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 $i\partial_tu+\La u=0$ where $\La=-Δ+a|x|^{-2}$ with $a\geq0$ 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 \begin{equation*} iu_{t} +\La u = -|u|^2u,\ \ (t,x)\in \mathbb{R}\times\mathbb{R}^{3}. \end{equation*} for radial initial data $u_0\in H_a^s$ with $s>\frac{11}{13}$ 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.