Sharp Strichartz type estimates for the Schrödinger equation associated with harmonic oscillator

In this article we study the Schrödinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the L2→Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2 \rightarrow L^p$$\end{document} operator norm estimates of the spectral projections associated to the harmonic oscillator proved in [Jeong, Lee and  Ryu, 2022 arxiv:2205.03036v1, 2022, and Koch and Tataru 2005 Duke Math. J. 128:369–392, 2005]. Our Strichartz type estimates are sharp in sense of regularity of initial data.