  Let $G=-\Delta-|x|^2\partial_{t}^2$ denote the Grushin operator on
$\mathbb{R}^{n+1}$. The aim of this paper is two fold. In the first part, due
to the non-dispersive phenomena of the Grushin-Schr\"odinger equation on
$\mathbb{R}^{n+1}$, we establish a local dispersive estimate by defining the
Grushin-Schr\"odinger kernel on a suitable domain. As a corollary we obtain a
local Strichartz estimate for the Grushin-Schr\"odinger equation. In the next
part, we prove a restriction theorem with respect to the scaled Hermite-Fourier
transform on $\mathbb{R}^{n+2}$ for certain surfaces in
$\mathbb{N}_0^n\times\mathbb{R^*}\times \mathbb{R}$ and derive anisotropic
Strichartz estimates for the Grushin-Schr\"{o}dinger equation and for the
Grushin wave equation as well.
