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On the eigenvalues of the Chandrasekhar-Page angular equation

Feb, 2004
29 pages
Published in:
  • J.Math.Phys. 46 (2005) 012504
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Abstract:
In this paper we study for a given azimuthal quantum number κ\kappa the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters μ:=am\mu:=am and ν:=aω\nu:=a\omega, where aa is the angular momentum per unit mass of a black hole, mm is the rest mass of the Dirac particle and ω\omega is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family A(κ:μ,ν)A(\kappa:\mu,\nu) associated to this eigenvalue problem is considered. At first we prove that for fixed ∣κ∣≥1/2|\kappa|\geq{1/2} the spectrum of A(κ:μ,ν)A(\kappa:\mu,\nu) is discrete and that its eigenvalues depend analytically on (\mu,\nu)\in\C^2. Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to μ\mu and ν\nu, whose characteristic equations can be reduced to a Painleve III equation. In addition, we derive a power series expansion for the eigenvalues in terms of ν−μ\nu-\mu and ν+μ\nu+\mu, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed (\mu,\nu)\in\C^2 the eigenvalues of A(κ:μ,ν)A(\kappa:\mu,\nu) are the zeros of a holomorphic function Θ\Theta 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.