Observability and quantum limits for the Schr\"odinger equation on the sphere

TL;DR

This paper investigates the behavior of quantum states on the sphere, exploring how observability and control are affected by geometric and potential conditions, with new insights into eigenmodes and Radon transform effects.

Contribution

It provides new results on semiclassical measures for Schrödinger evolution on Zoll manifolds, especially the sphere, and links these to observability issues in control theory.

Findings

Examples where observability fails for evolution but holds for stationary states.

New results on the Radon transform of potentials and their impact.

Analysis of eigenmodes of the Schrödinger operator on the sphere.

Abstract

In this note, we describe our recent results on semiclassical measures for the Schr{\"o}dinger evolution on Zoll manifolds. We focus on the particular case of eigenmodes of the Schr{\"o}dinger operator on the sphere endowed with its canonical metric. We also recall the relation of this problem with the observability question from control theory. In particular, we exhibit examples of open sets and potentials on the 2-sphere for which observability fails for the evolution problem while it holds for the stationary one. Finally, we give some new results in the case where the Radon transform of the potential identically vanishes.