Quasi-relativistic quantum particle on a circle with complex magnetic field

We consider a quasi-relativistic particle on a circle with complex magnetic field. First, the momentum of the particle with real and complex magnetic vector potential is investigated. Thence, we obtain the magnetic quasi-relativistic operator with real potential and find its spectrum. Since operators with complex potential are non-self-adjoint, we derive the condition under which the magnetic quasi-relativistic operator can be obtained as a square root of an m-accretive operator. Furthermore, its properties are investigated, and we attempt to construct such operator. Finally, we derive the condition under which the operator is quasi-Hermitian.