Single-particle density of states for the Aharonov-Bohm potential and the instability of matter with anomalous magnetic moment in 2+1 dimensions

In the nonrelativistic case I found that whenever the relation mc/e < X(α, gm) is satisfied, where α is a flux, gm is magnetic moment, and X(α, gm) is some function that is nonzero only for gm > 2 (note that gm = 2.00232 for the electron), then the matter is unstable against formation of the flux α. The result persists down to gm = 2 provided the Aharonov-Bohm potential is supplemented with a short range attractive potential. I also show that whenever a bound state is present in the spectrum it is always accompanied by a resonance with the energy proportional to the absolute value of the binding energy. In the relativistic case one again finds the resonance when the bound state is present but the instability disappear as long as the minimal coupling is considered. For the Klein-Gordon equation with the Pauli coupling which exists in (2+1) dimensions without any reference to a spin the matter is again unstable for gm > 2. The results are obtained by calculating the change of the density of states induced by the Aharonov-Bohm potential. The KreinFriedel formula for this long-ranged potential is shown to be valid when supplemented with zeta function regularization. PACS : 03.65.Bz, 03-70.+k, 03-80.+r, 05.30.Fk e-mail address : [email protected] Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS In this letter a change of the density of states in the whole space is calculated for the Schrödinger, the Klein-Gordon, and the Dirac equations with the Aharonov-Bohm (AB) potential Ar = 0, Aφ = Φ/2πr. This enables to discuss the stability properties of matter against the spontaneous creation of a magnetic field. In the relativistic case and for the normal magnetic moment we reconfirm previous result, known as the diamagnetic inequality [1] that the matter is stable. The latter was proven under the assumption of minimal coupling which implicitely assumes the normal magnetic moment. However, in the nonrelativistic case a window exists for the magnetic moment gm > 2 in which the inequality is violated leading to the instability of matter against a magnetic field formation. The reason is the formation of bound states which decouple from the Hilbert space by taking away negative energy. Note that one has the unitary equivalence between a spin 1/2 charged particle in a 2D magnetic field and a spin 1/2 neutral particle with an anomalous magnetic moment in a 2D electric field [2]. One also has a formal similarity between the scattering of electron in the AB potential and in the spacetime of a gravitational vortex in 2 + 1 dimensions [3]. Moreover, one encounters the AB potential (of non-magnetic origin) in the cosmic-string scenarious [4] and our results apply to the this cases as well. 1. The Schrödinger equation.By using the separation of variables, assuming e = −|e|, the total Hamiltonian is written as a direct sum of channel radial Hamiltonians Hl [5, 6] Hl = − d dr2 − 1 r d dr + ν r2 , (1) where ν = |l + α|, α being the total flux Φ in the units of the flux quantum Φo = hc/|e|, with the spectrum given by ψl(r, φ) = J|l+α|(kr)e , (2) with k = √ 2mE/h̄. The eigenfunction expansion for the Green function in the polar coordinates x = (rx, φx) is G(x,y, E) = m πh̄ ∫ ∞ o kdk q2 − k2 ∞