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

### Michael Solomyak

Weizmann Institute of Science, Rehovot, Israel

## Abstract

A survey of estimates on the number $N_{−}(M_{αG})$ of negative eigenvalues (bound states) of the Sturm–Liouville operator $M_{αG}u=−u_{′′}−αG$ on the half-line, as depending on the properties of the function $G$ and the value of the coupling parameter $α>0$. The central result is Theorem 5.1, giving a sharp sufficient condition for the semi-classical behavior $N_{−}(M_{αG})=O(α_{1/2})$, and the necessary and sufficient conditions for a “super-classical” growth rate $N_{−}(M_{αG})=O(α_{q})$ 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.

## Cite this article

Michael Solomyak, On a class of spectral problems on the half-line and their applications to multi-dimensional problems. J. Spectr. Theory 3 (2013), no. 2, pp. 215–235

DOI 10.4171/JST/43