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

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.