  It was proved by H. Bahouri, P. G{\'e}rard and C.-J. Xu in [9] that the
Schr{\"o}dinger equation on the Heisenberg group $\mathbb{H}^d$, 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 $\mathbb{H}^d$ for the linear
Schr{\"o}dinger equation, by a refined study of the Schr{\"o}dinger kernel
$S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through
several examples. Our approach, based on the explicit formula of the heat
kernel on $\mathbb{H}^d$ derived by B. Gaveau in [20], 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 $\mathbb{H}^d$.
