Catálogo de publicaciones - libros

We summarize our work in collaboration with Maciej Zworski [16], on the semiclassical density of resonances for a quantum open system, in the case when the associated classical dynamics is uniformly hyperbolic, and the set of trajectories is a . The system we consider is not a Hamiltonian flow, but rather a “symplectic map with a hole” on a compact phase space (the 2-torus). Such a map can be considered as a model for the Poincaré section associated with a scattering Hamiltonian on ℝ, at some positive energy; the “hole” represents the points which never return to the Poincaré section, that is, which are scattered to infinity. We then quantize this open map, obtaining a sequence of subunitary operators, the eigenvalues of which are interpreted as resonances.

In this survey article based on the papers [7, 10], and [8], we consider the 3D Schröxzdinger operator with constant magnetic field of intensity > 0, perturbed by an electric potential which decays fast enough at infinity, and discuss various asymptotic properties of the corresponding spectral shift function.

We report on some recent work with M. R. Douglas and B. Shiffman on vacuum statistics for flux compactifications in string/M theory.