JournalsjstVol. 3, No. 2pp. 215–235

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

Abstract

\begin{abstract} A survey of estimates on the number N(MαG)N_-(\mathbf M_{\alpha G}) of negative eigenvalues (bound states) of the Sturm--Liouville operator MαGu=uαG\mathbf M_{\alpha G}u=-u''-\alpha G on the half-line, as depending on the properties of the function GG and the value of the coupling parameter α>0\alpha>0. The central result is Theorem 5.1, giving a sharp sufficient condition for the semi-classical behavior N(MαG)=O(α1/2)N_-(\mathbf M_{\alpha G})=O(\alpha^{1/2}), and the necessary and sufficient conditions for a ``super-classical'' growth rate N(MαG)=O(αq)N_-(\mathbf M_{\alpha G})=O(\alpha^q) with any given q>1/2q>1/2. Similar results for the problem on the whole R\mathbb 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