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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.03580 |
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| _version_ | 1866908631248666624 |
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| author | Thomas-Markarian, Jaden Arrieta, Rodrigo Yang, Shu-Ching Parzygnat, Arthur J. Johnson, Steven G. |
| author_facet | Thomas-Markarian, Jaden Arrieta, Rodrigo Yang, Shu-Ching Parzygnat, Arthur J. Johnson, Steven G. |
| contents | 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{ö}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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for localized vibrational modes in one- and two-dimensional discrete lattices Thomas-Markarian, Jaden Arrieta, Rodrigo Yang, Shu-Ching Parzygnat, Arthur J. Johnson, Steven G. Other Condensed Matter 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{ö}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. |
| title | Sufficient conditions for localized vibrational modes in one- and two-dimensional discrete lattices |
| topic | Other Condensed Matter |
| url | https://arxiv.org/abs/2511.03580 |