We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

中文翻译：

---

改进了薛定谔算子的质量临界 Strichartz 估计

我们为一类与时间相关的 Schr\"{o}dinger 算子开发了 $L^2$ 规律性的精细 Strichartz 估计。这种改进开始表征 Strichartz 估计的近优化器，并在全局中发挥关键作用。质量临界 NLS 理论。一方面，由于大量对称性，谐波分析在 $L^2$ 临界设置中非常微妙，而另一方面，现有方法采用的时空傅立叶分析常系数方程不适用于非平移不变情况，尤其是在本文中考虑的势能如此大的情况下。使用相空间技术，我们简化为证明（伴随）双线性傅立叶限制估计的某些类似物。然后我们扩展陶's 对抛物面的双线性限制估计到更一般的 Schr\"{o}dinger 算子。作为一个特定的应用，由此产生的逆 Strichartz 定理和剖面分解构成了研究 $L^2$- 的大数据解决方案的关键谐波分析输入在维度 $\ge 2$ 中具有谐振子势的临界 NLS。本文基于 Killip、Visan 和作者在一维空间中的近期工作。