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

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.