The Aharonov-Bohm Hamiltonian: self-adjointness, spectral and scattering properties

Abstract: This work provides an introduction and overview on some basic mathematical aspects of the single-flux Aharonov-Bohm Schr\"odinger operator. The whole family of admissible self-adjoint realizations is characterized by means of four different methods: von Neumann theory, boundary triplets, quadratic forms and Kre{\u\i}n's resolvent formalism. The relation between the different parametrizations thus obtained is explored, comparing the asymptotic behavior of functions in the corresponding operator domains close to the flux singularity. Special attention is devoted to those self-adjoint realizations which preserve the same rotational symmetry and homogeneity under dilations of the basic differential operator. The spectral and scattering properties of all the Hamiltonian operators are finally described.