The heat kernel of a Schrödinger operator with inverse square potential

Kazuhiro Ishige, Yoshitsugu Kabeya, El Maati Ouhabaz

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


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 languageEnglish
Pages (from-to)381-410
Number of pages30
JournalProceedings of the London Mathematical Society
Issue number2
Publication statusPublished - 2017 Aug


  • 35J08 (primary)
  • 35J10
  • 47E05

ASJC Scopus subject areas

  • Mathematics(all)


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