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

TL;DR

This paper surveys estimates on the number of negative eigenvalues of a Sturm-Liouville operator on the half-line, providing sharp conditions for their growth rate and discussing applications to multi-dimensional spectral problems.

Contribution

It introduces sharp sufficient and necessary conditions for the eigenvalue count growth rate, extending classical semi-classical estimates to broader settings.

Findings

Sharp conditions for semi-classical eigenvalue growth $O(eta^{1/2})$

Necessary and sufficient conditions for super-classical growth $O(eta^q)$, $q>1/2$

Applications to multi-dimensional spectral problems

Abstract

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.