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.

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. *

tion for elliptic Dirichlet problems in exterior domains, where the radiation conditions at infinity and the near-singularity properties of fundamental solutions were made use of in an essential manner. But recently Agmon j^l] proposed a new method without explicit recourse to these tools and involving a priori estimates, which is valid for elliptic operators in the

We study the spectrum of spherically symmetric Dirac operators m three-dimensional space with potentials tending to infinity at infinity under weak regularity assumptions. We prove that purely absolutely continuous spectrum covers the whole real line if the potential dominates the mass, or scalar potential, term. In the situation where the potential and the scalar potential are identical, the positive part of the spectrum is purely discrete : we show that the negative half-line is filled with purely absolutely continuous spectrum m this case. § 1. IntroductionIn a recent paper [20] the spectral properties of the three-dimensional Dirac operator (where p -~i F, i 2 -~~ 1, l n is the n Xn unit matrix, and ao -j8, a\, a^ 0.3 are Hermitian 4x4 matrices satisfying the anti-commutation relations aja k +a k aj = 25 Jk (/, *e{0, 1, 2, 3})) were studied under the condition that the real-valued coefficient function m tends to °° (or -°°) as \x -*°°. For constant m, H is the Hamilton operator describing a relativistic quantum mechanical particle of mass m moving in an external force field of (real-valued) potential q. As a non-constant function, m can also take the role of a so-called scalar potential, which has been discussed in the physical literature as a model of quark confinement (cf. the references in [20], Thaller [14] p. 305).

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