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 $\Omega \subset {\mathbb{R}}^3$ 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 (${\partial }_{\zeta }^2\Phi =2t$) 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 $\eta =0$. This yields sharp global Strichartz estimates featuring an explicit derivative loss exponent of $\rho (q)=\frac{3}{2}\left(\frac{1}{2}-\frac{1}{q}\right)$.