Hamiltonian self-adjoint extensions for (2+1)-dimensional Dirac particles

Abstract

We study the stationary problem of a charged Dirac particle in (2+1) dimensions in the presence of a uniform magnetic field B and a singular magnetic tube of flux Phi = 2 pi kappa/e. The rotational invariance of this configuration implies that the subspaces of definite angular momentum l+1/2 are invariant under the action of the Hamiltonian H. We show that, for l different from the integer part of kappa, the restriction of H to these subspaces, H_l is essentially self-adjoint, while for l equal to the integer part of kappa, H_l admits a one-parameter family of self-adjoint extensions (SAE). In the later case, the functions in the domain of H_l are singular (but square-integrable) at the origin, their behavior being dictated by the value of the parameter gamma that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of kappa and gamma, as well as its closure.

RevTex, 7 pages, 1 figure (references added - version to appear in Jour. Phys. A: Math. and Gen.)

References in corpus (2)

• Self-adjointness of the two-dimensional massless Dirac Hamiltonian and vacuum polarization effects in the background of a singular magnetic vortex
• Exact Solution of the one-impurity quantum Hall problem

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