Abstract
In this paper we study for a given azimuthal quantum number K the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters mu:= am and v:= aomega, 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 ...
Abstract
In this paper we study for a given azimuthal quantum number K the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters mu:= am and v:= aomega, 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 (K; mu, V) associated to this eigenvalue problem is considered. At first we prove that for fixed K is an element of R
(-(1)/(2),(1)/(2)) the spectrum of A (K: mu, v) is discrete and that its eigenvalues depend analytically on (mu, V) is an element of C-2. Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to I-L and v, 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 v - mu and v + A, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed (mu, v) is an element of C-2 the eigenvalues of A (K; mu, v) 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. (C) 2005 American Institute of Physics.
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