Quantum transfer operators and quantum scattering

These notes describe a new method to investigate the spectral properties if quantum scattering Hamiltonians, developed in collaboration with J. Sjöstrand and M.Zworski. This method consists in constructing a family of "quantized transfer operators" $\{M(z,h)\}$ associated with a classical Poincaré section near some fixed classical energy E. These operators are finite dimensional, and have the structure of "open quantum maps". In the semiclassical limit, the family $\{M(z,h)\}$ encode the quantum dynamics near the energy E. In particular, the quantum resonances of the form $E+z$, for $z=O(h)$, are obtained as the roots of $\det(1-M(z,h))=0$.

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