Scattering phase asymptotics with fractal remainders

Colin Guillarmou (1) , Semyon Dyatlov (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 Department of Mathematics [Berkeley]

For a Riemannian manifold $(M,g)$ which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate. For Axiom~A geodesic flows, this gives a polynomial improvement over the known remainders. We also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis, and provide similar asymptotics for hyperbolic quotients using the Selberg zeta function.

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Colin Guillarmou, Semyon Dyatlov. Scattering phase asymptotics with fractal remainders. Communications in Mathematical Physics, 2013, 324 (2), pp.425--444. ⟨hal-00701772⟩