Restriction theorem for the Fourier-Hermite transform and solution of the Hermite-Schr\"odinger equation

TL;DR

This paper establishes a restriction theorem for the Fourier-Hermite transform and derives Strichartz estimates for the Hermite operator, providing insights into the behavior of solutions to the Hermite-Schrödinger equation.

Contribution

It introduces a restriction theorem for the Fourier-Hermite transform and applies it to obtain Strichartz estimates for the Hermite operator, including the optimal behavior of constants.

Findings

Proved the restriction theorem for the Fourier-Hermite transform.

Derived Strichartz estimates for the Hermite operator.

Identified the optimal behavior of constants in the estimates.

Abstract

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.