Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces

TL;DR

This paper investigates how the complex zeros of eigenfunctions on real analytic surfaces intersect with geodesics, showing that under ergodic and generic conditions, these intersections become uniformly distributed along the geodesic.

Contribution

It establishes new results on the distribution of complex nodal intersections with geodesics under ergodic and generic asymmetry conditions, extending understanding of eigenfunction behavior.

Findings

Intersections condense along the geodesic in ergodic cases

Intersections become uniformly distributed with respect to arc-length

Results apply to both periodic and non-periodic geodesics

Abstract

We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic and if the geodesic is periodic and satisfies a generic asymmetry condition, then the intersection points condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the `origin' is allowed to move with $\lambda_j$. The proof uses the quantum ergodic restriction theorem due to J. Toth and the author (see also Dyatlov-Zworski).