Quadratic forms for Aharonov-Bohm Hamiltonians

We consider a charged quantum particle immersed in an axial magnetic field, comprising a local Aharonov-Bohm singularity and a regular perturbation. Quadratic form techniques are used to characterize different self-adjoint realizations of the reduced two-dimensional Schrödinger operator, including the Friedrichs Hamiltonian and a family of singular perturbations indexed by $2\times 2$ Hermitian matrices. The limit of the Friedrichs Hamiltonian when the Aharonov-Bohm flux parameter goes to zero is discussed in terms of $\mathrm{\Gamma }$ - convergence.