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

## TL;DR

This paper analyzes singular self-adjoint extensions of the Schrödinger operator for spin-1/2 particles in one dimension, identifying boundary conditions that include spin-flip interactions and relate to Rashba spin-momentum coupling.

## Contribution

It introduces new boundary conditions for point-like interactions that incorporate spin-flip mechanisms and connects these to Rashba Hamiltonians in a non-relativistic limit.

## Key findings

- Boundary conditions with spin-flip interactions are derived.
- Point-like Rashba interactions are modeled as singular self-adjoint extensions.
- Transmissivity depends on the Rashba coupling strength.

## Abstract

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