Spectrum of a family of spin-orbit coupled Hamiltonians with singular perturbation

TL;DR

This paper rigorously analyzes the spectral properties of a three-dimensional Rashba spin-orbit coupled Hamiltonian with singular perturbations, revealing how spin-orbit interaction influences eigenvalues and spectral structure.

Contribution

It introduces a comprehensive method to construct self-adjoint extensions of the Rashba Hamiltonian with singular, spin-dependent contact interactions in three dimensions.

Findings

Eigenvalue asymptotics for weak spin-orbit coupling

Conditions for eigenvalues above the spectral threshold

Detailed spectral analysis considering spin-orbit effects

Abstract

The present study is the first such attempt to examine rigorously and comprehensively the spectral properties of a three-dimensional ultracold atom when both the spin-orbit interaction and the Zeeman field are taken into account. The model operator is the Rashba spin-orbit coupled operator in dimension three. The self-adjoint extensions are constructed using the theory of singular perturbations, where regularized rank two perturbations describe spin-dependent contact interactions. The spectrum of self-adjoint extensions is investigated in detail laying emphasis on the effects due to spin-orbit coupling. When the spin-orbit-coupling strength is small enough, the asymptotics of eigenvalues is obtained. The conditions for the existence of eigenvalues above the threshold are discussed in particular.