Open problems: Lieb-Thirring type inequalities for Schrödinger operators with a complex-valued potential
type: Item
instance of: scholarly article
title: Open problems: Lieb-Thirring type inequalities for Schrödinger operators with a complex-valued potential
zbMATH Open document ID: 1279.47071
author: Marcel Hansmann
author: Guy Katriel
author: Michael Demuth
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: The Lieb-Thirring inequality states that, if \(H=-\Delta +V\), \(V \in L^{p}(\mathbb{R}^{d};\mathbb{R})\), \(p\geq 1\) if \(d=1\), \(p>1\) if \(d=2\) and \(p \geq d/2\) if \(d \geq3\), then \[ \sum_{\lambda \in \sigma_{d}(H)} | \lambda | ^{p-d/2} \leq C(p,d) \| V \| ^{p}_{L^{p}}, \] where \(C(p,d)\) depends only on \(p\) and on \(d\). The authors raise the following question: is it true that, if \(V\) is complex valued, then the inequality \[ \sum_{\lambda \in \sigma_{d}(H)} \frac{\operatorname{dist} (\lambda, [0, \infty))^{p}} { | \lambda | ^{d/2}} \leq C(p,d) \| V \| ^{p}_{L^{p}}\tag{1} \] holds? They review several results which may be considered weakened versions of (1).
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 00A07
zbMATH DE Number: 6153954
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Lieb-Thirring inequality
zbMATH Keywords: complex valued potential
MaRDI profile type: Publication
full work available at URL: s00020-012-2021-5
OpenAlex ID: W1996368962
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DOI: 10.1007/S00020-012-2021-5
description: scientific article; zbMATH DE number 6153954