On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions d=2,3

2025/06/21 by Yilin Song, Song, Yilin, Ruixiao Zhang +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2506.17731

openalex publication_date 2025/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions d=2,3, \begincases i∂_tu-Hu=|u|²u,amp;(t,x)∈ℝ×ℝ^d,
u(0,x)=u₀(x), \endcases where H=-Δ+|x|². Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the ε-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.

Citations

• On the growth of Sobolev norm for the cubic NLS on two dimensional product space
• A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on ℝ² perturbed by a real smooth potential decaying to zero at infinity
• Growth of Sobolev Norms for 2d NLS with harmonic potential
• Long time dynamics for defocusing cubic NLS on three dimensional product space
• Strichartz estimates for the Schrodinger equation on irrational tori
• Growth of solutions to NLS on irrational tori
• The Energy-Critical Quantum Harmonic Oscillator
• Modified scattering for the cubic Schrödinger equation on product spaces and applications
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
• A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms
• A bilinear oscillatory integral estimate and bilinear refinements to\n Strichartz estimates on closed manifolds
• Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrÖdinger Equations
• Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces
• Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R³

Related

• On the growth of Sobolev norm for the cubic NLS on two dimensional product space
• Growth of Sobolev norms in the cubic defocusing nonlinear Schr "odinger\n equation
• Growth of Sobolev Norms for 2d NLS with harmonic potential
• Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential