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On scattering theory for field equations in the Kerr metric

Friday 2nd September 2005 - 10:00 to 11:00
INI Seminar Room 1
Session Title: 
Black hole mini-symposium

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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons