In this paper we study for a given azimuthal quantum number κ the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters µ := am and ν := aω, where a is the angular momentum per unit mass of a black hole, m is the rest mass of the Dirac particle and ω is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family A(κ; µ, ν) associated to this eigenvalue problem is considered. At first we prove that for fixed κ ∈ R \ (− 1 2 , 1 2 ) the spectrum of A(κ; µ, ν) is discrete and that its eigenvalues depend analytically on (µ, ν) ∈ C 2 . Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to µ and ν, whose characteristic equations can be reduced to a Painlevé III equation. In addition, we derive a power series expansion for the eigenvalues in terms of ν − µ and ν + µ, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed (µ, ν) ∈ C 2 the eigenvalues of A(κ; µ, ν) are the zeros of a holomorphic function Θ 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.

Starting with the whole class of type-D vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero rest-mass fields with spin s = ±2 (gravitational waves), s = ±1 (electromagnetic waves) and s = ±1/2 (neutrinos) is an Heun equation in disguise.

We show that not all quasinormal modes of a massless scalar field in the Schwarzschild metric are encoded in the corresponding characteristic equation expressed by means of a continued fraction. We provide an analytical formula for these new hitherto missing quasinormal modes, and in doing that we also construct a generalization of the Gauss convergence criterion.

We give an alternative proof of the completeness of the Chandrasekhar ansatz for the Dirac equation in the Kerr-Newman metric. Based on this, we derive an integral representation for smooth compactly supported functions which in turn we use to derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the above class of functions. As a by-product, we also obtain the propagator for the Dirac equation in the Minkowski space-time in oblate spheroidal coordinates. * )

We review different methods of generating potentials such that the one-dimensional Schrödinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with previous studies, and complement the subject with new findings. Our main result is to derive new classes of potentials such that the ODSE can be transformed into the Heun equation and its confluent cases. The generalized Heun equation is also considered.