Quantum transfer operators and quantum scattering

Abstract

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 $\det (1-M(z,h))=0$.

18 pages, 3 figures

References in corpus (1)

• Quantum-to-classical crossover of quasi-bound states in open quantum systems