Abstract: Quantum resonances describe states which have initial non-zero energies but positive rates of decay. For instance most of chemical reactions proceed via metastable states corresponding to (often many) quantum resonances. In pure maths, the zeros of the Riemann zeta functions are the quantum resonances for the Laplacian on the modular surface. More generally, they coincide with the zeros of the Selberg zeta function of hyperbolic quotients.

The talk will describe the general motivation and mathematical modeling of quantum resonances, in the physical context and in the context of hyperbolic quotients. It will then focus on be the recent theoretical and numerical advances in the understanding of fractal Weyl laws for resonances of classically chaotic systems. These are power laws for the density of quantum states in the semiclassical or high energy limits with the power given by the dimension of the "trapped set" of the classical hyperbolic dynamics.