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arXiv · 1602.04172

The heat kernel of a Schrödinger operator with inverse square potential

Kazuhiro Ishige·Yoshitsugu Kabeya·El Maati Ouhabaz
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Abstract

We consider the Schr{ö}dinger operator H = --$Δ$ + V (|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x, y, t) of the type 0 \textless{} p(x, y, t) $\le$ C t -- N 2 U (min{|x|, $\sqrt$ t})U (min{|y|, $\sqrt$ t}) U ($\sqrt$ t) 2 exp -- |x -- y| 2 Ct for all x, y $\in$ R N and t \textgreater{} 0, where U is a positive harmonic function of H. Third, if U 2 is an A 2 weight on R N , then we prove a lower bound of a similar type.

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Kazuhiro Ishige, Yoshitsugu Kabeya, El Maati Ouhabaz. 2016-02-12. The heat kernel of a Schrödinger operator with inverse square potential. https://doi.org/10.1112/plms.12041

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