Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An obstruction to small-time controllability of the bilinear Schr{ö}dinger equation

Ivan Beschastnyi, Ugo Boscain, Mario Sigalotti

Abstract

In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and quantum systems using the WKB method. In particular, we investigate the conjecture that if the classical system is not small-time controllable, then the corresponding quantum system is not small-time controllable either.

An obstruction to small-time controllability of the bilinear Schr{ö}dinger equation

Abstract

In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and quantum systems using the WKB method. In particular, we investigate the conjecture that if the classical system is not small-time controllable, then the corresponding quantum system is not small-time controllable either.

Paper Structure

This paper contains 8 sections, 9 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Classical small-time controllability
  3. Small-time controllability in phase space
  4. Obstructions to small-time controllability of quantum systems
  5. WKB method
  6. Relation to the standard calculus of variations
  7. Proof of Theorem \ref{['thm:main1']} in the case when $N_2$ is a point
  8. Proof of Theorem \ref{['thm:main1']}

Key Result

Theorem 1.1

Assume that $M = N_1 \times N_2$ is the product of two connected complete Riemannian manifolds $N_1,N_2$. Consider a Schrödinger equation of the form eq:schro on $M$ and assume that the potentials $V, W: M\to \mathbb{R}$ satisfy Assumption ass. Suppose that there exists a nonempty open set $\Omega\i where $d_{1}V(x,y)$ denotes the differential at $x$ of $V(\cdot,y):N_1\to \mathbb{R}$. Under these

Theorems & Definitions (16)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 6 more