A new proof of the Cwikel-Lieb-Rosenbljum bound
type: Item
instance of: scholarly article
title: A new proof of the Cwikel-Lieb-Rosenbljum bound
zbMATH Open document ID: 0581.35014
DOI: 10.1216/RMJ-1985-15-1-117
author: Joseph G. Conlon
published in: Rocky Mountain Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Concerning the operator \(-\Delta +V\) acting on \(L^ 2(R^ 3)\) where V is a potential in \(L^{3/2}(R^ 3)\) and the dimension N(V) of the spectral projection of \(-\Delta +V\) on (-\(\infty,0]\) there is known the inequality \[ (1.1)\quad N(V)\leq C\int_{R^ 3}| V_- (x)|^{3/2}dx. \] \textit{E. Lieb} [Proc. Symp. Pure Math. 36, 241-252 (1980; Zbl 0445.58029)] gives as the best values of \(C=.116\). Here the author presents a new derivation of (1.1) with the constant \(C=.168\). To derive this result the author considers the following problem: Assuming V(x)\(\leq 0\) \(\forall x\in R^ 3\) and \(V(x)=-W^ 2(x),W(x)\geq 0\) \(\forall x\in R^ 3\), the operator A on \(L^ 2(R^ 3)\) generated by the integral kernel \(a(x,y)=W(x)W(y)[4\pi | x-y|)^{-1}\) is a positive Hilbert-Schmidt operator and has a discrete spectrum \(\mu_ 1\mu_ 2...\geq 0\). The author proves that \(\sum^{N}_{i=1}\mu_ i\leq C^{2/3}N^{1/3}\| W\|^ 2_ 3\) whence the inequality {\#}\(\{\) \(\mu\) \({}_ i\geq \lambda^{-1}\); \(i=1,2,...\}\leq C\lambda^{3/2}\| W\|^ 3_ 3\) \(\forall \lambda >0\) and finally the inequality (1.1) with the mentioned value \(C=.168\).
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 3929524
zbMATH Keywords: Cwikel-Lieb-Rosenbljum bound
MaRDI profile type: Publication
full work available at URL: rmj-1985-15-1-117
OpenAlex ID: W2021896287
Recommended article: A New Bound on Quantum Wielandt Inequality
Recommended article: On General Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities
Recommended article: New bounds on the Lieb-Thirring constants
Recommended article: Q4870614
Recommended article: The Lieb-Thirring inequality revisited
Recommended article: A new proof of Wigner's theorem
Recommended article: A proof of the Lieb-Thirring inequality via the Besicovitch covering lemma
Recommended article: The Lieb–Thirring inequalities: Recent results and open problems
Recommended article: Lieb-Robinson Bounds in Quantum Many-Body Physics
Recommended article: From Lieb-Robinson bounds to automorphic equivalence
description: scientific article; zbMATH DE number 3929524