Exact Control for Schr\"odinger Equation on Torus

TL;DR

This paper establishes observability results for the Schrödinger equation on a 2D torus, providing explicit bounds on the observability constant based on the size of the observation region.

Contribution

It offers the first explicit dependence of the observability constant on the size of the observation ball for the Schrödinger equation on a torus.

Findings

Explicit observability constant dependence on the observation radius.

Proof of observability from a small ball on the torus.

Quantitative estimates for control of Schrödinger dynamics.

Abstract

For standard torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, we prove observability for free Schr\"odinger equation from a ball of radius $\epsilon$ with explicit dependence of the observability constant on $\epsilon$.