  In this article, we prove the restriction theorem for the Fourier-Hermite
transform and obtain the Strichartz estimate for the system of orthonormal
functions for the Hermite operator $H=-\Delta+|x|^2$ on $\mathbb{R}^n$ as
application. Further, we show the an optimal behavior of the constant in the
Strichartz estimate as limit of a large number of functions.
