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In this paper we study for a given azimuthal quantum number $\kappa$ theeigenvalues of the Chandrasekhar-Page angular equation with respect to theparameters $\mu:=am$ and $\nu:=a\omega$, where $a$ is the angular momentum perunit 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 thispurpose, a self-adjoint holomorphic operator family $A(\kappa;\mu,\nu)$associated to this eigenvalue problem is considered. At first we prove that forfixed $|\kappa|\geq{1/2}$ the spectrum of $A(\kappa;\mu,\nu)$ is discrete andthat its eigenvalues depend analytically on $(\mu,\nu)\in\C^2$. Moreover, itwill be shown that the eigenvalues satisfy a first order partial differentialequation with respect to $\mu$ and $\nu$, whose characteristic equations can bereduced to a Painleve III equation. In addition, we derive a power seriesexpansion for the eigenvalues in terms of $\nu-\mu$ and $\nu+\mu$, and we givea recurrence relation for their coefficients. Further, it will be proved thatfor fixed $(\mu,\nu)\in\C^2$ the eigenvalues of $A(\kappa;\mu,\nu)$ are thezeros of a holomorphic function $\Theta$ which is defined by a relativelysimple limit formula. Finally, we discuss the problem if there exists a closedexpression for the eigenvalues of the Chandrasekhar-Page angular equation.Comment: 29 page

On the eigenvalues of the Chandrasekhar–Page angular equation