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

Abstract

A survey of estimates on the number $N_{-}({\mathbf{M}}_{\alpha G})$ of negative eigenvalues (bound states) of the Sturm–Liouville operator ${\mathbf{M}}_{\alpha G}u=-u^{\prime \prime }-\alpha G$ 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.