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Physical realizations of non-standard singular self-adjoint extensions forone-dimensional Schr\"odinger operator in terms of the mass-jump areconsidered. It is shown that corresponding boundary conditions can be realizedfor the Hamiltonian with the position-dependent effective mass in twoqualitatively different profiles of the effective mass inhomogeneity: themass-jump and the mass-bump. The existence of quantized magnetic flux in a caseof the mass-jump is proven by explicit demonstration of the Zeeman-likesplitting for states with the opposite projections of angular momentum.

Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schr\"odinger operator in one dimension