Control for Schr\"odinger equation on hyperbolic surfaces

TL;DR

This paper proves that any nonempty open set on a hyperbolic surface allows for observability and control of the Schrödinger equation, extending known results from flat tori to hyperbolic geometries.

Contribution

It establishes control and observability results for the Schrödinger equation on hyperbolic surfaces, a significant extension beyond previously known flat cases.

Findings

Any nonempty open set on a hyperbolic surface provides control for the Schrödinger equation.

The proof leverages the main estimate from Dyatlov-Jin and standard control theory techniques.

This extends control results from flat tori to hyperbolic surfaces.

Abstract

We show that the any nonempty open set on a hyperbolic surface provides observability and control for the time dependent Schr\"odinger equation. The only other manifolds for which this was previously known are flat tori. The proof is based on the main estimate in Dyatlov-Jin and standard arguments of control theory.