# Singular spin-flip interactions for the 1D Schr\"{o}dinger operator

## TL;DR

This paper studies singular boundary conditions for 1D Schrödinger operators with two-component wave functions, identifying a specific subset compatible with spin interpretation, and proposes physical realizations involving spin-momentum interactions.

## Contribution

It characterizes the boundary conditions compatible with spin interpretation for 1D Schrödinger operators and introduces physical models for spin-momentum interactions.

## Key findings

- Only a 2-parameter subset of boundary conditions supports spin interpretation.
- Identifies point-like spin-momentum (Rashba) interactions within this subset.
- Proposes physical realizations involving electrical field inhomogeneity coupling.

## Abstract

We consider singular self-adjoint extensions for one-dimensional Schr\"{o}dinger operator for two-component wave function within the framework of the distribution theory for discontinuous test functions \cite{funcan_deltadistr_kurasov_jmathan1996}. We show that among $\mathds{C}^{4}$-parameter set of boundary conditions with state mixing there is only $\mathds{R}^2$-parameter subset compatible with the spin interpretation of the two-component structure of the wave function. For the spin interpretation of such wave function they can be identified as the point-like spin-momentum (Rashba) interactions. We suggest their physical realizations based on the regularized form of the Hamiltonian which couples the electrical field inhomogeneity to spin.