Perturbation theory for convolution semigroups
type: Item
instance of: scholarly article
title: Perturbation theory for convolution semigroups
zbMATH Open document ID: 1194.47049
author: Mustapha Mokhtar-Kharroubi
published in: Journal of Functional Analysis
publication date: 2010-07-07T00:00:00Z
review text: The paper under review is an extension in various directions of the results from the author's previous paper [J.~Funct.\ Anal.\ 256, No.\,6, 1998--2025 (2009; Zbl 1168.47035)] devoted to Schrödinger operators \(-\Delta+V\) with singular negative potentials \(V\). In the present paper, the author deals with the generators \(T_p\) of convolution semigroups (not necessarily symmetric) \[ S_p(t): L^p ({\mathbb R}^N)\to L^p ({\mathbb R}^N),\;f\mapsto \int f(x-y) m_t(dy), \] where \(\{m_t\}_{t\geq 0}\) are Borel sub-probability measures on \({\mathbb R}^N\) such that \(m_0 = \delta_0\) (Dirac measure), \(m_t \star m_s = m_{t+s}\) and \(m_t\to m_0\) (\(t\to 0_+\)). The typical examples of such functions are Gaussian semigroups, \(\alpha\)-stable semigroups, relativistic Schrödinger semigroups, etc. The author provides a general perturbation theory of their generators by real potentials \(V = V_+ - V_-\), where \(V_+, V_-\) are singular nonnegative measurable functions. In the investigation of \(L^1\)-perturbation theory, the smallness of the Kato norm with respect to \(T_1^+=T_1+V_+\) is substituted by the weaker assumption that \(V_-\) is \(T_1^+\)-weakly compact. This assumption implies the following spectral radius \(r_\sigma\) inequality \[ \lim_{\lambda\to\infty} r_\sigma[V_-(\lambda-T_1^+)^{-1}]<1. \] The proof of the generation result is based on Desch's theorem. Feynman-Kac formulas are also proved in this framework. In the \(L^2\) setting, Dirichlet form-perturbation theory is related to \(L^1\)-theory. Using interpolation arguments, it is shown (even in the non-symmetric case) how to extend corresponding semigroups to \(L^p\) spaces as a \(C_0\)-semigroup. Finally, various perturbation problems for subordinate semigroups and measure perturbations of the Hamiltonian \(T_2\) are considered.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D08
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 5731618
zbMATH Keywords: convolution semigroup
zbMATH Keywords: characteristic exponent
zbMATH Keywords: \(L^{1}\)-relative weakly compact perturbation
zbMATH Keywords: Kato class measures
zbMATH Keywords: subordination
zbMATH Keywords: \(\alpha \)-stable semigroup
zbMATH Keywords: relativistic Schrödinger semigroup
zbMATH Keywords: subordinate Brownian semigroup
zbMATH Keywords: Feynman-Kac formula
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.03.018
OpenAlex ID: W2091523796
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DOI: 10.1016/J.JFA.2010.03.018
description: scientific article; zbMATH DE number 5731618