Schrödinger and Dirac Operators with Aharonov–Bohm and Magnetic-Solenoid Fields

Abstract

We construct all self-adjoint Schrödinger and Dirac operators (Hamiltonians) with both the pure Aharonov-Bohm (AB) field and the so-called magnetic-solenoid field (a collinear superposition of the AB field and a constant magnetic field). We perform a spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulae. In constructing the Hamiltonians and performing their spectral analysis, we follow, respectively, the von Neumann theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals.

Bound states of massive fermions in Aharonov–Bohm-like fields

“…The additional term must influence the behavior of the solutions at the origin and it can be taken into account by means of boundary conditions at the point r = 0. In the nonrelativistic AC problem the boundary condition (13) can be given by [16] (see also [44,45])…”

“…The special case γ = 0 can be of some interest (the analogous case was considered in [17,44] for the nonrelativistic AB problem in 2+1 dimensions). One can show that for |ξ | = ∞ the energy spectrum is continuous and nonnegative and also that for −∞ < ξ < 0 there exists (in addition to the continuous part of the spectrum) one negative level…”