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Globally integrable quantum systems and their perturbations

Dario Bambusi, Beatrice Langella

Abstract

In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear perturbations of a globally integrable quantum system. In particular, we give a spectral result ensuring stability of most of its eigenvalues under relatively bounded perturbations, and two results controlling the growth of Sobolev norms when it is subject either to linear unbounded time dependent perturbations or a small nonlinear Hamiltonian nonlinear perturbation.

Globally integrable quantum systems and their perturbations

Abstract

In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear perturbations of a globally integrable quantum system. In particular, we give a spectral result ensuring stability of most of its eigenvalues under relatively bounded perturbations, and two results controlling the growth of Sobolev norms when it is subject either to linear unbounded time dependent perturbations or a small nonlinear Hamiltonian nonlinear perturbation.
Paper Structure (25 sections, 15 theorems, 85 equations, 2 figures)

This paper contains 25 sections, 15 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical Integrable and Superintegrable Systems.
  3. Integrable systems
  4. Superintegrable systems
  5. Conclusion
  6. Quantizing Hamiltonians with periodic flow
  7. Conclusion
  8. Two purely quantum examples
  9. Globally integrable quantum systems
  10. Definitions
  11. Examples of globally integrable quantum systems
  12. Semiclassical integrable systems
  13. Perturbations of globally integrable quantum systems
  14. Linear Perturbations: spectral stability and growth of Sobolev norms
  15. Stability of the majority of eigenvalues
  16. ...and 10 more sections

Key Result

Theorem 2.3

Let $h$ be an integrable system; let $z_0\in T^*M$ be s.t. the differentials $df_j(z_0)$ are independent and denote $m_j=f_j(z_0)$. If the level surface is compact, then $S_0$ is diffeomorphic to an $n$ dimensional torus and there exists a neighborhood ${\mathcal{U}}$ of $S_0$ foliated in invariant $n$ dimensional tori. Furthermore there exist ${\mathcal{A}} \subseteq \mathbb R^n$ and a symplecti

Figures (2)

  • Figure 1: Representation of the path $\gamma$.
  • Figure :

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 28 more