Scattering and self-adjoint extensions of the Aharonov-Bohm hamiltonian

TL;DR

This paper analyzes the self-adjoint extensions and scattering properties of the Aharonov-Bohm Hamiltonian for planar solenoids with magnetic fields, characterizing boundary conditions compatible with quantum mechanics.

Contribution

It provides a complete characterization of boundary conditions leading to self-adjoint Hamiltonians and compares scattering under common boundary conditions.

Findings

Characterized all self-adjoint boundary conditions on the solenoid boundary.

Compared scattering behavior for Dirichlet, Neumann, and Robin boundary conditions.

Established the mathematical framework for quantum scattering in magnetic flux scenarios.

Abstract

We consider the hamiltonian operator associated with planar sec- tions of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior. First, in the Sobolev space $\mathcal H^2$, we characterize all generalized boundary conditions on the solenoid bor- der compatible with quantum mechanics, i.e., the boundary conditions so that the corresponding hamiltonian operators are self-adjoint. Then we study and compare the scattering of the most usual boundary con- ditions, that is, Dirichlet, Neumann and Robin.