Bound state solutions of the Dirac equation in the extreme Kerr geometry

Abstract

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 ω = − 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.

The Problem of Embedded Eigenvalues for the Dirac Equation in the Schwarzschild Black Hole Metric

“…According to [6], on the region r > 2M and on the hypersurfaces t = const, we can introduce the following inner product:…”

“…More precisely, they exploited the conservation and positivity of the Dirac current to show that, because of the matching conditions, the only way in which a bound state solution of the Dirac equation can be normalizable is that each term in the Fourier expansion of the spinor field in time, and the angular variable around the axis of symmetry, be identically zero. The case of the extreme Kerr metric has been treated by [6], where necessary and sufficient conditions for the existence of bound states have been derived. However, since such conditions are expressed in terms of a complicated set of equations and inequalities, the author did not further study the problem of whether or not there exist values of the particle energy satisfying the aforementioned conditions.…”

The Dirac equation in Kerr–Newman–Ads black hole background

“…See e.g. [19,24,12,13,14] for the Kerr-Newman case. The following system of coupled eigenvalue equations have to be satisfied simultaneously in L 2 ((0, π),…”

A variational principle for block operator matrices and its application to the angular part of the Dirac operator in curved spacetime

“…Together with λ − (18) and (19) in the case x ∈ D(S 1 (p(x))). It remains to show (18) in the case x / ∈ D(S 1 (p(x))), i.e., for elements x ∈ D(T * 12 ) such that ( in contradiction to the assumption on x. Moreover, since T 11 is relatively bounded with respect to T * 12 , also the limit lim n→∞ T 11 x n exists.…”