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Emergence of Gaussian fields in noisy quantum chaotic dynamics

Maxime Ingremeau, Martin Vogel

Abstract

We study the long time Schrödinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schrödinger operators $P_h^α=-h^2Δ_g +h^αQ_ω$, $0<α\leq 1$, where the semiclassical Laplace-Beltrami operator $-h^2Δ_g$ on $X$ is subject to a small random perturbation $h^αQ_ω$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_ω$ is bounded, but oscillates and decorrelates at scale $h^β$, $0< β< \frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $α,β$, in probability with respect to $ω$ the long time propagation $$\mathrm{e}^{\frac{i}{h}t_h P_h^α} f_h, \quad o(|\log h|)=t_h\to\infty, ~~h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $ω$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$.

Emergence of Gaussian fields in noisy quantum chaotic dynamics

Abstract

We study the long time Schrödinger evolution of Lagrangian states on a compact Riemannian manifold of negative sectional curvature. We consider two models of semiclassical random Schrödinger operators , , where the semiclassical Laplace-Beltrami operator on is subject to a small random perturbation given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of is bounded, but oscillates and decorrelates at scale , . We prove a quantitative result that, under appropriate conditions on , in probability with respect to the long time propagation rescaled to the local scale of around a uniformly at random chosen point on , converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and -almost sure version of this convergence along sufficiently fast decaying subsequences .
Paper Structure (49 sections, 44 theorems, 489 equations, 3 figures)

This paper contains 49 sections, 44 theorems, 489 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Lagrangian states
  4. Noisy propagation of Lagrangian states
  5. Randomization, local weak limits and the Berry Gaussian field
  6. Random smooth functions.
  7. Randomization and local weak limits.
  8. The Berry Gaussian field.
  9. The main results
  10. Ideas of the proof and organization of the paper
  11. Notations and conventions
  12. Cotangent space
  13. Universal cover
  14. Semiclassical analysis on smooth manifolds
  15. Semiclassical pseudo-differential operators
  16. ...and 34 more sections

Key Result

Theorem 2.9

Let $X$ be a compact connected Riemannian manifold with negative sectional curvature and without boundary. Let $P_h^\delta$ be as in eq:SchroedingerOp and suppose that Hypotheses HypPot, Hyp:BetaDelta and Hyp:Prob are satisfied. Let $D>0$. There exists $\eta=\eta(D)>0$ such that the following holds. with $\lambda_a= \frac{\|a\|^2}{\mathrm{Vol}(X)}$.

Figures (3)

  • Figure 1: Admissible parameters $\alpha$ and $\beta$, see Hypothesis \ref{['Hyp:BetaDelta']}. The dark grey region is admissible for the \ref{['eq:Pot']} and the \ref{['eq:Pseudo']}, while the light grey region is only admissible in the \ref{['eq:Pseudo']}.
  • Figure 2: Construction of the point $\zeta_\delta^{s,t}(x)$
  • Figure 3: An example where the phases $\textcolor{black}{\widetilde{\phi}_{t,\delta}( \widetilde{x})}$ and $\textcolor{black}{\widetilde{\phi}_{t,\delta}( \widetilde{x}')}$ are not independent. Here, $\Lambda_0$ is a piece of the unstable manifold of the periodic point $\rho_0$ to which $\rho$ and $\rho'$ both belong. There are two lifts of $x$, $\widetilde{x}$ and $\widetilde{x}'$ such that $\rho_{\widetilde{x}}$ and $\rho_{\widetilde{x}'}$ originate respectively from the points $\rho$ and $\rho'$; since $\rho$ and $\rho'$ belong to the same geodesic, the phases $\textcolor{black}{\widetilde{\phi}_{t,\delta}( \widetilde{x})}$ and $\textcolor{black}{\widetilde{\phi}_{t,\delta}( \widetilde{x}')}$ are dependent, since the integrals in \ref{['eq:PhiIntegrale2']} defining them contain a part which is equal.

Theorems & Definitions (96)

  • Definition 2.1
  • Example 2.3
  • Remark 2.5
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Remark 2.11
  • Corollary 2.12
  • Lemma 3.1
  • ...and 86 more