(untitled) (Communications in Mathematical Research, 2022)
It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group H^d, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on H^d for the linear Schrödinger equation, by a refined study of the Schrödinger kernel S_t on H^d. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on H^d derived by Gaveau [19], is achieved by combining complex analysis and Fourier-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 H^d.