How Lagrangian states evolve into random waves
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 177-212.

In this paper, we consider a compact connected manifold (X,g) of negative curvature, and a family of semi-classical Lagrangian states f h (x)=a(x)e iϕ(x)/h on X. For a wide family of phases ϕ, we show that f h , when evolved by the semi-classical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry’s random waves conjecture for Lagrangian states.

Dans cet article, nous considérons une variété riemannienne connexe, compacte, de courbure sectionnelle négative, et une famille d’états lagrangiens semi-classiques f h (x)=a(x)e iϕ(x)/h sur X. Pour une grande famille de phases ϕ, nous montrons que f h que l’on fait évoluer par l’équation de Schrödinger pendant un temps long ressemble à un champ aléatoire gaussien. Ceci peut être vu comme un analogue de la conjecture des ondes aléatoires de Berry pour les états lagrangiens.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.181
Classification: 35P99, 81B50, 81Q20
Keywords: Quantum chaos, semi-classical analysis, Berry’s conjecture, random waves
Mot clés : Chaos quantique, analyse semi-classique, conjecture de Berry, ondes aléatoires
Maxime Ingremeau 1; Alejandro Rivera 2

1 Laboratoire J.A. Dieudonné, Université Côte d’Azur Parc Valrose, 06108 Nice Cedex 2, France
2 École Polytechnique Fédérale de Lausanne, Chair of Random Geometry CH-1015 Lausanne, Suisse
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Maxime Ingremeau; Alejandro Rivera. How Lagrangian states evolve into random waves. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 177-212. doi : 10.5802/jep.181. https://jep.centre-mersenne.org/articles/10.5802/jep.181/

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