Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvature

TL;DR

This paper establishes lower bounds on the number of nodal domains for sums of plane waves and distorted plane waves on non-positively curved surfaces, matching Courant's upper bounds in order.

Contribution

It provides a criterion for amplitudes and directions of plane waves that guarantees optimal lower bounds on nodal domains, extending to distorted plane waves on certain surfaces.

Findings

Derived criteria for plane wave amplitudes and directions

Established optimal lower bounds for nodal domains

Applied results to distorted plane waves on non-positive curvature surfaces

Abstract

In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant's upper bound. As an application, we will obtain optimal lower bounds for the number of nodal domains of distorted plane waves on some families of surfaces of non-positive curvature.