On the eigenvalues of the Chandrasekhar-Page angular equation

In this paper we study for a given azimuthal quantum number κ the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters μ:=am and ν:=aω, where a is the angular momentum per unit mass of a black hole, m is the rest mass of the Dirac particle and ω is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family A(κ; μ, ν) associated to this eigenvalue problem is considered. At first we prove that for fixed κ ∈ R\(-1/2, 1/2) the spectrum of A(κ; μ., ν) is discrete and that its eigenvalues depend analytically on (μ, ν) ∈ C ². Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to μ and ν, whose characteristic equations can be reduced to a Painlevé III equation. In addition, we derive a power series expansion for the eigenvalues in terms of ν-μ and ν+μ, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed (μ, ν) ∈ C ² the eigenvalues of A(κ; μ, ν) are the zeros of a holomorphic function Θ which is defined by a relatively simple limit formula. Finally, we discuss the problem if there exists a closed expression for the eigenvalues of the Chandrasekhar-Page angular equation.