Localization of Neumann Eigenfunctions near Irregular Boundaries

Abstract

It has been empirically observed that eigenfunctions of Laplace's equation $-\Delta \varphi =\lambda \varphi$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.

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• Geometrical structure of Laplacian eigenfunctions
• Effective confining potential of quantum states in disordered media
• Localization of Laplacian eigenfunctions in circular, spherical and elliptical domains
• Localized Eigenfunctions: Here You See Them, There You Don't
• Detecting localized eigenstates of linear operators