Controllability of the discrete-spectrum Schrodinger equation driven by an external field

TL;DR

This paper proves approximate controllability of the bilinear Schrödinger equation with discrete non-resonant spectrum, applicable to various domains and potential bounds, using finite-dimensional Galerkin methods.

Contribution

It introduces a novel approach combining Galerkin approximations with controllability analysis for Schrödinger equations with discrete spectra.

Findings

Controllability established for harmonic oscillator and 3D potential well.

Method applicable to bounded and unbounded domains.

Results extend to density matrix controllability.

Abstract

We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.