Observability of the Schr{\"o}dinger equation with subquadratic confining potential in the Euclidean space

TL;DR

This paper characterizes the conditions under which the Schrödinger equation with subquadratic confining potential in Euclidean space is observable from certain open sets, using semiclassical analysis to estimate optimal observation times.

Contribution

It provides a new observability criterion based on Hamiltonian flow and semiclassical analysis for Schrödinger equations with subquadratic potentials, including explicit conditions for harmonic oscillators.

Findings

Observability depends on the regularity of the observation set.

The optimal observation time can be accurately estimated.

For harmonic potentials, arithmetical properties of frequencies are key.

Abstract

We consider the Schr{\"o}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{\"o}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.