On the Eigenvalues of the Chandrasekhar-Page Angular Equation

Abstract

In this paper we study for a given azimuthal quantum number $\kappa$ the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters $\mu :=am$ and $\nu :=a\omega$, where $a$ is the angular momentum per unit mass of a black hole, $m$ is the rest mass of the Dirac particle and $\omega$ is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family $A(\kappa ;\mu ,\nu )$ associated to this eigenvalue problem is considered. At first we prove that for fixed $|\kappa |\ge 1/2$ the spectrum of $A(\kappa ;\mu ,\nu )$ 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 $\mu$ and $\nu$, 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 $\nu -\mu$ and $\nu +\mu$, 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 $A(\kappa ;\mu ,\nu )$ are the zeros of a holomorphic function $\Theta$ 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.

29 pages