An optimal \(L^p\)-bound on the Krein spectral shift function.
type: Item
instance of: scholarly article
title: An optimal \(L^p\)-bound on the Krein spectral shift function.
zbMATH Open document ID: 1054.47020
DOI: 10.1007/BF02868474
author: Dirk Hundertmark
author: B. Simon
published in: Journal d'Analyse Mathématique
publication date: 2003-09-11T00:00:00Z
review text: Let \(A,B\) be bounded self-adjoint operators such that \(C=A-B\) is a trace class operator and \(\xi_{A,B}\) be the Krein spectral shift function for the pair \((A,B)\). The authors give two different proofs of the inequality \[ \int F(| \xi_{A,B}(\lambda )| )\,d\lambda \leq \int F(| \xi_{| C| ,0}(\lambda )| )\,d\lambda =\sum\limits_{j=1}^\infty [F(j)-F(j-1)]\mu_j(C), \] where \(\mu_j(C)\) are the singular values of \(C\) and \(F\) is an arbitrary non-negative convex function on \([0,\infty )\) such that \(F(0)=0\). For the case \(F(t)=t^p\), \(p\geq 1\), this result improves a bound found by \textit{J. M. Combes}, \textit{P. D. Hislop} and \textit{S. Nakamura} [Commun. Math. Phys. 218, 113--130 (2001; Zbl 1042.82024)].
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 1931228
zbMATH Keywords: Krein spectral shift function
zbMATH Keywords: trace class perturbation
MaRDI profile type: Publication
cites work: Q5287255
cites work: The \(L^p\)-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators.
cites work: Q4482931
cites work: SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION
cites work: THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS
cites work: Q4836262
cites work: Q3996935
full work available at URL: https://doi.org/10.1007/bf02868474
OpenAlex ID: W2018587388
description: scientific article; zbMATH DE number 1931228