Lyapunov control of a quantum particle in a decaying potential

Abstract

A Lyapunov-based approach for the trajectory generation of an N-dimensional Schrödinger equation in whole ${\mathbb{R}}^N$ 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.