The heat kernel of a Schrödinger operator with inverse square potential

Kazuhiro Ishige (1) , Yoshitsugu Kabeya (2) , El Maati Ouhabaz (3)

1 Tohoku University [Sendai]
2 Osaka Prefecture University
3 IMB - Institut de Mathématiques de Bordeaux

We consider the Schrödinger operator H = −∆ + 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) ≤ C t − N 2 U (min{|x|, √ t})U (min{|y|, √ t}) U (√ t) 2 exp − |x − y| 2 Ct for all x, y ∈ R N and t > 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.

Domaines: Equations aux dérivées partielles [math.AP]

Kazuhiro Ishige, Yoshitsugu Kabeya, El Maati Ouhabaz. The heat kernel of a Schrödinger operator with inverse square potential. Proceedings of the London Mathematical Society, 2017. ⟨hal-01273288⟩