Sturm-Liouville operator $\BM_{\a G}u=-u"-\a G$ on the
half-line, as depending on the properties of the function $G$ and the value of
the coupling parameter $\a>0$. The central result is \thmref{S1/2a} giving a
sharp sufficient condition for the semi-classical behavior $N_-(\BM_{\a
G})=O(\a^{1/2})$, and the necessary and sufficient conditions for a
"super-classical" growth rate $N_-(\BM_{\a G})=O(\a^q)$ with any given $q>1/2$.
Similar results for the problem on the whole $\R$ are also presented.
Applications to the multi-dimensional spectral problems are discussed.
