On the eigenvalues of the Chandrasekhar–Page angular equation

Abstract

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.

Scattering for massive Dirac fields on the Kerr metric

“…For further properties on the angular eigenfunctions S kj ω we refer to Finster et al (2000), and to Batic et al (2005). As shown in Lemma 6.1-2 in Batic and, Schmid (2006) in the limit u → −∞ the radial solutions behave for u ≤ u 1 < 0 as follows…”

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

“…From [23], we already know that the second inequality in (49) is satisfied for the eigenfunctions associated with the eigenvalues of the angular problem. Therefore, in the next two sections, we will investigate whether or not there exist solutions of the radial problem such that the first inequality in (49) is satisfied.…”

“…The angular system has been thoroughly studied in [23], where eigenvalues and associated eigenfunctions have been computed. Note that after separation of variables of the Dirac equation, ω ∈ R will be an energy eigenvalue of (42), if there exists some λ ∈ R and non-trivial solutions…”

“…The spectrum of the angular problem is purely discrete and the eigenvalues are given as follows [23]:…”

The generalized Heun equation in QFT in curved spacetimes

“…Let us briefly recall that in the case a = 0 the components S ± of the angular eigenfunctions can be expressed in terms of spin-weighted spherical harmonics [9,10] whereas for a = 0 they satisfy a generalized Heun equation [11]. We show now how to transform the radial equations for the components R ± of the radial functions in such a way that they are reduced to a GHE.…”