A Liouville property for Schrödinger operators
type: Item
instance of: scholarly article
title: A Liouville property for Schrödinger operators
zbMATH Open document ID: 0938.31007
author: A. A. Grigor'yan
author: Wolfhard Hansen
published in: Mathematische Annalen
publication date: 1999-10-17T00:00:00Z
review text: The authors consider the problem whether a Schrödinger equation of type \[ \Delta u - u\mu = 0 \] admits a bounded nonvanishing solution or not depending on how the measure \(\mu\) is chosen. Here \(\Delta\) is the Laplace-Beltrami operator on a Riemannian manifold and \(\mu\) is a Kato measure or, more generally, a smooth measure, i.e. a countable sum of Radon measures charging no polar sets. The measure \(\mu\) is called big if no bounded non-trivial solutions (of appropriate smoothness depending on the kind of the measure) exist. The main result of the article is a complete characterization of ``bigness'' of \(\mu\), namely \(\mu\) is not big if and only if it is the sum of two measures, one having support in a so-called non-thick set, the other having a Green potential which is not identically \(\infty\). Moreover, the authors give a sufficient criterion of Wiener type, which is easily applicable to check whether a measure \(\mu\) is big. Since their approach is mainly based on fundamental properties of harmonic and superharmonic functions, they are able to extend their result in a second part of the paper to the more abstract setting of harmonic spaces. Then in this way it is also possible to cover much more general operators than the Laplace operator, including Hörmander type operators and general elliptic and parabolic operators.
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 1336963
zbMATH Keywords: bounded solutions
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: smooth measures
zbMATH Keywords: harmonic space
reviewed by: Walter Hoh
MaRDI profile type: Publication
full work available at URL: s002080050241
OpenAlex ID: W2092723063
DOI: 10.1007/S002080050241
Recommended article: A Liouville-type theorem for Schrödinger operators
Recommended article: A Liouville type theorem for the Schrödinger operator
Recommended article: Generalized Liouville property for Schrödinger operator on Riemannian manifolds
Recommended article: Liouville type theorems for Schrödinger systems
Recommended article: GENERALIZED LIOUVILLE PROPERTY FOR SCHRÖDINGER OPERATOR ON GRAPHS
Recommended article: A Liouville type result for Schrödinger equation on half-spaces
Recommended article: Certain Liouville properties of eigenfunctions of elliptic operators
Recommended article: General properties of the Liouville operator
Recommended article: Q3361233
Recommended article: The Liouville operator
description: scientific article; zbMATH DE number 1336963