Spectral Analysis of the Schrödinger Operator for the Incommensurate System
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908822390439936 |
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| author | Li, Yan Song, Yujian Zhou, Aihui |
| author_facet | Li, Yan Song, Yujian Zhou, Aihui |
| contents | Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schrödinger operators. However, the absence of periodicity in these systems poses significant challenges for obtaining the spectral information. In this paper, by embedding the system into higher dimensions together with introducing a regularization technique, we prove that the spectrum of the Schrödinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schrödinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the existence of Bloch-type solutions to the Schrödinger equation for the incommensurate system, which can be well approximated by the Bloch solutions to the equations associated with the regularized operators. Our analysis provides a theoretical support for understanding and computing the incommensurate systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08308 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Analysis of the Schrödinger Operator for the Incommensurate System Li, Yan Song, Yujian Zhou, Aihui Mathematical Physics Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schrödinger operators. However, the absence of periodicity in these systems poses significant challenges for obtaining the spectral information. In this paper, by embedding the system into higher dimensions together with introducing a regularization technique, we prove that the spectrum of the Schrödinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schrödinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the existence of Bloch-type solutions to the Schrödinger equation for the incommensurate system, which can be well approximated by the Bloch solutions to the equations associated with the regularized operators. Our analysis provides a theoretical support for understanding and computing the incommensurate systems. |
| title | Spectral Analysis of the Schrödinger Operator for the Incommensurate System |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2602.08308 |