Asymptotic Boundary Observability for the Schrödinger Equation on Simplices

Abstract

We consider the Schrödinger equation $(i{\partial }_t+\Delta )u=0$ on an $n$-dimensional simplex with Dirichlet boundary conditions. We use a commutator argument along with integration by parts to obtain an observability asymptotic for any one face of the simplex. Rather than the typical observability inequality, we are able to do better as we instead prove a large-time asymptotic.