  In this article we study the Schr\"odinger equation associated with Harmonic
oscillator in the form of Strichartz type inequality. We give simple proofs for
Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm
estimates of the spectral projections associated harmonic oscillator proved in
\cite{KT}. Our Strichartz type estimates are sharp in sense of regularity of
initial data.

Analysis of PDEs;Classical Analysis and ODEs