Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

Abstract: We prove two types of nodal results for density one subsequences of an orthonormal basis ${\phi_j}$ of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections $Z_{\phi_j} \cap H$ of the nodal set $Z_{\phi_j}$ of $\phi_j$ with a smooth curve $H$. Using recent results on quantum ergodic restriction theorems and prior results on periods of eigenfunctions over curves, we prove that under an asymmetry assumption on $H$, the number of intersection points tends to infinity for a density one subsequence of the $\phi_j$. . We also prove that the number of zeros of the normal derivative $\partial_{\nu} \phi_j$ on $H$ tends to infinity. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an isometric involution. Using (and generalizing) a geometric argument of Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd eigenfunctions tends to infinity for a density one subsequence of eigenfunctions.