Pointwise convergence for single-particle Schrödinger equation and Schrödinger equation with infinitely many particles on the torus and waveguide manifolds

Abstract

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.