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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A probabilistic approach to bounded/positive solutions for Schrödinger operators with certain classes of potentials


Author: Ross G. Pinsky
Journal: Trans. Amer. Math. Soc. 360 (2008), 6545-6554
MSC (2000): Primary 60H30, 35J10
Published electronically: June 26, 2008
MathSciNet review: 2434298
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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the equation

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

for $ d\ge3$. For certain classes of potentials $ V$, we use probabilistic tools to study the bounded solutions and the positive solutions for (*). A primary motivation is to offer probabilistic intuition for the results.


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Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel
Email: pinsky@math.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04473-5
PII: S 0002-9947(08)04473-5
Keywords: Liouville theorem, bounded solutions, positive solutions, Schr\"odinger equation
Received by editor(s): June 26, 2006
Received by editor(s) in revised form: January 16, 2007
Published electronically: June 26, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.