Properties of the Dirac–Weyl operator with a strongly singular gauge potentiala)

Considered is a quantum system of a charged particle moving in the plane R^2 under the influence of a perpendicular magnetic field concentrated on some fixed isolated points in R^2. Such a magnetic field is represented as a finite linear combination of the two-dimensional Dirac delta distributions and their derivatives, so that the gauge potential of the magnetic field also may be strongly singular at those isolated points. Properties of the Dirac–Weyl operator with such a singular gauge potential are investigated. It is seen that some of them depend on whether the magnetic flux is locally quantized or not. Particular attention is paid to the zero-energy state. For each of the self-adjoint realizations of the Dirac–Weyl operator, the number of the zero-energy states is computed. It is shown that, in the present case, a theorem of Aharonov and Casher [Phys. Rev. A 19, 2461 (1979)], which relates the total magnetic flux to the number of zero-energy states, does not hold. It is also proven that the spectrum of every self-adjoint extension of the minimal Dirac–Weyl operator is equal to R