Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains
Abstract
Dispersive and Strichartz estimates are fundamental tools for establishing the well-posedness and long-time behavior of solutions to nonlinear partial differential equations. While these estimates are well-understood in the boundaryless Euclidean setting, the presence of a geometric boundary introduces severe analytical complexities, such as the continuous formation of caustics. In this work, we establish sharp local-in-time dispersive estimates for the semiclassical Schrödinger equation inside a three-dimensional cylindrical domain Ω⊂ ℝ3 subject to homogeneous Dirichlet boundary conditions. This paper provides the first comprehensive microlocal treatment for this anisotropic geometric setting, extending the optimal strictly convex boundary results of Ivanovici \cite{Ivanovici2023} to the parabolic setting. The primary analytical challenge in our cylindrical model stems from the fact that the boundary curvature is non-uniform, depending explicitly on the tracking angle of classical trajectories and vanishing identically along the flat longitudinal axis. Crucially, we demonstrate that the quadratic structure of the Schrödinger phase function (∂ζ2 Φ= 2t) establishes global non-degeneracy, completely bypassing the arduous low-frequency trajectory ray-tracing mandatory in hyperbolic wave equations. By exploiting this structural advantage alongside a streamlined Littlewood-Paley dyadic block decomposition, we prove that the zero-frequency axial tail can be consistently integrated down to the flat limit η=0. This yields sharp global Strichartz estimates featuring an explicit derivative loss exponent of ρ(q) = 3/2(1/2−1/q).
Related
- Sharp Dispersive and Strichartz Estimates for the Neumann Cylindrical Wave Model
- Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential
- Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Strichartz estimates to fractional Schrödinger equations
BibTeX
@misc{meas2026dispersive,
title = {{Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains}},
author = {Len Meas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13697},
eprint = {2609.13697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13697},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains
AU - Len Meas
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13697
UR - https://arxiv.org/abs/2609.13697
AB - Dispersive and Strichartz estimates are fundamental tools for establishing the well-posedness and long-time behavior of solutions to nonlinear partial differential equations. While these estimates are well-understood in the boundaryless Euclidean setting, the presence of a geometric boundary introduces severe analytical complexities, such as the continuous formation of caustics. In this work, we establish sharp local-in-time dispersive estimates for the semiclassical Schrödinger equation inside a three-dimensional cylindrical domain Ω⊂ ℝ^3 subject to homogeneous Dirichlet boundary conditions. This paper provides the first comprehensive microlocal treatment for this anisotropic geometric setting, extending the optimal strictly convex boundary results of Ivanovici \cite{Ivanovici2023} to the parabolic setting. The primary analytical challenge in our cylindrical model stems from the fact that the boundary curvature is non-uniform, depending explicitly on the tracking angle of classical trajectories and vanishing identically along the flat longitudinal axis. Crucially, we demonstrate that the quadratic structure of the Schrödinger phase function (∂_ζ^2 Φ= 2t) establishes global non-degeneracy, completely bypassing the arduous low-frequency trajectory ray-tracing mandatory in hyperbolic wave equations. By exploiting this structural advantage alongside a streamlined Littlewood-Paley dyadic block decomposition, we prove that the zero-frequency axial tail can be consistently integrated down to the flat limit η=0. This yields sharp global Strichartz estimates featuring an explicit derivative loss exponent of ρ(q) = 3/2(1/2−1/q).
ER -