On global approximate controllability of a quantum particle in a box by moving walls

TL;DR

This paper proves that a quantum particle in a box can be approximately controlled to reach any state by moving the box walls, using weak solutions of Schrödinger's equation and stability analysis.

Contribution

It establishes the global approximate controllability of a quantum particle in a box with moving walls, a novel result in quantum control theory.

Findings

Any initial quantum state can be driven arbitrarily close to any target state.

The controllability is achieved through boundary manipulation of the box.

The proof uses stability theorems for the Schrödinger equation.

Abstract

We study a system composed of a free quantum particle trapped in a box whose walls can change their position. We prove the global approximate controllability of the system. That is, any initial state can be driven arbitrarily close to any target state in the Hilbert space of the free particle with a predetermined final position of the box. To this purpose we consider weak solutions of the Schr\"odinger equation and use a stability theorem for the time-dependent Schr\"odinger equation.