Geometric conditions for the exact controllability of fractional free and harmonic Schr\"odinger equations

TL;DR

This paper establishes precise geometric criteria for the exact controllability of one-dimensional fractional Schrödinger equations, using advanced harmonic analysis and spectral techniques.

Contribution

It provides the first comprehensive set of necessary and sufficient geometric conditions for controllability of fractional Schrödinger equations.

Findings

Derived conditions from Logvinenko-Sereda theorem for fractional free Schrödinger equations.

Used Hautus test and Plancherel-Rotach formula for fractional harmonic Schrödinger equations.

Established a link between geometric properties and controllability in fractional quantum systems.

Abstract

We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schr\"odinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schr\"odinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schr\"odinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.