Control for Schr\"odinger equations on 2-tori: rough potentials

TL;DR

This paper proves that control and observability of the Schr"odinger equation on 2-tori hold even with rough potentials in L^2, extending previous results that required smoother potentials.

Contribution

It extends control results for Schr"odinger equations on 2-tori to include potentials in L^2, broadening the class of potentials for which observability is established.

Findings

Control holds for potentials in L^2 on 2-tori.

Extends previous results from smooth to rough potentials.

Open problem remains for higher dimensions with L^ potentials.

Abstract

For the Schr\"odinger equation, $ (i \partial_t + \Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \Omega $ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T] \times \Omega)} $. We show that the same result remains true for $ (i \partial_t + \Delta - V) u = 0 $ where $ V \in L^2 (\T^2) $, and $ \T^2 $ is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for $ V \in C (\T^2) $ and conjectured for $ V \in L^\infty (\T^2) $. The higher dimensional generalization remains open for $ V \in L^\infty (\T^n) $.