On Growth of Sobolev norms for cubic Schr\"odinger equation with harmonic potential in dimensions $d=2,3$

TL;DR

This paper investigates the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions 2 and 3, establishing new bilinear estimates and using the upside-down I-method.

Contribution

It introduces bilinear Strichartz estimates without epsilon-loss and applies the upside-down I-method to prove polynomial Sobolev norm growth in higher dimensions.

Findings

Established bilinear Strichartz estimates without epsilon-loss.

Proved polynomial growth of Sobolev norms in dimension 3.

Extended results of previous work in dimension 2 to dimension 3.

Abstract

In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schr\"odinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-\Delta+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three.