On the absence of positive eigenvalues of Schrödinger operators with rough potentials
type: Item
instance of: scholarly article
title: On the absence of positive eigenvalues of Schrödinger operators with rough potentials
zbMATH Open document ID: 1055.35098
author: Alexandru D. Ionescu
author: David Jerison
published in: Geometric and Functional Analysis. GAFA
publication date: 2004-03-04T00:00:00Z
review text: Let \(V: \mathbb{R}^N\to\mathbb{R}\) denote a real-valued potential and \[ H= -\Delta+ V\tag{1} \] the associated Schrödinger operator. In this paper the authors consider the problem of proving the absence of positive eigenvalues of the operator \(H\) for a certain class of rough potential \(V\). To this end the authors for simplicity assume \(N\geq 3\) and \(V\in L^{N/2}(\mathbb{R}^N)\). They show that, absence of positive eigenvalues for (1) is a straight forward consequence of a Carleman inequality of the form \[ \| W_m u\|_{L^{p'}(\mathbb{R}^N)}\leq C\| W_n(\Delta+ 1)u\|_{L^p(\mathbb{R}^N)}, \] for a sequence \[ W_m(x)= \begin{cases} | x|^m &\text{for }| x|\leq R_m,\\ R^m_m\;&\text{for }| x|\geq R_m,\end{cases} \] with the property \(R_m\to \infty\) as \(m\to\infty\).
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 2050578
zbMATH Keywords: absence of positive eigenvalues
zbMATH Keywords: Carleman inequality
zbMATH Keywords: Schrödinger operator
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s00039-003-0439-2
OpenAlex ID: W2036645494
DOI: 10.1007/S00039-003-0439-2
description: scientific article; zbMATH DE number 2050578