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Proceedings of the American Mathematical Society

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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

$\displaystyle \frac12\Delta u-Vu=0$   in$\displaystyle R^d,$ (*)

for $ d\ge3$, where $ V\gneq0$. One says that the Liouville property holds if the only bounded solution to (*) is 0. Under a certain growth condition on $ V$, we give a necessary and sufficient analytic condition for the Liouville property to hold. We then apply this to the case that $ V$ 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.