The Spectral Position of Neumann Domains on the Torus

Abstract

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. Morse-Smale complexes). This partition is generated by gradient flow lines of the eigenfunction-these bound the socalled Neumann domains. We prove that the Neumann Laplacian ∆ defined on a single Neumann domain is self-adjoint and possesses a purely discrete spectrum. In addition, we prove that the restriction of the eigenfunction to any one of its Neumann domains is an eigenfunction of ∆. As a comparison, similar statements for a nodal domain of an eigenfunction (with the Dirichlet Laplacian) are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, and hence is not necessarily continuous, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.

Neumann Domains on Quantum Graphs

“…One may expect that the restriction of an eigenfunction to a Neumann domain, which was shown to be an eigenfunction of that domain, would correspond to the first nonzero eigenvalue of the domain. This is not the case in general, as can be seen for manifolds in [10] for example. The question we may ask, given a generic eigenfunction f of eigenvalue k 2 and a Neumann domain Ω is what is the position of k 2 in the spectrum of Ω, namely the spectral position which we define as follows:…”

“…Remark 2.5. As the goal of this paper is to define and analyze Neumann domains on graphs, we should mention another class of eigenfunctions used when considering Neumann domains on manifolds [5,11,10,30,12], which is Morse eigenfunctions. A Morse eigenfunction is such that at no point both the function and its derivative vanish.…”

“…Throughout this paper if Ω is a Neumann domain of an eigenfunction f of eigenvalue k 2 we will write N (Ω) instead of N (Ω, k). Another parameter that relates the spectrum and geometry of a domain is the rescaled area to perimeter ratio ρ which was introduced in [24], and was used in [10]…”

“…To the best of our knowledge, this is the first work on Neumann domains on graphs 2 . Even on manifolds, Neumann domains are a very recent topic of research within spectral theory and is currently mentioned only in [37,30,12,10,5]. Partial results of the current paper were already announced in [5] which reviews the Neumann domain research on manifolds and on graphs.…”

Neumann Domains on Graphs and Manifolds

“…The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in [2,6,7,46,62] and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.…”