Sufficient conditions for localized vibrational modes in one- and two-dimensional discrete lattices

TL;DR

This paper rigorously proves that even weak localized perturbations in one- and two-dimensional discrete lattices can induce localized vibrational modes, extending understanding of phonon localization in crystal models.

Contribution

It provides a rigorous proof that weak perturbations decreasing mass induce localized vibrational modes in 1D and 2D lattices, using a variational approach.

Findings

Weak perturbations produce localized vibrational modes.

Localized mass decreases lead to phonon localization.

The variational proof is adaptable to other lattice models.

Abstract

This paper presents a rigorous proof that arbitrarily weak perturbations produce localized vibrational (phonon) modes in one- and two-dimensional discrete lattices, inspired by analogous results for the Schr{\"o}dinger and Maxwell equations, and complementing previous explicit solutions for specific perturbations (e.g., decreasing a single mass). In particular, we study monatomic crystals with nearest-neighbor harmonic interactions, corresponding to square lattices of masses and springs, and prove that arbitrary localized perturbations that decrease the net mass lead to localized vibrating modes. The proof employs a straightforward variational method that should be extensible to other discrete lattices, interactions, and perturbations.