By Batic D., Schmid H., Winklmeier M.
During this paper we learn for a given azimuthal quantum quantity okay the eigenvalues ofthe Chandrasekhar-Page angular equation with recognize to the parameters mªamand nªav, the place a is the angular momentum consistent with unit mass of a black gap, m isthe leisure mass of the Dirac particle and v is the power of the particle (as measuredat infinity). For this objective, a self-adjoint holomorphic operator kin Ask ;m ,ndassociated to this eigenvalue challenge is taken into account. in the beginning we turn out that for fixedkPR\ s−12 , 12 d the spectrum of Ask ;m ,nd is discrete and that its eigenvalues dependanalytically on sm ,ndPC2. additionally, will probably be proven that the eigenvalues satisfya first order partial differential equation with appreciate to m and n, whose characteristicequations could be lowered to a Painlevé III equation. additionally, we derive apower sequence enlargement for the eigenvalues by way of n −m and n +m, and we givea recurrence relation for his or her coefficients. additional, will probably be proved that for fixedsm ,ndPC2 the eigenvalues of Ask ;m ,nd are the zeros of a holomorphic functionality Qwhich is outlined through a comparatively uncomplicated restrict formulation. eventually, we speak about the problemif there exists a closed expression for the eigenvalues of the Chandrasekhar-Page angular equation.
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Then each function = 0j , j = 1 − k , . . , k − 1, satisfies the partial differential equation (29) in S. Proof: Let j ͕1 − k , . . , k − 1͖ be fixed. The monodromy eigenvalues of A͑ ; , ͒ are exactly the zeros of the polynomial P͑ ; · ; , ͒, and since all zeros of P͑ ; · ; , ͒ are simple, the implicit function theorem implies that 0j ͑ ; , ͒ depends analytically on ͑ , ͒ in S. In order to show that the function = 0j satisfies the PDE (29), we make use of the unique continuation property of analytical functions.