Restriction theorem for the Fourier-Hermite transform associated with the normalized Hermite polynomials and the Ornstein-Uhlenbeck-Schrödinger equation

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 $L=-\frac{1}{2}\mathrm{\Delta }+⟨x,\mathrm{\nabla }⟩$ 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.