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
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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.References
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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