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 Hs 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 ℝn× 𝕋m 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 Hs for s > d/(d+2). On the other hand, we adopt Bourgain counterexample approach to show that this convergence fails for s< 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.
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BibTeX
@misc{bhimani2026pointwise,
title = {{Pointwise convergence for single-particle Schrödinger equation and Schrödinger equation with infinitely many particles on the torus and waveguide manifolds}},
author = {Divyang G. Bhimani and Subhash R. Choudhary},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15363},
eprint = {2609.15363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15363},
}
RIS
TY - UNPB
TI - Pointwise convergence for single-particle Schrödinger equation and Schrödinger equation with infinitely many particles on the torus and waveguide manifolds
AU - Divyang G. Bhimani
AU - Subhash R. Choudhary
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15363
UR - https://arxiv.org/abs/2609.15363
AB - 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 ℝ^n× 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 > d/(d+2). On the other hand, we adopt Bourgain counterexample approach to show that this convergence fails for s< 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.
ER -