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.

We investigate the existence of time-periodic solutions of the Dirac equation in the Kerr-Newman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable t, and the Chandrasekhar separation ansatz is applied so that the question of existence of a time-periodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no time-periodic solutions in the nonextreme case. Then, it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses ͑m N ͒ NN for which a time-periodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the Kerr-Newman metric.

Abstract. In this article we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of Quantum Field Theory in curved space-times. In particular, we relate the separated radial part of a massive Dirac equation in the Kerr-Newman metric and the static perturbations for the non-extremal ReissnerNordström solution to a GHE.

Second order equations of the formz(t) + A 0 z(t) + Dż(t) = 0 are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix A = 0 I −A 0 −D associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of A in the phase space.

375.0ptWe investigate the local energy decay of solutions of the Dirac equation in the non-extreme Kerr-Newman metric. First, we write the Dirac equation as a Cauchy problem and define the Dirac operator. It is shown that the Dirac operator is selfadjoint in a suitable Hilbert space. With the RAGE theorem, we show that for each particle its energy located in any compact region outside of the event horizon of the Kerr-Newman black hole decays in the time mean. *

Abstract. We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph G with a finite number n of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of G. Special attention is paid to Robin matching conditions with parameter τ ∈ R ∪ {∞}. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on τ , and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for R → ∞, the difference of the number of eigenvalues in the intervals [0, R) and [−R, 0) deviates from some integer κ0, which we call dislocation index, at most by n + 2.Mathematics Subject Classification (2010). 81Q10, 81Q35, 47A10, 47A75.

The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared with numerical values from the literature.

In this paper we investigate the one-dimensional harmonic oscillator with a leftright boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point.The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.