Uniform complex time heat Kernel estimates without Gaussian bounds

Abstract

The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e−z(−Δ)α2{e}^{-z{\left(-\Delta )}^{\frac{\alpha }{2}}} for α>0,z∈C+\alpha \gt 0,z\in {{\mathbb{C}}}^{+}. To this end, we establish the asymptotic estimates for P(z,x)P\left(z,x) with zz satisfying 0<ω≤∣θ∣<π20\lt \omega \le | \theta | \lt \frac{\pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H=(−Δ)α2+VH={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V, where VV belongs to higher-order Kato class.

Keywords

• complex time heat kernel estimates
• fractional schrödinger operators
• primary 35k08
• secondary 35j10