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This paper applies Hermite function techniques to give elementary proofs ofKato type smoothing estimates for the Schr\"odinger equation with quadraticpotential in R^n+1. This is equivalent to proving a uniform L^2(R^n) toL^2(R^n) boundedness for a family of singularized Hermite projection kernels.As an applicationas the above estimate, we also prove the R^9 collapsingvariable type Strichartz estimate.

Classical Proofs Of Kato Type Smoothing Estimates for The Schr\"odinger Equation with Quadratic Potential in R^n+1 with application