Quantized Poincarè maps in chaotic scattering

Quantized Poincarè maps in chaotic scattering

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Stephane Nonnenmacher, IAS
Fine Hall 110

I will sketch a recent approach to study the resonance spectrum of scattering Schrödinger operators, in cases where the trapped set of the corresponding classical dynamics (near some positive energy) is a fractal chaotic repeller. In that situation, we are interested in the distribution of resonances in the vicinity of the real axis, in the semiclassical limit. Our approach tends to mimic the Poincarè section method used to study the corresponding classical flows. Namely, we show that resonances can be defined through an implicit equation involving an appropriately defined quantized Poincarè map. The subsequent study of this "quantum map" allows to recover known properties of the resonance spectrum (fractal bounds on the number of resonances in strips, spectral gap), and hopefully more.This is joint work with J. Sjoestrand and M. Zworski.