On the eigenvalues of the Chandrasekhar-Page angular equation

Batic D., Schmid H., Winklmeier M.

In this paper we study for a given azimuthal quantum number k the eigenvalues ofthe Chandrasekhar-Page angular equation with respect to the parameters mªamand nªav, where a is the angular momentum per unit mass of a black hole, m isthe rest mass of the Dirac particle and v is the energy of the particle (as measuredat infinity). For this purpose, a self-adjoint holomorphic operator family Ask ;m ,ndassociated to this eigenvalue problem is considered. At first we prove that for fixedkPR s−12 , 12 d the spectrum of Ask ;m ,nd is discrete and that its eigenvalues dependanalytically on sm ,ndPC2. Moreover, it will be shown that the eigenvalues satisfya first order partial differential equation with respect to m and n, whose characteristicequations can be reduced to a Painlevé III equation. In addition, we derive apower series expansion for the eigenvalues in terms of n −m and n +m, and we givea recurrence relation for their coefficients. Further, it will be proved that for fixedsm ,ndPC2 the eigenvalues of Ask ;m ,nd are the zeros of a holomorphic function Qwhich is defined by a relatively simple limit formula. Finally, we discuss the problemif there exists a closed expression for the eigenvalues of the Chandrasekhar-Page angular equation.