Dispersive estimates for the semi-classical Schrödinger equation in a strictly convex domain

Oana Ivanovici (1)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$ and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the optimal 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. The result is optimal and implies corresponding Strichartz estimates.

Domaines: Equations aux dérivées partielles [math.AP]

Oana Ivanovici. Dispersive estimates for the semi-classical Schrödinger equation in a strictly convex domain. 2020. ⟨hal-02950431v1⟩