Lyapunov control of a quantum particle in a decaying potential

TL;DR

This paper introduces a Lyapunov-based control method for stabilizing a quantum particle in an N-dimensional decaying potential, addressing challenges posed by the system's spectral properties and ensuring approximate stabilization.

Contribution

It develops a Lyapunov control strategy tailored for quantum particles in decaying potentials, analyzing convergence and stabilization in the presence of mixed spectra.

Findings

Lyapunov control achieves approximate stabilization in discrete spectral states.

Dispersion in the continuous spectrum poses a major challenge for stabilization.

Control strategy encodes distance to target and penalizes passage through continuous spectrum.

Abstract

A Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\"o}dinger equation in whole $\RR^N$ is proposed. For the case of a quantum particle in an $N$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.