A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields
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| Format: | Preprint |
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2024
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| _version_ | 1866908663938023424 |
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| author | Cornean, Horia D. Helffer, Bernard Purice, Radu |
| author_facet | Cornean, Horia D. Helffer, Bernard Purice, Radu |
| contents | We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields Cornean, Horia D. Helffer, Bernard Purice, Radu Mathematical Physics Primary: 81Q10, 81Q15. Secondary: 35S05 We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control. |
| title | A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields |
| topic | Mathematical Physics Primary: 81Q10, 81Q15. Secondary: 35S05 |
| url | https://arxiv.org/abs/2411.14171 |