Dispersion for the Schrödinger equation inside strictly convex domains: the general case
Abstract
We consider a general smooth bounded strictly convex domain Ω⊂ℝd of dimension d≥2 and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of 1/4 in the exponent of h/t with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.
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BibTeX
@misc{ivanovici2026dispersion,
title = {{Dispersion for the Schrödinger equation inside strictly convex domains: the general case}},
author = {Oana Ivanovici and Fabrice Planchon},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06305},
eprint = {2610.06305},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05780759v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Dispersion for the Schrödinger equation inside strictly convex domains: the general case
AU - Oana Ivanovici
AU - Fabrice Planchon
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.48550/arxiv.2610.06305
UR - https://hal.science/hal-05780759v1
AB - We consider a general smooth bounded strictly convex domain Ω⊂ℝ^d of dimension d≥2 and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of 1/4 in the exponent of h/t with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.
ER -