Asymptotic Boundary Observability for the Schr\"odinger Equation on Simplices

TL;DR

This paper establishes a large-time asymptotic boundary observability result for the Schrödinger equation on an n-dimensional simplex, advancing understanding of wave control in geometric domains.

Contribution

It introduces a novel asymptotic boundary observability result for the Schrödinger equation on simplices, using commutator and integration by parts techniques.

Findings

Asymptotic boundary observability for Schrödinger on simplices

Improved understanding over traditional observability inequalities

Method applicable to complex geometric domains

Abstract

We consider the Schr\"{o}dinger 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.