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We consider singular self-adjoint extensions for the Schr\"{o}dinger operatorof spin-$1/2$ particle in one dimension. The corresponding boundary conditionsat a singular point are obtained. There are boundary conditions with thespin-flip mechanism, i.e. for these point-like interactions the spin operatordoes not commute with the Hamiltonian. One of these extensions is the analog ofzero-range $\delta$-potential. The other one is the analog of so called$\delta^{(1)}$-interaction. We show that in physical terms such contactinteractions can be identified as the point-like analogues of RashbaHamiltonian (spin-momentum coupling) due to material heterogeneity of differenttypes. The dependence of the transmissivity of some simple devices on thestrength of the Rashba coupling parameter is discussed. Additionally, we showhow these boundary conditions can be obtained in the non-relativistic limit ofDirac Hamiltonian.

Point-like Rashba interactions as singular self-adjoint extensions of the Schr\"{o}dinger operator in one dimension