Classical Proofs Of Kato Type Smoothing Estimates for The Schrödinger Equation with Quadratic Potential in $\mathbb{R}^{n+1}$ with application

Abstract

In this paper, we consider the Schrödinger equation with quadratic potential \begin{equation*} i\frac{\partial }{\partial t}u =-\triangle u+ \left \vert x \right \vert ^{2}u \text{ }in\text{ }\mathbb{R}^{n+1}, \text{ }u(x,0)=f(x)\in L^{2}(\mathbb{R} ^{n}). \end{equation*} Using Hermite functions and some other classical tools, we give an elementary proof of the Kato-type smoothing estimate: for $i\neq j\neq k,$ $ \delta \in \lbrack 0,1],$ and $n\geqslant 3,$ \begin{equation*} \int _{0}^{2\pi } \int _{\mathbb{R}^{n}}\frac{ \left \vert u(x,t) \right \vert ^{2}}{ \left ( x_{i}^{2}+x_{j}^{2}+x_{k}^{2} \right ) ^{\delta }} dxdt\leqslant C \left \Vert f \right \Vert _{2}^{2}. \end{equation*} This is equivalent to proving a uniform $L^{2}(\mathbb{R}^{n})$ boundedness result for a family of singularized Hermite projection kernels. As an application of the above estimate, we also prove the $\mathbb{R}^{9}$ collapsing variable-type Strichartz estimate \begin{equation*} \int _{0}^{2\pi } \int _{\mathbb{R}^{3}} \left \vert u(\mathbf{x}, \mathbf{x},\mathbf{x},t) \right \vert ^{2}d\mathbf{x}dt\leqslant C \left \Vert (-\triangle + \left \vert x \right \vert ^{2})f \right \Vert _{2}^{2} \ \ \ \text{ where $\mathbf{x\in }\mathbb{R}^{3}$.} \end{equation*}