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

Aitor Balmaseda, Davide Lonigro, Juan Manuel Pérez-Pardo

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 Schrödinger equation and use a stability theorem for the time-dependent Schrödinger equation.

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

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 Schrödinger equation and use a stability theorem for the time-dependent Schrödinger equation.
Paper Structure (12 sections, 12 theorems, 105 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Dynamics of a particle in a moving box
  4. Approximate controllability of a particle in a moving box
  5. Existence of solutions and stability
  6. Proving controllability by moving walls
  7. Step one: an auxiliary control system
  8. Step two: weak controllability by piecewise smooth functions
  9. Step three: from the auxiliary system to the moving walls setup
  10. Case (i): $\delta \neq0$.
  11. Case (ii): $\delta=0$.
  12. Conclusions

Key Result

Proposition 2.6

Let $I\subset\mathbb{R}$ be a compact interval, and let $t\in I\mapsto\ell(t),d(t)\in\mathbb{R}$ two functions in $\mathrm{C}^2(I)$ with $\ell(t)>0$ for all $t\in I$. Then the time-dependent Hamiltonian $\tilde{H}_{\ell(t),d(t)}$, cf. Eqs. eq:domtransf, is self-adjoint, semibounded from below unifor

Figures (1)

  • Figure 1: Two particular cases covered by Theorem \ref{['thm:main']}. (a) A purely dilating cavity, to which Theorem \ref{['thm:main']}(ii) applies: approximate controllability between states with the same parity holds. (b) A translating and dilating cavity, to which Theorem \ref{['thm:main']}(i) applies: approximate controllability between arbitrary states holds.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Corollary 2.7
  • proof
  • Remark 2.8
  • Theorem 2.9
  • ...and 24 more