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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A probabilistic approach to the Liouville property for Schrödinger operators with an application to infinite configurations of balls
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by Rachel Hess-Green and Ross G. Pinsky PDF
Proc. Amer. Math. Soc. 138 (2010), 4487-4496 Request permission

Abstract:

Consider the equation \begin{equation}\frac 12\Delta u-Vu=0\ \text {in}\ R^d,\tag {*} \end{equation} for $d\ge 3$, where $V\gneq 0$. 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
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 4487-4496
  • MSC (2010): Primary 60H30, 35J10
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10452-5
  • MathSciNet review: 2680073