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 $|κ|\geq{1/2}$ the spectrum of $A(κ;μ,ν)$ is discrete and that its eigenvalues depend analytically on $(μ,ν)\in\C^2$. 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 Painleve 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 $(μ,ν)\in\C^2$ 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.