Arithmetic, Zeros, and Nodal Domains on the Sphere

Abstract

We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelöf hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelöf.

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. Aronovich and Smilansky [2] studied the nodal intersections of random monochromatic waves on the plane [3] with various reference curves.…”

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds

“…To put the nodal results into context, it is proved in varying degrees of generality in [8,9,11,12,13,14,17,22] that in dimension 2, the number of nodal domains of an orthonormal basis {u j } of Laplace eigenfunctions on certain surfaces with ergodic geodesic flow tends to infinity with the eigenvalue along almost the entire sequence of eigenvalues. By the first item of Theorem 1.5, the same is true for their lifts to the unit tangent bundle SX as invariant eigenfunctions of the Kaluza-Klein metric.…”

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

“…(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) .…”