Discrete vs. continuous in the semiclassical limit
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914830541127680 |
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| author | Becker, Simon Wittsten, Jens Zworski, Maciej |
| author_facet | Becker, Simon Wittsten, Jens Zworski, Maciej |
| contents | We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr-Sommerfeld quantization conditions for potentials exhibiting non-degenerate wells, and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter $h$ and show that it can be discontinuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete vs. continuous in the semiclassical limit Becker, Simon Wittsten, Jens Zworski, Maciej Spectral Theory Mathematical Physics Analysis of PDEs We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr-Sommerfeld quantization conditions for potentials exhibiting non-degenerate wells, and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter $h$ and show that it can be discontinuous. |
| title | Discrete vs. continuous in the semiclassical limit |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2406.05934 |