Kernel estimates and the regularized trace of the semigroup generated by a potential perturbation of the bi-Laplacian

摘要

The purpose of this note is to present asymptotic formulae and bounds for the integral kernel GV (x, y, t) of the semigroup U(t) = exp(tHV ) with generator HV = P(i)+V (x), where P(ξ) = |ξ|~4/4 and V (x) ∈ L1(R3). The bounds established and the techniques involved can be extended to the case of a positive-definite form P(ξ) and a bounded integrable potential in any dimension.