The Rozenblum-Lieb-Cwikel inequality for Markov generators
type: Item
instance of: scholarly article
title: The Rozenblum-Lieb-Cwikel inequality for Markov generators
zbMATH Open document ID: 0910.47017
DOI: 10.1007/BF02788029
author: Daniel Levin
author: Michael Solomyak
published in: Journal d'Analyse Mathématique
publication date: 1999-04-12T00:00:00Z
review text: Let \(N(V)\) be the number of negative eigenvalues of the Schrödinger operator \(H_0+V\), \(H_0 = -\Delta\) in \(L^2({\mathbb R}^d)\). According to the well-known Rozenblum-Lieb-Cwikel inequality \[ N(V) \leq \int V_{+}^{d/2} dx, \] where \(V_{+} = (| V| +V)/2\). Several proofs of this inequality due to Rozenblum, Lieb, Cwikel, Li-Yau, Conlon are known. The authors show that the approach of Li and Yau works in the more general situation, when \(H_0\) is a Markov generator in \(L^2(\Omega,\sigma)\), where \((\Omega,\sigma)\) is a measure space with sigma-finite measure. Several applications of this result are included. In particular, the cases of Riemannian manifolds and hyperbolic spaces are discussed.
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1038445
zbMATH Keywords: negative eigenvalues
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Rozenblum-Lieb-Cwikel inequality
zbMATH Keywords: Markov generator
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: hyperbolic spaces
reviewed by: Michael Perelmuter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1038445