Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns
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| Format: | Preprint |
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2025
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| _version_ | 1866917170016944128 |
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| author | Fujii, Toshiyuki Hiroshima, Fumio Tanda, Satoshi |
| author_facet | Fujii, Toshiyuki Hiroshima, Fumio Tanda, Satoshi |
| contents | The Hamiltonian J of the Josephson junction is introduced as a self-adjoint operator on l2 tensor l2. It is shown that J can also be realized as a self-adjoint operator HS1 on L2(S1) tensor L2(S1), from which a Mathieu operator given by "-d^2/dθ^2 - 2α cos θ" is derived. A fiber decomposition of HS1 with respect to the total particle number is established, and the action on each fiber is analyzed. In the presence of a magnetic field, a phase shift defines the magnetic Josephson junction Hamiltonian HS1(Φ) and the Josephson current IS1(Φ). For a constant magnetic field inducing a local phase shift Φ(x), the corresponding local current IS1(Φ(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns Fujii, Toshiyuki Hiroshima, Fumio Tanda, Satoshi Mathematical Physics The Hamiltonian J of the Josephson junction is introduced as a self-adjoint operator on l2 tensor l2. It is shown that J can also be realized as a self-adjoint operator HS1 on L2(S1) tensor L2(S1), from which a Mathieu operator given by "-d^2/dθ^2 - 2α cos θ" is derived. A fiber decomposition of HS1 with respect to the total particle number is established, and the action on each fiber is analyzed. In the presence of a magnetic field, a phase shift defines the magnetic Josephson junction Hamiltonian HS1(Φ) and the Josephson current IS1(Φ). For a constant magnetic field inducing a local phase shift Φ(x), the corresponding local current IS1(Φ(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally. |
| title | Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2510.00447 |