Observability of Schr\"odinger equations in Euclidean space

TL;DR

This paper introduces a new geometric control condition for observing Schr"odinger equations in Euclidean space, demonstrating its effectiveness and weaker requirements compared to previous conditions, with extensions to fractional Schr"odinger equations.

Contribution

The paper proposes the comb geometric control condition, a novel criterion for observability, and establishes its equivalence to observability under certain conditions, extending to fractional Schr"odinger equations.

Findings

The comb geometric control condition ensures observability of Schr"odinger equations.

It is strictly weaker than the observation set being open and periodic.

For fractional Schr"odinger equations, the condition is equivalent to observability for uniformly continuous observation functions.

Abstract

In this paper we introduce a new dynamical condition, the comb geometric control condition, which is sufficient for observability of the Schr\"odinger equation in Euclidean space. We provide examples which show this condition is strictly weaker than the observation set being open and periodic. We also prove for the fractional Schr\"odinger equation that for observation functions which are uniformly continuous, the geometric control condition is equivalent to observability and implies arbitrary time observability. The proofs rely on uncertainty principles for frequency localized functions which are proved using a semiclassical propagation of singularities approach.