Differential and Integral Equations

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

Xuwen Chen

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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*}

Article information

Source
Differential Integral Equations, Volume 24, Number 3/4 (2011), 209-230.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019031

Mathematical Reviews number (MathSciNet)
MR2757458

Zentralblatt MATH identifier
1240.35429

Subjects
Primary: 35B45: A priori estimates 35Q41: Time-dependent Schrödinger equations, Dirac equations 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 33C4

Citation

Chen, Xuwen. Classical Proofs Of Kato Type Smoothing Estimates for The Schrödinger Equation with Quadratic Potential in $\mathbb{R}^{n+1}$ with application. Differential Integral Equations 24 (2011), no. 3/4, 209--230. https://projecteuclid.org/euclid.die/1356019031


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