Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space

TL;DR

This work investigates the observability of the Schrödinger equation on $\mathbb{R}^d$ with confining potentials growing at most quadratically, linking quantum observability to the classical Hamiltonian flow via semiclassical analysis. The authors prove a robust main theorem: if a high-energy classical observable $${\mathfrak{K}}_{p_0}^{\infty}(\omega,T_0)$$ is positive, then the Schrödinger equation is observable from a thickened set $\omega_R$ (with $R$ efficiently chosen) for times $T>T_0$, and they quantify the cost and the dependence on subprincipal perturbations. They develop a detailed analysis of the underlying classical dynamics, establish an Egorov-type link between quantum and classical evolutions, and apply the framework to conical and spherical observation sets, with sharp results in two-dimensional harmonic oscillators involving arithmetic properties of the frequencies. Applications include uniform observability of eigenfunctions and energy decay for damped waves, and the approach highlights a Kato-smoothed, high-energy perspective on observability in unbounded domains. The paper also discusses a natural semiclassical scaling for homogeneous potentials, clarifying why subquadratic growth is essential for the method and how the optimal observation time can be estimated from geometric and number-theoretic data. Overall, the results extend observability theory for Schrödinger-type equations to non-compact settings and provide precise, geometry-aware criteria and timescales for practical monitoring of quantum states.

Abstract

We consider the Schr{ö}dinger equation in $\mathbf{R}^d$, $d \ge 1$, with a confining potential growing at most quadratically. Our main theorem characterizes open sets from which observability holds, provided they are sufficiently regular in a certain sense. The observability condition involves the Hamiltonian flow associated with the Schr{ö}dinger operator under consideration. It is obtained using semiclassical analysis techniques. It allows to provide with an accurate estimation of the optimal observation time. We illustrate this result with several examples. In the case of two-dimensional harmonic potentials, focusing on conical or rotation-invariant observation sets, we express our observability condition in terms of arithmetical properties of the characteristic frequencies of the oscillator.

Figures (4)

• Figure 1: Case of a potential with critical points.
• Figure 2: Typical projected trajectories of two-dimensional harmonic oscillators. Shading indicates the course of the trajectory.
• Figure 3: The above projected trajectory is responsible for the lower bound on the optimal observation time in \ref{['eq:optimaltimeconicset']}. It is obtained with an oscillator such that $\frac{\nu_2}{\nu_1} = 3.9$. For $\frac{\nu_2}{\nu_1} = 4$, one can choose the initial datum so that the curve goes back to the upper-right quadrant, passing through the origin, without crossing the two cones. This yields a larger lower bound on the optimal time, corresponding to the jump from $\lfloor 3.9 \rfloor = 3$ to $\lfloor 4 \rfloor = 4$ in Formula \ref{['eq:optimaltimeconicset']}.
• Figure 4: The above curve is a projected trajectory of a harmonic oscillator with $\frac{\nu_2}{\nu_1} = \frac{4}{3}$, that does not intersect the observation set $\omega(I)$. The existence of a sequence of energy layers $\{p = E_n\}$, $E_n \to + \infty$, containing such curves would imply that observability from $\omega(I)$ fails.

Theorems & Definitions (18)

• proof
• proof
• proof : Proof of Lemma \ref{['lem:rootspolynomial']}
• proof : Proof of Proposition \ref{['prop:timeincylinder']}
• proof : Proof of Corollary \ref{['cor:classicalnonobs']}
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• proof