  We consider physical interpretations of non-trivial boundary conditions of
self-adjoint extensions for one-dimensional Schr\"odinger 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 Ganella et. al., (Journal of Physics A:
Mathematical and Theoretical 42, 465207 (2009)) along with the quantized
magnetic flux. Real physical system which can be modeled by such boundary
conditions is the localized quantazied flux in the Josephson junction of two
superconductors with different effective masses of the elementary excitations.
