Self-adjoint realizations of Aharonov-Bohm Hamiltonians: classical results and recent advances

The Aharonov-Bohm effect relates to the phase shift experienced by non-relativistic charged quantum particles interacting with magnetic fields confined inside ideal solenoids. The prototypical model is described by a Schrödinger operator with a singular vector potential, yielding a finite magnetic flux concentrated along a line. In this seminar we first review classical results, using Von-Neumann theory and resolvent techniques to characterize all admissible one-body Hamiltonians as self-adjoint realizations of the said Schrödinger operator. Next, we discuss recent advances usingquadratic form methods to treat related models, including magnetic perturbations and multiple fluxes configurations. We also mention connections to anyonic particles systems. Based on joint works with Michele Correggi (Politecnico di Milano).