Bilinear estimates for Schrödinger equation with repulsive inverse-square potential and its application
Abstract
In this paper, we establish the bilinear Strichartz estimates for the following linear Schrödinger equation with repulsive inverse-square potential i∂tu+La u=0 where La=−Δ+a|x|−2 with a≥0 for non-radial data. Our proof relies on the physical space method (interaction Morawetz estimate). Such type of method was first introduced in Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) {\bf 42} (2009), 261--290] in flat case, and Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, \textbf{39}(2023), no. 4, 1405-1436] for harmonic oscillator. Our bilinear estimate extends results obtained in Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire {\bf 39} (2022), 1--58] and Pusateri-Soffer [Mem. Amer. Math. Soc. {\bf 299} (2024), v+107 pp] to the critical case. As an application, we establish the global well-posedness for cubic NLS with inverse-square potential iut +La u = −|u|2u, (t,x)∈ ℝ×ℝ3. for radial initial data u0∈ Has with s>11/13 and a>0 where Has is the Sobolev space adapted to the Schrödinger operator with inverse-square potential. The proof relies on Bourgain's high-low frequency decomposition argument and our bilinear estimates. This is the first result on the low regularity well-posedness for NLS with scaling-critical potential.
Related
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential
- Maximal estimates of the Schrödinger operators with quadratic potential for orthonormal systems
- Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential
- Small-time approximate controllability of bilinear Schrödinger equations on compact normal homogeneous spaces
BibTeX
@misc{deng2026bilinear,
title = {{Bilinear estimates for Schrödinger equation with repulsive inverse-square potential and its application}},
author = {Mingming Deng and Yilin Song and Ruixiao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15183},
eprint = {2609.15183},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15183},
}
RIS
TY - UNPB
TI - Bilinear estimates for Schrödinger equation with repulsive inverse-square potential and its application
AU - Mingming Deng
AU - Yilin Song
AU - Ruixiao Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15183
UR - https://arxiv.org/abs/2609.15183
AB - In this paper, we establish the bilinear Strichartz estimates for the following linear Schrödinger equation with repulsive inverse-square potential i∂_tu+La u=0 where La=−Δ+a|x|^−2 with a≥0 for non-radial data. Our proof relies on the physical space method (interaction Morawetz estimate). Such type of method was first introduced in Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) {\bf 42} (2009), 261--290] in flat case, and Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, \textbf{39}(2023), no. 4, 1405-1436] for harmonic oscillator. Our bilinear estimate extends results obtained in Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire {\bf 39} (2022), 1--58] and Pusateri-Soffer [Mem. Amer. Math. Soc. {\bf 299} (2024), v+107 pp] to the critical case. As an application, we establish the global well-posedness for cubic NLS with inverse-square potential iu_t +La u = −|u|^2u, (t,x)∈ ℝ×ℝ^3. for radial initial data u_0∈ H_a^s with s>11/13 and a>0 where H_a^s is the Sobolev space adapted to the Schrödinger operator with inverse-square potential. The proof relies on Bourgain's high-low frequency decomposition argument and our bilinear estimates. This is the first result on the low regularity well-posedness for NLS with scaling-critical potential.
ER -