Internal control for a non-local Schr\"odinger equation involving the fractional Laplace operator

TL;DR

This paper investigates the controllability of a nonlocal Schr"odinger equation with fractional Laplacian, showing controllability for certain fractional powers and non-controllability for others, using multiplier and spectral analysis methods.

Contribution

It establishes the boundary controllability results for the fractional Schr"odinger equation depending on the fractional power s, including explicit spectral computations in one dimension.

Findings

Controllability holds for s in [1/2, 1)

Controllability fails for s < 1/2

Spectral analysis in 1D supports the results

Abstract

We analyze the interior controllability problem for a nonlocal Schr\"odinger equation involving the fractional Laplace operator $(-\Delta)^s$, $s\in(0,1)$, on a bounded $C^{1,1}$ domain $\Omega\subset\mathbb{R}^n$. The controllability from a neighborhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. The results follow applying the multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian, and from an explicit computation of the spectrum of the operator in the one-dimensional case.