On the nodal lines of Eisenstein series on Schottky surfaces

TL;DR

This paper studies the high-energy behavior of Eisenstein series on certain hyperbolic surfaces, providing bounds on nodal line intersections and nodal domains, advancing understanding of eigenfunction nodal sets in hyperbolic geometry.

Contribution

It establishes equidistribution results for Eisenstein series restrictions and derives optimal bounds on nodal line intersections and nodal domains on Schottky surfaces.

Findings

Proves equidistribution of Eisenstein series on geodesic segments.

Provides optimal bounds for intersections of nodal lines with geodesics.

Derives upper bounds on the number of nodal domains.

Abstract

On convex co-compact hyperbolic surfaces with Hausdorff dimension of the limit set less than 1/2, we investigate high energy behaviour of Eisenstein Series. Eisenstein Series are non-L^2 eigenfunctions of the hyperbolic Laplacian which parametrize the continous spectrum. We prove an equidistribution result for restrictions to geodesics segments and use it to obtain optimal lower and upper bounds for the number of intersections of nodal lines with a fixed geodesic segment as the frequency goes to infinity. Upper bounds on the number of nodal domains are also derived.