Uncertainty Principle and Geometric Condition for the Observability of Schr\"{o}dinger Equations

TL;DR

This paper establishes geometric conditions for the exact observability of Schrödinger equations with inverse-square potentials, utilizing a Logvinenko-Sereda theorem for the Hankel transform, which is key to understanding control and measurement of quantum systems.

Contribution

It introduces necessary and sufficient geometric conditions for observability of Schrödinger equations with inverse-square potentials, based on a new Logvinenko-Sereda type theorem for the Hankel transform.

Findings

Derived geometric conditions for observability.

Connected observability to support properties of Hankel transform.

Provided a criterion for bounding functions via their restriction to subsets.

Abstract

We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schr\"odinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $\Omega$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $\Omega$, with a constant independent of the position of the interval.