Observability of the Schrödinger equation with subquadratic confining potential in the Euclidean space

Prouff, Antoine

doi:10.2140/apde.2025.18.1147

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Observability of the Schrödinger equation with subquadratic confining potential in the Euclidean space

Antoine Prouff

Vol. 18 (2025), No. 5, 1147–1229

DOI: 10.2140/apde.2025.18.1147

Abstract

We consider the Schrödinger equation in $ℝ^{d}$ , $d \geq 1$ , 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 Schrödinger operator under consideration. It is obtained using semiclassical analysis techniques. It allows us to provide 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.

Keywords

Schrödinger equation, Schrödinger group, observability, semiclassical analysis, quantum-classical correspondence principle, harmonic oscillators

Mathematical Subject Classification

Primary: 35J10, 35Q40, 81Q20, 81S30, 93B07

Secondary: 35S05, 47D08

Milestones

Received: 30 June 2023

Revised: 30 January 2024

Accepted: 31 March 2024

Published: 10 May 2025

Authors

Antoine Prouff
Department of Mathematics Purdue University West Lafayette, IN United States

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Vol. 18, No. 5, 2025