Physical structure of point-like interactions for one-dimensional Schrödinger operator and the gauge symmetry

We consider physical interpretations of non-trivial boundary conditions of self-adjoint extensions for one-dimensional Schrödinger operator of free spinless particle. Despite its model and rather abstract character this question is worth of investigation due to application for one-dimensional nanostructures. The main result is the physical interpretation of peculiar self-adjoint extension with discontinuity of both the probability density and the derivative of the wave function. We show that this case differs very much from other three which were considered before and corresponds to the presence of mass-jump in a sense of works of Gadella et al. [Self-adjoint Hamiltonians with a mass jump: general matching conditions, Phys. Lett. A 362 (4) (2007) 265-268; A delta well with a mass jump, J. Phys. A: Math. Theor. 42 (46) (2009) 465207] along with the quantized magnetic flux. Real physical system which can be modeled by such boundary conditions is the localized quantized flux in the Josephson junction of two superconductors with different effective masses of the elementary excitations.