A probabilistic approach to the Liouville property for Schrödinger operators with an application to infinite configurations of balls

Authors:
Rachel Hess-Green and Ross G. Pinsky

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4487-4496

MSC (2010):
Primary 60H30, 35J10

Published electronically:
June 10, 2010

MathSciNet review:
2680073

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the equation

in | (*) |

for , where . One says that the Liouville property holds if the only bounded solution to (*) is 0. Under a certain growth condition on , we give a necessary and sufficient analytic condition for the Liouville property to hold. We then apply this to the case that is the indicator function of an infinite collection of small balls.

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Additional Information

**Rachel Hess-Green**

Affiliation:
Department of Mathematics, Technion–Israel Institute of Technology, Haifa, 32000, Israel

Email:
rachelg@tx.technion.ac.il

**Ross G. Pinsky**

Affiliation:
Department of Mathematics, Technion–Israel Institute of Technology, Haifa, 32000, Israel

Email:
pinsky@math.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-2010-10452-5

Keywords:
Liouville property,
bounded solutions,
Schrödinger equation

Received by editor(s):
October 6, 2009

Received by editor(s) in revised form:
February 11, 2010

Published electronically:
June 10, 2010

Communicated by:
Richard C. Bradley

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.