Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group (Communications in Mathematical Research, 2022)
It was proved by Bahouri et al. [9] that the Schrödinger equation onthe Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totallynon-dispersive evolution equation: for this reason global dispersive estimatescannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through severalexamples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis andFourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$