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 {ϕj } of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections Zϕ j ∩H of the nodal set Zϕ j of ϕj 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 the number of intersection points tends to infinity for a density one subsequence of the ϕj, and furthermore that the number of such points where ϕj |H changes sign tends to infinity. We also prove that the number of zeros of the normal derivative ∂νϕj on H tends to infinity, also with sign changes. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an orientation-reversing isometric involution with non-empty and separating fixed point set. 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.

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

“…Remark 1.6. In [JZ13], the same assertion has been obtained when M is a negatively curved surface, but for a density one subsequence of {u n }. The argument of [JZ13] to detect a sign change of an eigenfunction u n on a curve β is to compare β u n (s)ds and β |u n (s)|ds.…”

“…Graph structure of the nodal set and Euler's inequality. In this section we briefly review the topological argument in [GRS13,JZ13] on bounding the number of nodal domains from below by the number of zeros on Fix(τ ). We refer the readers to [JZ13] for details.…”

“…In this section we briefly review the topological argument in [GRS13,JZ13] on bounding the number of nodal domains from below by the number of zeros on Fix(τ ). We refer the readers to [JZ13] for details. Firstly note that if there exists a segment of η ⊂ Fix(τ ) such that η ⊂ Z u , then because normal derivative of u vanishes along Fix(τ ), any point on η is a singular point, contradicting the upper bound on the number of singular points in [Don92].…”

“…the number of singular points) of the eigenfunction along Fix(τ ). Such an argument is first developed in [GRS13] and we review in §4.1 in terms of the nodal graphs and Euler's inequality as in [JZ13]. We then use Bochner's theorem and a Rellich type identity to deduce from QUE that even (resp.…”

“…(See [GRS13,Jun13,BR13,JZ13,Mag13,GRS14], where such an idea is used to prove a lower bound for the number of sign changes in various contexts.) In order to bound u n L 1 (β) from below using Hölder's inequality, the authors use the Quantum Ergodic Restriction (QER) theorem [TZ13,DZ13] for the lower bound of u n L 2 (β) and the point-wise Weyl law with an improved error term [Bér77] for the upper bound of u n L ∞ (β) .…”