A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929501565353984 |
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| author | Band, Ram Beckus, Siegfried Biber, Barak Raymond, Laurent Thomas, Yannik |
| author_facet | Band, Ram Beckus, Siegfried Biber, Barak Raymond, Laurent Thomas, Yannik |
| contents | We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schrödinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $V\in\mathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels Band, Ram Beckus, Siegfried Biber, Barak Raymond, Laurent Thomas, Yannik Mathematical Physics Spectral Theory We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schrödinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $V\in\mathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there. |
| title | A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels |
| topic | Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2409.10920 |