Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring estimates for Schrödinger operators on Riemannian manifolds
type: Item
instance of: scholarly article
title: Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring estimates for Schrödinger operators on Riemannian manifolds
zbMATH Open document ID: 1192.58018
author: E. M. Ouhabaz
author: César Poupaud
published in: Acta Applicandae Mathematicae
publication date: 2010-05-19T00:00:00Z
review text: Let \(\Delta=\delta d\) be the Laplace operator on the \(L^2\) space of a complete Riemannian manifold. Consider the Schroedinger operator \(S:=\Delta+V\) where \(V\) is a potential function. Assume the positive part \(V_+\) of \(V\) is locally integrable and the negative part \(V_-\) is form-small. Let \(\{\lambda_1,\dots,\lambda_N\}\) be the eigenvalues of \(\Delta+V\) with negative real part. Let \(p(t,x,y)\) be the heat kernel of \(\Delta\). Let \(G\) be a non-trivial continuous convex function on \([0,\infty)\) which satisfies some suitable growth conditions and let \(g(1)\) be the Laplace transform at \(1\) of \(z^{-1}G(z)\). If the potential is real-valued and if \(\gamma>0\), the authors establish the estimate: \[ \sum_{j=1}^N(-\lambda_j)^\gamma\leq{\Gamma(\gamma+1)\over g(1)} \int_0^\infty {dt\over t^{\gamma+1}}\int_Mp(t,x,x)G(tV_-(x))dvol\,; \] there is a suitable inequality as well when \(\gamma=0\). More generally, if \(V\) is complex valued, then for \(\gamma\geq1\), the authors establish \[ \sum_{j=1}^N(-\Re(\lambda_j))^\gamma\leq {\Gamma(\gamma+1)\over g(1)} \int_0^\infty {dt\over t^{\gamma+1}}\int_Mp(t,x,x)G(t\Re(V)_-(x))dvol\,. \] The estimates in fact hold true for more general operators as they are based on an abstract result and an approximation argument that does not explicitly use the Laplacian.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 5709155
zbMATH Keywords: Cwikel-Lieb-Rozenblum estimates
zbMATH Keywords: Lieb-Thirring estimates
zbMATH Keywords: Rozenblum-Solomyak estimate
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: Laplacian
zbMATH Keywords: Laplace-Beltrami operator
Wikidata QID: Q115384838
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: s10440-009-9519-0
OpenAlex ID: W1991890521
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DOI: 10.1007/S10440-009-9519-0
description: scientific article; zbMATH DE number 5709155