On the nodal lines of Eisenstein series on Schottky surfaces

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.