On the Nodal Lines of Eisenstein Series on Schottky Surfaces

Abstract

On convex co-compact hyperbolic surfaces ${X=\Gamma \backslash \mathbb{H}^{2}}$, we investigate the behavior of nodal curves of real valued Eisenstein series ${F_\lambda(z,\xi)}$, where ${\lambda}$ is the spectral parameter, ${\xi}$ the direction at infinity. Eisenstein series are (non-${L^2}$) eigenfunctions of the Laplacian ${\Delta_X}$ satisfying ${\Delta_X F_\lambda=(\frac{1}{4}+\lambda^2)F_\lambda}$. As ${\lambda}$ goes to infinity (the high energy limit), we show that, for generic ${\xi}$, the number of intersections of nodal lines with any compact segment of geodesic grows like ${\lambda}$, up to multiplicative constants. Applications to the number of nodal domains inside the convex core of the surface are then derived.