Resolvent convergence in norm for Dirac operator with Aharonov–Bohm field

Abstract

We consider the Hamiltonian for relativistic particles moving in the Aharonov–Bohm magnetic field in two dimensions. The field has δ-like singularity at the origin, and the Hamiltonian is not necessarily essentially self-adjoint. The self-adjoint realization requires one parameter family of boundary conditions at the origin. We approximate the point-like field by smooth ones and study the problem of norm resolvent convergence to see which boundary condition is physically reasonable among admissible boundary conditions. We also study the effect of perturbations by scalar potentials. Roughly speaking, the obtained result is that the limit self-adjoint realization is different even for small perturbation of scalar potentials according to the values of magnetic fluxes. It changes at half-integer fluxes. The method is based on the resolvent analysis at low energy on magnetic Schrödinger operators with resonance at zero energy and the resonance plays an important role from a mathematical point of view. However it has been neglected in earlier physical works. The emphasis here is placed on this natural aspect.