Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension

Highlights: • Self-adjoint extensions in terms of variable mass are considered. • The local $\delta$'-potential corresponds to the "mass-jump". • The non-local $\delta$'-potential correspond to the "mass-bump". • A quantized magnetic flux is present in the "mass-jump" case. • The "mass-bump" case is of pure potential, i.e. "electrostatic" nature. - Abstract: Physical realizations of non-standard singular self-adjoint extensions for one-dimensional Schrödinger 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.