Intersection bounds for nodal sets of planar Neumann eigenfunctions with interior analytic curves

Abstract

Let Ω ⊂ R 2 be a bounded piecewise smooth domain and ϕ λ be a Neumann (or Dirichlet) eigenfunction with eigenvalue λ 2 and nodal set N ϕ λ = {x ∈ Ω; ϕ λ (x) = 0}. Let H ⊂ Ω be an interior C ω curve. Consider the intersection numberWe first prove that for general piecewise-analytic domains, and under an appropriate "goodness" condition on H (see Theorem 1.1),as λ → ∞. Then, using Theorem 1.1, we prove in Theorem 1.2 that the bound in (1) is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, provided Ω is convex and H has strictly positive geodesic curvature.

Nodal intersections and L p restriction theorems on the torus

“…Thus we find that we are reduced to giving an upper bound on the restriction L 4 -norm. In § 7 we show [7] show that for a bounded, piecewise-analytic convex domain with ergodic billiard flow and C an analytic interior curve with strictly positive geodesic curvature, the upper bound (1.3) holds for a density-one subsequence of eigenfunctions. For eigenfunctions on a compact hyperbolic surface, Jung [11] has recently obtained an upper bound analogous to (1.3) when C is a geodesic circle.…”

“…We want to estimate the number of nodal intersections (1.1) N F,C = #{x : F (x) = 0} ∩ C that is the number of zeros of F on C . If C is real analytic, then upper bounds of the form N F,C ≪ λ can be obtained from a result of Toth and Zelditch [13] (see also [4], [7]) once we have an exponential restriction lower bound C |F | 2 ≫ e −cλ ||F || 2 2 for the L 2norm of F restricted to C, in terms of the L 2 -norm ||F || 2 2 = T 2 |F (x)| 2 dx. In the case of the torus, for any smooth C with non-vanishing curvature we have earlier obtained a uniform L 2 -restriction bound [1] (1.2) C |F | 2 ≫ ||F || 2 2 (the implied constants depending only on the curve C) and hence by [13] we get an upper bound for C analytic (1.3) N F,C ≪ λ In our paper [4] we also obtained a lower bound for N F,C when the curve C has non-vanishing curvature:…”

Nodal intersections for random eigenfunctions on the torus

“…It is expected that in many situations, there is an upper bound of the form Z(F ) ≪ λ, and general criteria for this to happen exist [26,12], though it is difficult to verify these criteria in most situations. As for lower bounds, nothing seems to be known in general, see [14] for results on Hecke eigenfunctions on hyperbolic surfaces (and [21] for analogous results on the sphere), and [16,17] for results on density one subsequences for hyperbolic surfaces.…”

Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty}$ bounded planar domains

“…Our second theorem deals with bounds for eigenfunction restrictions. This problem has been the focus of many papers over the past decade and has deep and interesting connection to the study of the asymptotics of eigenfunction nodal set and, in particular, intersection bounds [CTZ,DZ,ET,G,GRS,HZ,JJ,JJZ,TZ,TZ13,TZ12] Specifically, in the case of L 2 -normalized Neumann eigenfunctions with φ h L 2 (Ω) = 1, the h-Sobolev estimates give…”