Computation of unitary group for the Rashba spin-orbit coupled operator, with application to point-interactions

TL;DR

This paper derives an explicit formula for the unitary group of the 3D Rashba spin-orbit operator, computes the Green function for two particles, and shows spin-dependent point-interactions are supersingular, lacking self-adjoint extensions.

Contribution

It provides a new explicit formula for the unitary group of the Rashba operator and analyzes the nature of spin-dependent point-interactions as supersingular perturbations.

Findings

Explicit formula for the unitary group of the Rashba operator

Green function for the two-particle operator derived

Spin-dependent point-interactions are supersingular, with no self-adjoint extension

Abstract

We compute an explicit formula for the one-parameter unitary group of the single-particle Rashba spin-orbit coupled operator in dimension three. As an application, we derive the formula for the Green function for the two-particle operator, and then prove that the spin-dependent point-interaction is of class $\mathcal{H}_{-4}$. The latter is thus the example of a supersingular perturbation for which no self-adjoint operator can be constructed.