Stephane Nonnenmacher

University Paris-Sud University Paris-Sud

Delocalization of the Laplace eigenmodes on Anosov surfaces

Mathematical Physics Seminar

9th February 2024, 1:45 pm – 3:30 pm
Fry Building, G.07

The eigenmodes of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$ can exhibit various localization properties in the high frequency regime, which strongly depend on the properties of the geodesic flow (the classical dynamics on $(M,g)$). We focus on situations where this flow is strongly chaotic (Anosov), for instance if the sectional curvature of $(M,g)$ is negative. Studying the Laplace (=quantum) eigenmodes is in the realm of Quantum Chaos.

In this situation, the Quantum Unique Ergodicity conjecture states that the eigenmodes become equidistributed on $M$ in the high-frequency limit: for any open set $\Omega$ on $M$, the $L^2$ masses on $\Omega$ of the eigenstates converge to the relative volume of $\Omega$. In the two-dimensional case, we show the weaker property of full delocalization: the $L^2$ masses of the eigenstates on $\Omega$ are bounded from below, independently of the frequency. This is in contrast with, e.g., the case of eigenstates on the round sphere, which may be strongly concentrated near a closed geodesic.

The proof uses various methods of semiclassical analysis, the structure of stable and unstable manifolds of the Anosov flow, and a recent Fractal Uncertainty Principle due to Bourgain-Dyatlov. Joint work with S.Dyatlov and L.Jin.

Organiser: Thomas Bothner

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