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arXiv · 2510.00447

Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns

Toshiyuki Fujii·Fumio Hiroshima·Satoshi Tanda
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Abstract

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/d{\theta}^2 - 2{\alpha} cos {\theta}" 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({\Phi}) and the Josephson current IS1({\Phi}). For a constant magnetic field inducing a local phase shift {\Phi}(x), the corresponding local current IS1({\Phi}(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally.

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Toshiyuki Fujii, Fumio Hiroshima, Satoshi Tanda. 2025-10-01. Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns. https://arxiv.org/abs/2510.00447

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