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

Abstract

\begin{abstract} A survey of estimates on the number $N_-(\mathbf M_{\alpha G})$ of negative eigenvalues (bound states) of the Sturm--Liouville operator $\mathbf M_{\alpha G}u=-u''-\alpha G$ on the half-line, as depending on the properties of the function $G$ and the value of the coupling parameter $\alpha>0$. The central result is Theorem 5.1, giving a sharp sufficient condition for the semi-classical behavior $N_-(\mathbf M_{\alpha G})=O(\alpha^{1/2})$, and the necessary and sufficient conditions for a ``super-classical'' growth rate $N_-(\mathbf M_{\alpha G})=O(\alpha^q)$ with any given $q>1/2$. Similar results for the problem on the whole $\mathbb R$ are also presented. Applications to the multi-dimensional spectral problems are discussed.