Rigorous Methods for Bohr-Sommerfeld Quantization Rules
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908389206917120 |
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| author | Dong, Joanne Miller, Peter D. Young, Giorgio |
| author_facet | Dong, Joanne Miller, Peter D. Young, Giorgio |
| contents | In this work, we prove Bohr-Sommerfeld quantization rules for the self-adjoint Zakharov-Shabat system and the Schrödinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for $2\times 2$ traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigorous Methods for Bohr-Sommerfeld Quantization Rules Dong, Joanne Miller, Peter D. Young, Giorgio Classical Analysis and ODEs Pattern Formation and Solitons In this work, we prove Bohr-Sommerfeld quantization rules for the self-adjoint Zakharov-Shabat system and the Schrödinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for $2\times 2$ traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values. |
| title | Rigorous Methods for Bohr-Sommerfeld Quantization Rules |
| topic | Classical Analysis and ODEs Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2505.05342 |