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On scattering theory for field equations in the Kerr metric

Hafner, D (Bordeaux)
Friday 02 September 2005, 10:00-11:00

Seminar Room 1, Newton Institute


We show asymptotic completeness for the massless Dirac field and the non- superradiant modes of the Klein-Gordon field in the Kerr metric.

In the first part we treat massless Dirac fields. We introduce a new Newman- Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are reinterpreted to provide a solution to the Goursat problem on the Penrose compactified exterior.

In the second part we treat Klein-Gordon fields. We start with an abstract Hilbert space result. From a Mourre estimate for a positive selfadjoint oparator one can deduce a Mourre estimate for its square root. Using this result and the techniques explained in the first part of the talk, we can establish an asymptotic completeness result for the non-superradiant modes of the Klein-Gordon field. Because of the mass of the field the wave operators have to be Dollard modified at infinity.


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