On the observability of the Schrödinger equation in the torus from open sets

Technical Summary: Observability of the Schrödinger Equation on the Torus from Open Sets

Problem Statement
This paper investigates the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb{T}^d = \mathbb{R}^d/(2\pi\mathbb{Z})^d$ from an open subset $\omega \subset \mathbb{T}^d$. The central question is whether the global energy of a solution can be recovered by monitoring the solution only on the subregion $\omega$ over a finite time interval $(0, T)$. The authors analyze two distinct models:

1. The free Schrödinger equation ($i\partial_t u - \Delta u = 0$).
2. The Schrödinger equation with a bounded potential ($i\partial_t u - \Delta u + V(t, x)u = 0$), where $V \in L^\infty$.

The study addresses two specific regimes: quantitative estimates for small times and small observation sets (specifically products of intervals), and the general validity of observability for arbitrary nonempty open sets and bounded potentials in any dimension $d \ge 1$.

Methodology
The proofs for both main theorems rely on a synthesis of cluster decomposition techniques and mathematical induction, building upon methods introduced by Bourgain [Bou14] and further developed by Burq and Zhu [BZ25]. The strategy diverges slightly for the two main results:

• For the Free Equation (Theorem 1.1): The proof is divided into high-frequency and low-frequency components.

  • High-Frequency: The authors exploit the cluster structure of the set $S_N = \{(n, |n|^2) \in \mathbb{Z}^{d+1} : |n| > N\}$. They partition this set into disjoint clusters that are spatially separated and contained in lower-dimensional affine sublattices. An inductive argument on the dimension of these sublattices yields a high-frequency observability estimate.
  • Low-Frequency: The authors establish a quantitative unique continuation estimate. This is achieved by reducing the problem to an observability estimate for the heat equation via a quantitative Lebeau-Robbiano strategy, utilizing Turán-Nazarov inequalities and spectral estimates.
  • Synthesis: By combining the high-frequency estimate with the low-frequency unique continuation, and carefully quantifying the dependence on the measure of the observation set $|\omega|$ and time $T$, they derive the final estimate.

• For the Equation with Bounded Potential (Theorem 1.2): The approach involves establishing a weak observability inequality first, followed by a compactness-uniqueness argument (inspired by [BLR92]) to remove the remainder term.

  • Weak Observability: The solution $u$ is time-truncated to a function $U$ on $\mathbb{T}^{d+1}$. The authors perform a frequency decomposition, isolating frequencies near the hypersurface $m = |n|^2$. They utilize a cluster decomposition for the set $S'_N = \{(n, m) : |n| > N, |m - |n|^2| \le A\}$ to prove an observability estimate for the localized high-frequency part.
  • Removal of Remainder: A compactness-uniqueness argument is applied. The authors show that the subspace of initial data producing zero observation on $\omega \times (0, T)$ is finite-dimensional and stable under the operator $-\Delta + V$. By invoking the unique continuation property (either for time-independent potentials or assumed for time-dependent ones), they prove this subspace is trivial, thereby upgrading the weak inequality to a full observability estimate.
  • Direct Treatment: A key methodological advantage noted is that the authors treat the bounded potential $V \in L^\infty$ directly, avoiding the need to approximate $V$ by smooth potentials, which is a common technique in prior literature.

Key Contributions and Results

1. Quantitative Estimate for the Free Equation (Theorem 1.1):
   The authors establish a quantitative observability estimate for the free Schrödinger equation when the observation set $\omega$ is a product of small intervals and the time $T$ is small. The estimate takes the form:
   $$\|u_0\|_{L^2}^2 \le C \exp\left( \frac{C}{(|\omega|T)^{C'}} \right) \int_0^T \|e^{-it\Delta}u_0\|_{L^2(\omega)}^2 dt$$
   This result extends Tao's [Tao21] work from dimension $d=2$ and small balls to higher dimensions and product sets. While the authors acknowledge their constant's dependence on $|\omega|$ and $T$ is not asymptotically sharp compared to Tao's (due to the absence of a $\log \log \epsilon^{-1}$ term), their method provides explicit quantification for both the measure of the set and the time interval.

2. Observability for Bounded Potentials (Theorem 1.2):
   The paper resolves a well-known conjecture in the field by proving that observability holds for the Schrödinger equation with a merely bounded potential $V \in L^\infty(\mathbb{T}^d)$ in any dimension $d \ge 1$, for every time $T > 0$ and every nonempty open subset $\omega$.

   • For time-independent potentials $V(x)$, observability holds unconditionally.
   • For time-dependent potentials $V(t, x)$, observability holds provided the potential satisfies a unique continuation property.
     The authors note that their method does not yet extend to arbitrary measurable sets of positive measure (a stronger conjecture by Burq and Zhu [BZ25]), though it handles products of measurable sets.

Significance and Claims
The paper claims to resolve a long-standing conjecture regarding the observability of the Schrödinger equation with $L^\infty$ potentials on the torus across all dimensions. By avoiding the approximation of potentials by smooth functions, the methodology offers a more direct treatment of rough potentials.

The authors position their work as a refinement of existing techniques, specifically leveraging Bourgain's cluster decomposition to handle the geometric and spectral complexities of the torus. They explicitly state that while their quantitative estimates for the free equation are not sharp in terms of the dependence on $T$ and $|\omega|$ (a problem they note has received limited attention in the Schrödinger literature compared to the heat equation), the results provide a unified framework for both free and potential-driven equations.

The results have standard applications in control theory, specifically yielding resolvent estimates for the stationary Schrödinger operator and controllability results via the Hilbert Uniqueness Method, though the authors defer the explicit formulation of these applications to existing literature.