The Aharonov-Bohm Hamiltonian: Self-adjointness, Spectral, and Scattering Properties

This chapter provides an introduction and overview on some basic mathematical aspects of the single-flux Aharonov-Bohm Schrödinger 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 Kreı̆n’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 are invariant under rotations and homogeneous of degree − 2 under dilations, like the basic differential operator. The spectral and scattering properties of all the Hamiltonian operators are finally described.