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

On the Nodal Lines of Eisenstein Series on Schottky Surfaces

“…On compact non positively curved surfaces with boundary, Jung and Zelditch [11], show that #(N λ ∩C 0 ) goes to infinity as λ goes to infinity, when C is a boundary curve. On the other hand, a similar statement holds [10] on a negatively curved surface (without boundary) and when C satisfies a non symmetry condition. On the modular surface PSL 2 (Z)\H 2 , Jung [9] obtains effective lower bounds of the type…”

“…The first remark that we have in mind is that by adapting straightforwardly the combinatorial arguments used in [7,11,10] we can obtain a lower bound for the number of connected components of X 0 \ N λ , for generic ξ, which says that for large λ, M ξ (λ) ≥ C −1 λ.…”

“…Clearly the main input here is the lower bound given by Theorem 1.1 and the graph theoretic arguments from [10], which are a generalization of the more elementary ideas pioneered in [7]. However that kind of lower bound is rather irrelevant, because we cannot rule out the fact that these connected components could very well come from a single nodal domain which would intersect several times the convex core X 0 .…”

“…Most of the above mentioned works are motivated by the study of nodal domains. It is a notoriously challenging problem to count them and [11,10,7] provide the very first (deterministic) examples of eigenfunctions where one is actually able to show that the number of nodal domains goes to infinity at high frequency. As a corollary of theorem 1.1, we prove the following.…”

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

“…We further denote by Σ ϕ λ = {x ∈ Z ϕ λ : dϕ λ (x) = 0} the singular set of ϕ λ . The main result of this article, developing the method of [JZ13], gives a rather general sufficient condition on surface M with nonempty boundary ∂M = ∅ and ergodic billiard flow under which the number of nodal domains tends to infinity along a full density subsequence (i..e of 'almost all' eigenfunctions of any orthonormal basis). A significant gain in the billiard case is that we do not require the surface (or eigenfunctions) to have a symmetry.…”