Localized low-frequency Neumann modes in 2d-systems with rough boundaries

TL;DR

This paper investigates how boundary conditions affect vibrational mode localization in 2D systems, revealing that Neumann boundaries lead to many localized low-frequency modes due to integral constraints.

Contribution

It demonstrates the significant impact of boundary conditions on mode localization and explains the prevalence of localized states under Neumann conditions through integral properties.

Findings

Localized states are common under Neumann conditions

Number of localized states increases with boundary roughness

Neumann boundary integral constraint causes low-frequency localization

Abstract

We compute the relative localization volumes of the vibrational eigenmodes in two-dimensional systems with a regular body but irregular boundaries under Dirichlet and under Neumann boundary conditions. We find that localized states are rare under Dirichlet boundary conditions but very common in the Neumann case. In order to explain this difference, we utilize the fact that under Neumann conditions the integral of the amplitudes, carried out over the whole system area is zero. We discuss, how this condition leads to many localized states in the low-frequency regime and show by numerical simulations, how the number of the localized states and their localization volumes vary with the boundary roughness.