Intrinsic ultracontractivity of a Schrödinger semigroup in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: Intrinsic ultracontractivity of a Schrödinger semigroup in \(\mathbb R^N\)
zbMATH Open document ID: 1181.47049
author: Bénédicte Alziary
author: Peter Takáč
published in: Journal of Functional Analysis
publication date: 2009-06-30T00:00:00Z
review text: Let \(\mathcal{A}:=-\Delta +q(x)\cdot\) denote the Schrödinger operator with a potential \(q:\mathbb{R}^N\to \mathbb{R}\) assumed to be continuous, have positive infimum, and tend to infinity as \(|x|\to \infty\). The authors present a general condition on \(q\) for the Schrödinger semigroup \(\{e^{-\mathcal{A}t}; \;t\geq 0\}\) on \(L^2(\mathbb{R}^N)\) generated by \(-\mathcal{A}\) to be intrinsically ultracontractive. The condition is shown to be necessary if \(q\) is radially symmetric. The two main ingredients in their proofs are a logarithmic Sobolev inequality and a new version of Rosen's inequality.
reviewed by: Ti-Jun Xiao
Mathematics Subject Classification ID: 47D08
zbMATH DE Number: 5572179
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: heat semigroup
zbMATH Keywords: ground state
zbMATH Keywords: intrinsic ultracontractivity
zbMATH Keywords: logarithmic Sobolev and Rosen's inequalities
zbMATH Keywords: WKB-type asymptotic formula
zbMATH Keywords: resolvent compactness
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.02.013
OpenAlex ID: W1970412383
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DOI: 10.1016/J.JFA.2009.02.013
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description: scientific article; zbMATH DE number 5572179