Classical Proofs Of Kato Type Smoothing Estimates for The Schrödinger Equation with Quadratic Potential in Rⁿ⁺¹ with application

In this paper, we consider the Schrödinger equation with quadratic potential {equation*} i∂ /∂ tu =- u+ x ²u in Rⁿ⁺¹, u(x,0)=f(x)∈ L²(R ⁿ). {equation*} Using Hermite functions and some other classical tools, we give an elementary proof of the Kato-type smoothing estimate: for i≠ j≠ k, δ ∈ 0,1], and n 3, {equation*} ∫ ₀2π ∫ _{Rⁿ}{ u(x,t) ²}{ ( xᵢ²+xⱼ²+xₖ² ) δ} dxdt C f ₂². {equation*} This is equivalent to proving a uniform L²(Rⁿ) boundedness result for a family of singularized Hermite projection kernels. As an application of the above estimate, we also prove the R⁹ collapsing variable-type Strichartz estimate {equation*} ∫ ₀2π ∫ _{R³} u(x, x,x,t) ²dxdt C (- + x ²)f ₂² \ \ \ where x∈ R³. {equation*}