  Physical realizations of non-standard singular self-adjoint extensions for
one-dimensional Schr\"odinger operator in terms of the mass-jump are
considered. It is shown that corresponding boundary conditions can be realized
for the Hamiltonian with the position-dependent effective mass in two
qualitatively different profiles of the effective mass inhomogeneity: the
mass-jump and the mass-bump. The existence of quantized magnetic flux in a case
of the mass-jump is proven by explicit demonstration of the Zeeman-like
splitting for states with the opposite projections of angular momentum.
