Bound State Solutions of the Dirac Equation in the Extreme Kerr Geometry

In this paper we consider bound state solutions, i.e., normalizable time-periodic solutions of the Dirac equation in the exterior region of an extreme Kerr black hole with mass $M$ and angular momentum $J$. It is shown that for each azimuthal quantum number $k$ and for particular values of $J$ the Dirac equation has a bound state solution, and that the energy of this Dirac particle is uniquely determined by $\omega = -\frac{kM}{2J}$. Moreover, we prove a necessary and sufficient condition for the existence of bound states in the extreme Kerr-Newman geometry, and we give an explicit expression for the radial eigenfunctions in terms of Laguerre polynomials.