Dispersive estimates for the Schrödinger equation in a strictly convex domain and applications

2020/09/29 by Oana Ivanovici, Ivanovici, Oana · 1 citation
Mathematics · Physics and Astronomy · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2009.13810

openalex publication_date 2020/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an anisotropic model case for a strictly convex domain of dimension d≥ 2 with smoothboundary and we describe dispersion forthe semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semi-classical flow : a loss of (\frac ht)^1/4 occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Schödinger equation on such a 3D model convex domain, hence matching known results on generic compact boundaryless manifolds.

Cited by

• Strichartz and dispersive estimates for quantum bouncing ball model: exponential sums and Van der Corput methods in 1d semi-classical Schrödinger equations

Related

• Dispersion for the wave equation inside strictly convex domains II: the general case
• Sharp resolvent and time decay estimates for dispersive equations on asymptotically Euclidean backgrounds
• Global solutions for 1D cubic defocusing dispersive equations, Part IV: general dispersion relations
• Dispersion for the wave and Schrödinger equations outside a ball and counterexamples
• Semiclassical analysis of dispersion phenomena