March 9 – 13, 2015
Mathematically, quantum resonances appear as discrete eigenvalues of the quantization Ĥ of a Hamiltonian acting on a Hilbert space H on which Ĥ is not self-adjoint, or equivalently as poles of the resolvent (Ĥ − z)−1 of Ĥ on H. Quantum resonances arising from geometric situations –such as from the Laplacian on (asymptotically) hyperbolic manifolds and (locally) symmetric spaces– are linked to interesting geometrical, representation theoretic and analytic objects. In several physical contexts, they provide information about the time evolution of propagation of states with certain regularity properties or decay at infinity. They also come naturally in trace formulas relating geometric invariants and spectral invariants (like Selberg trace formula). They are therefore important data relating classical and quantum dynamics. The questions are their existence, distribution (counting functions, spectral gap), stability under perturbations.
Another aspect of resonances is linked to hyperbolic dynamics. Modern microlocal techniques allow to show that the vector field X generating a smooth Anosov flow φt on a compact manifold has a resolvent (X − z)−1 which extends meromorphically in z ∈ C on certain anisotropic Sobolev spaces associated to the unstable/stable foliations. The poles are called classical resonances. Their location (like obtaining a spectral gap) is fundamental to obtain precise description of the long time behaviour of correlations. Classical resonances also appear in trace formulas relating closed orbits of the flow and spectrum of X.
The last years have seen considerable progress in understanding the geometry and analysis of classical and quantum resonances, but many interesting (and difficult) questions in the subject are open. The purpose of this workshop is to gather researchers working on the different aspects of the geometric theory of resonances (spectral geometry, representation theory, harmonic or microlocal analysis, analytic number theory, mathematical physics) to present their latest results, share their points of view and ideas, strengthen or promote interactions.
Scientific & Organizing Committee
Euclidean scattering, resolvent estimates and evolution PDE’s
Riesz transform on manifolds with quadratic curvature decay