Defining the spectral position of a Neumann domain

TL;DR

This paper studies the spectral properties of Neumann domains derived from Laplacian eigenfunctions on 2D manifolds, proving self-adjointness and discreteness of the Neumann Laplacian spectrum, and exploring the spectral position problem.

Contribution

It establishes the self-adjointness and discrete spectrum of the Neumann Laplacian on Neumann domains and analyzes the spectral position problem, which is more complex than the nodal domain case.

Findings

Neumann Laplacian on a Neumann domain is self-adjoint with a discrete spectrum.

Restriction of an eigenfunction to a Neumann domain is an eigenfunction of the Neumann Laplacian.

The spectral position problem for Neumann domains is more involved than for nodal domains.

Abstract

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, 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.