Dirac fermions in a magnetic-solenoid field

We consider the Dirac equation with a magnetic-solenoid field (the superposition of a solenoid field and a collinear uniform magnetic field). Using von Neumann’s theory of the selfadjoint extensions of symmetric operators, we construct selfadjoint Dirac Hamiltonians for the case of the Aharonov-Bohm solenoid in 2 + 1 and 3 + 1 dimensions. Each extension of the family is specified by certain asymptotic boundary conditions at the AB solenoid. We find the spectrum and eigenfunctions for each value of the extension parameter. We also consider the regularized case of a finite-radius solenoid and study the dependence of the eigenfunctions on the behavior of the magnetic field inside the solenoid. In addition, we consider the regularization problem in the spinless case as well. The zero-radius limit yields a concrete selfadjoint Hamiltonian for the case of the magnetic-solenoid field with the Aharonov-Bohm solenoid. In addition, by the example of the radial Dirac Hamiltonian with the magnetic-solenoid field, we present a simplified alternative method for the selfadjoint extensions of a wide class of singular differential operators.