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

TL;DR

This paper presents elementary Hermite function-based proofs of Kato type smoothing estimates for Schrödinger equations with quadratic potentials, leading to new Strichartz estimates in higher dimensions.

Contribution

It introduces a novel elementary proof technique for smoothing estimates using Hermite functions and extends results to R^9 collapsing variable Strichartz estimates.

Findings

Elementary proofs of Kato smoothing estimates using Hermite functions

Uniform boundedness of singularized Hermite projection kernels

Derivation of R^9 collapsing variable Strichartz estimate

Abstract

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