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

TL;DR

This paper explores the mathematical properties of the Aharonov-Bohm Schrödinger operator, including self-adjointness, spectral, and scattering characteristics, using multiple analytical methods to compare different realizations.

Contribution

It provides a comprehensive characterization of all self-adjoint realizations of the Aharonov-Bohm Hamiltonian through four distinct mathematical approaches.

Findings

Classified all admissible self-adjoint extensions.

Analyzed the asymptotic behavior near the flux singularity.

Described spectral and scattering properties of the Hamiltonians.

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.