Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long-time propagation of coherent states in a normally hyperbolic setting

Roméo Taboada

Abstract

We present a method to find asymptotics for the evolution of coherent states (or Gaussian wavepackets with standard deviation $\sqrt{h}$) under semiclassical Schrödinger's equation for a given Hamiltonian. These results extend the work of Combescure and Robert, in which the evolution of coherent states can be approximated in the limit $h\to 0$ with deformed Gaussian wavepackets called squeezed coherent states. The description with squeezed states holds for times $t$ that can go to infinity as $h\to 0$, under the constraint $|t|\leq |\log h|/(6λ_0)$ where $λ_0$ is the maximal Lyapunov exponent of the classical dynamics. The breakdown of this approximation at time $|\log h|/(6λ_0)$ is related to the bending of evolved wavepackets: once propagated states spread at a scale $h^{1/3}$, squeezed states no longer provide an appropriate description. To obtain a representation of propagated states valid up to times $|t|\leq C|\log h|$ with a larger $C$ (for instance, up to Ehrenfest's time $|\log h|/(2λ_0)$ where spreading on macroscopic scales is allowed), we make additional assumptions on the flow $Φ_t$ associated to the classical dynamics, imposing constraints on directions of elongation. Namely, we work in a neighborhood of a normally hyperbolic $Φ_t$-invariant submanifold $K$, on which the dynamics is considered as slow in comparison with its transverse directions, along which $Φ_t$ is assumed to be hyperbolic. In this context, we describe the propagated state as a WKB state in transverse directions and a squeezed state along $K$. This description emphasizes the fact that propagated states should no longer be thought of as microlocalized on a point, but rather on an isotropic submanifold (corresponding to transverse unstable directions). Guillemin, Uribe, and Wang presented a similar class of wavefunctions microlocalized on an isotropic submanifold.

Long-time propagation of coherent states in a normally hyperbolic setting

Abstract

We present a method to find asymptotics for the evolution of coherent states (or Gaussian wavepackets with standard deviation ) under semiclassical Schrödinger's equation for a given Hamiltonian. These results extend the work of Combescure and Robert, in which the evolution of coherent states can be approximated in the limit with deformed Gaussian wavepackets called squeezed coherent states. The description with squeezed states holds for times that can go to infinity as , under the constraint where is the maximal Lyapunov exponent of the classical dynamics. The breakdown of this approximation at time is related to the bending of evolved wavepackets: once propagated states spread at a scale , squeezed states no longer provide an appropriate description. To obtain a representation of propagated states valid up to times with a larger (for instance, up to Ehrenfest's time where spreading on macroscopic scales is allowed), we make additional assumptions on the flow associated to the classical dynamics, imposing constraints on directions of elongation. Namely, we work in a neighborhood of a normally hyperbolic -invariant submanifold , on which the dynamics is considered as slow in comparison with its transverse directions, along which is assumed to be hyperbolic. In this context, we describe the propagated state as a WKB state in transverse directions and a squeezed state along . This description emphasizes the fact that propagated states should no longer be thought of as microlocalized on a point, but rather on an isotropic submanifold (corresponding to transverse unstable directions). Guillemin, Uribe, and Wang presented a similar class of wavefunctions microlocalized on an isotropic submanifold.
Paper Structure (63 sections, 41 theorems, 311 equations, 2 figures)

This paper contains 63 sections, 41 theorems, 311 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A few properties of coherent states
  3. Classical and quantum dynamics
  4. Construction of coherent states
  5. Exact propagation under elementary Hamiltonians
  6. Propagation of coherent states in more general cases
  7. Approximation in the semiclassical limit
  8. Identification of the threshold, the role of Ehrenfest's time
  9. An alternative description with WKB states, properties, and limitations
  10. Assumptions and Statement of results
  11. Assumptions on classical dynamics
  12. Propagation of coherent states centered on $K^\delta$
  13. Propagation of coherent states centered in a neighborhood of $K^\delta$
  14. Organization of the paper
  15. Preliminaries
  16. ...and 48 more sections

Key Result

Theorem 1

Let $\varepsilon\in (0, \frac{1}{6\lambda_{\text{max}}})$. Under the hypotheses (H0) and (H1'), there exists $h_0>0$ such that for any $0<h<h_0$, any $\rho\in K^\delta$ and $\kappa_\alpha$ chart adapted to $K^\delta$ containing $\rho$ and for any time $t(h)\in [\varepsilon|\log h|,(\frac{1}{2 \lambd where with $\sigma_c^{(t)}=\sup_{\rho\in K^\delta} \|d\Phi^t(\rho)\vert_{T_{\rho}K^\delta}\|$ if $

Figures (2)

  • Figure 1: Clouds of point spreading along a manifold and ellipsoids approximating them.
  • Figure 2: Coordinates associated with the point $\rho\in K$

Theorems & Definitions (88)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Lemma 1
  • Proposition 1
  • Corollary 1
  • ...and 78 more