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

2012/03/06 by Michael Solomyak, Solomyak, Michael
Computer Science · Mathematics · #34L15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP #msc:34L15

paper · pdf · doi:10.48550/arxiv.1203.1156

17 pages

arxiv created 2012/03/06 · openalex publication_date 2012/03/06 · arxiv updated 2012/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \thmrefS1/2a 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.

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