Global lower bound for the heat kernel of -Δ+\frac{𝑐}|𝑥|²

Zhang, Qi

doi:10.1090/S0002-9939-00-05757-9

Global lower bound for the heat kernel of $-\Delta +\frac {c}{|x|^2}$
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by Qi S. Zhang PDF

Proc. Amer. Math. Soc. 129 (2001), 1105-1112 Request permission

Abstract:

We obtain global in time and qualitatively sharp lower bounds for the heat kernel of the singular Schrödinger operator $-\Delta + \frac {a}{|x|^2}$ 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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