  In this article, we prove the analogue theorems of Stein-Tomas and Srtichartz
on the discrete surface restrictions of Fourier-Hermite transforms associated
with the normalized Hermite polynomials and obtain the Strichartz estimate for
the system of orthonormal functions for the Ornstein-Uhlenbeck operator
$L=-\frac{1}{2}\Delta+\langle x, \nabla\rangle$ on $\mathbb{R}^n$. Further, we
show an optimal behavior of the constant in the Strichartz estimate as limit of
a large number of functions.
