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Quantum chaos in the Benjamini-Schramm limit

Banff International Research Station for Mathematical Innovation and Discovery

One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random waves. We will show how a notion of convergence for sequences of manifolds called Benjamini-Schramm convergence can give a satisfying formulation of this conjecture. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a more general framework. Based on this formulation, we will expand the scope and consider a case where the frequencies stay bounded and the size of the manifold increases instead. We will formulate the corresponding random wave conjecture and present some results to support it, including a quantum ergodicity theorem. Based on joint works with Tuomas Sahlsten, Miklos Abert and Nicolas Bergeron.

Mathematics; Quantum theory; Global analysis, analysis on manifolds; Mathematical physics

video/mp4

Author affiliation: Bristol University

2019-03-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68669

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/