发表机构
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究环面和波导流形上单粒子及无穷多粒子薛定谔方程的点态收敛性，建立了时间极大Strichartz估计，给出了收敛的充分与必要条件，并推广至费米子系统。
AI 中文摘要
1980年，Carleson提出了一个问题：在Sobolev空间$H^s$中，初始数据需要具备何种最低正则性，才能保证单粒子线性薛定谔方程的解点态收敛于初始数据。本文研究了$d$维波导流形$\u211d^n\times \u2a3f^m$上的Carleson问题，并在环面和波导流形上，针对无穷多正交粒子系统（源于多体量子力学向热力学极限的过渡），开创了其类比研究。对于波导流形上的单粒子，我们建立了一个时间极大Strichartz估计，该估计保证了当$s > \frac{d}{d+2}$时，解在$H^s$中几乎处处收敛于初始数据。另一方面，我们采用Bourgain反例方法证明，当$s< \frac{d}{2(d+1)}$时，该收敛性不成立。对于无穷多粒子，我们在环面上的费米子系统中，推广了Compaan、Luca和Staffilani（2021）的经典时间极大Strichartz估计和点态收敛结果。对于波导流形，我们也建立了类似的结果。此外，我们还为费米子系统的点态收敛问题建立了必要条件（遵循Bourgain反例的思路）。这些是环面和波导流形上的首批结果，补充了Bez、Lee和Nakamura（2020）以及Bez、Kinoshita、Shinya和Shiraki（2024）在欧几里得空间上的工作。
英文摘要
In 1980, Carleson posed a question about the least regularity required for initial data in a Sobolev space $H^s$ to ensure pointwise convergence of the solution to the single particle linear Schrodinger equation. In this paper, we study Carleson problem on the $d$-dimensional waveguide manifold $\mathbb R^n\times \mathbb T^m$ and initiate its analogy for the system of infinitely many orthonormal particles (arising from the transition of many-body quantum mechanics to the thermodynamic limit) on torus and waveguide manifold. For a single particle on the waveguide manifold, we establish a maximal-in-time Strichartz estimate that yields almost everywhere convergence to the initial data in $H^s$ for $s > \frac{d}{d+2}$. On the other hand, we adopt Bourgain counterexample approach to show that this convergence fails for $s< \frac{d}{2(d+1)}.$ For infinitely many particles, we generalize the classical maximal-in-time Strichartz estimate and pointwise convergence result of Compaan, Luca and Staffilani (2021) in the setting of fermionic systems on the torus. Similar result is also established for the waveguide manifold. We also establish necessary condition (in the spirit of Bourgain's counterexample) for the pointwise convergence problem for fermionic systems. These are the first results in the setting of torus and waveguide manifold, and complement the works of Bez, Lee and Nakamura (2020) and Bez, Kinoshita, Shinya and Shiraki (2024) on Euclidean space.
Comments40 pages
