Observability of Schrödinger propagators on tori in rough settings

TL;DR

This work addresses the observability of Schrödinger propagators on $\mathbb{T}^d$ under rough potentials by reducing observability to integrability properties of free Schrödinger waves. It introduces a general $\chi$-observability criterion, leveraging cluster decompositions and dimensional reduction to establish strong observability from space–time domains of positive measure, at least in low dimensions and under mild integrability conditions on $V$. A key contribution is the construction of convenable Bourgain-type spaces and a robust uniqueness–compactness framework that upgrades weak to strong observability, with explicit results in dimension $d=1$ (and extensions to higher $d$ under certain assumptions). The results connect to Cantor–Lebesgue-type phenomena, control theory via exact controllability, and quantum-limit observability, offering new domains and weaker, yet nontrivial, integrability bounds that advance the conjectured L^∞ observability on tori. Overall, the paper provides a versatile framework for observability in rough toral settings, including periodized formulations and a detailed strategy that blends cluster analysis with unique continuation.

Abstract

On tori of arbitrary dimensions, Schrödinger propagators with bounded potentials are conjectured to be observable from space-time domains of positive Lebesgue measure. We reduce this conjecture to certain integrability bounds for free Schrödinger waves, thereby proving the conjecture on the one-dimensional torus and producing new examples of observation domains. These bounds are far weaker than Bourgain's conjectured periodic Strichartz estimates, yet remain highly nontrivial.

Theorems & Definitions (76)

• Conjecture 1.1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Conjecture 1.9
• Theorem 1.10