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Defining the spectral position of a Neumann domain

Ram Band, Graham Cox and Sebastian K. Egger

Vol. 16 (2023), No. 9, 2147–2171
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.

Keywords
Neumann domains, Neumann lines, nodal domains, Laplacian eigenfunctions, Morse–Smale complexes
Mathematical Subject Classification
Primary: 35P05, 58C40, 58J50
Secondary: 37C15, 57K20
Milestones
Received: 23 April 2021
Accepted: 28 February 2022
Published: 11 November 2023
Authors
Ram Band
Department of Mathematics
Technion - Israel Institute of Technology
Haifa
Israel
Institute of Mathematics
University of Potsdam
Potsdam
Germany
Graham Cox
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL
Canada
Sebastian K. Egger
Department of Mathematics
Technion - Israel Institute of Technology
Haifa
Israel

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