One-dimensional tunnel-junction formula for the schrödinger particle

We handle all the self-adjoint extensions of the minimal Schrödinger operator for the nonrelativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the nonrelativistic electron. We find concrete tunnel-junction formulae for the nonrelativistic electron passing through the junction, which reveals the phase factor caused by the tunneling through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.