Abstract
We consider the Schrödinger 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<p(x,y,t)≤Ct-N/2U(min{|x|,√t})U(min{|y|,√t})/U(√t)2exp(-|x-y|2/Ct)for all x, yϵRN and t>0, where U is a positive harmonic function of H. Third, if U2 is an A2 weight on RN, then we prove a lower bound of a similar type.
Original language | English |
---|---|
Pages (from-to) | 381-410 |
Number of pages | 30 |
Journal | Proceedings of the London Mathematical Society |
Volume | 115 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2017 Aug |
Keywords
- 35J08 (primary)
- 35J10
- 47E05
ASJC Scopus subject areas
- Mathematics(all)