On a class of spectral problems on the half-line and their applications to multi-dimensional problems

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.