Comments on “Differential cross section for Aharonov-Bohm effect with nonstandard boundary conditions”

We show that the violation of rotational symmetry for differential cross section for Aharonov-Bohm effect with nonstandard boundary conditions has been known for some time. Moreover, the results were applied to discuss the Hall effect and persistent currents of fermions in a plane pierced by a flux tube. PACS: 03.65.Bz Nonrelativistic scattering theory PACS: 72.20.My Galvanomagnetic and other magnetotransport effects PACS: 72.20.Dp General theory of electronic transport, scattering mechanisms (Europhys. Lett. 47, 273-274 (1999)) In their recent paper [1], Šťov́ıček and Váňa discuss the differential cross section dσ/dθ for the AharonovBohm effect with nonstandard boundary conditions. The latter mean in general that a wave function does not vanish at the position of a singular flux tube. Their main result is that, compared with the standard case, two new features occur: a violation of rotational symmetry and a more significant backward scattering. They discuss the violation of rotational symmetry in the rotationally invariant case (the case w = 0 in [1]) and for a new class of self-adjoint extensions which do not commute with the angular momentum (see also [2]). In the latter case, the scattering cross section depends on the incident direction θ0 and the outgoing direction θ separately and is not invariant under θ0, θ → θ0 + γ, θ + γ. In the rotationally invariant case, the scattering cross section depends on the scattering angle φ = θ − θ0, nevertheless, dσ(φ)/dθ 6= dσ(−φ)/dθ in general. We should like to point out that the discussion of the scattering in the presence of an Aharonov-Bohm potential with the nonstandard boundary conditions has a much longer history (see, for example, [3]) than references in [1] tend to indicate. Also the violation of rotational symmetry for the differential cross section in the rotationally invariant case has been pointed out some time ago [4]. In Ref. [4], various physical quantities were calculated for a two-parameter family of self-adjoint extensions which respect the rotational invariance (i.e., no coupling between different angular momentum channels), including the local density of states, persistent currents for both spinless and spin one-half fermions in a plane pierced by a flux tube (see also [5]), the second virial coefficient of interacting anyons, etc., and the number of bound states for a flux tube of nonzero radius. Moreover, in contrast to the Šťov́ıček and Váňa paper [1], the results were applied to discuss the Hall effect in the presence of magnetic vortices [4]. Let e, m, E be as in [1] the charge, the mass, and the energy of the scattered particle and let us concentrate on the rotationally invariant case. Let α be the total flux through the flux tube in units of the flux quantum Φ0 = hc/|e|. Let us write α = n+ η, where n is an integer and η is the nonintegral part of α, 0 ≤ η < 1. Under certain conditions [4], one ends up with a bound state in either one or in both channels l = −n and −n− 1 [4]. If El is the corresponding bound state energy, the conventional phase shift δ l receives an additional contribution △l(E) and the resulting phase shift is δl(E) = δ 0 l +△l(E) = 1 2 π(|l| − |l + α|) +△l(E), (1) where δ l corresponds to the conventional Aharonov-Bohm scenario and △l = arctan ( sin(|l + α|π) cos(|l + α|π) −A l )