Curved Ingham inequalities and observability of the toroidal Schr{\"o}dinger equation

TL;DR

This paper demonstrates that solutions to the toroidal Schrödinger equation can be observed from curved space-time trajectories, using new bounds for trigonometric sums inspired by Ingham inequalities, with implications for observability of different frequency components.

Contribution

It introduces novel curved Ingham inequalities and applies them to establish observability of the Schrödinger equation on a torus from curved trajectories.

Findings

Solutions are observable from curved trajectories of zero Lebesgue measure.

New bounds for trigonometric sums along curves are established.

Observability of low- and high-frequency components is achieved separately.

Abstract

We prove that solutions of the toroidal Schr{\"o}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restrictions on the zero sets of the trigonometric sums under consideration.