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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Global lower bound for the heat kernel of $-\Delta+\frac{c}{\vert x\vert^2}$


Author: Qi S. Zhang
Journal: Proc. Amer. Math. Soc. 129 (2001), 1105-1112
MSC (1991): Primary 35K10, 35K65
Published electronically: October 11, 2000
MathSciNet review: 1814148
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Abstract:

We obtain global in time and qualitatively sharp lower bounds for the heat kernel of the singular Schrödinger operator $-\Delta + \frac{a}{\vert x\vert^2}$ with $a>0$. Here $\Delta$ is either the Laplace-Beltrami operator or the Laplacian on the Heisenberg group. This complements a recent paper by P. D. Milman and Yu. A. Semenov in which an upper bound was found. The above potential is interesting because it is a border line case where both the strong maximum principle and Gaussian bounds fail.


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Additional Information

Qi S. Zhang
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05757-9
PII: S 0002-9939(00)05757-9
Received by editor(s): June 30, 1999
Published electronically: October 11, 2000
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society