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Boundedness of the number of nodal domains of eigenfunctions

Banff International Research Station for Mathematical Innovation and Discovery

The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with non-empty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty. In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic: for any given orthonormal eigenbasis one can find a subsequence of density 1 where the number of nodal domains is identically 2. This highlights how underlying dynamics can impact the nodal counting. I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1. I also will present an explicit orthonormal eigenbasis on the 3 torus where all of them have only two nodal domains.

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

video/mp4

Author affiliation: Texas A&M University

2019-03-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68675

Unreviewed

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