We prove a Weyl upper bound on the number of scattering resonances in strips
for manifolds with Euclidean infinite ends. In contrast with previous results,
we do not make any strong structural assumptions on the geodesic flow on the
trapped set (such as hyperbolicity) and instead use propagation statements up
to the Ehrenfest time. By a similar method we prove a decay statement with high
probability for linear waves with random initial data. The latter statement is
related heuristically to the Weyl upper bound. For geodesic flows with positive
escape rate, we obtain a power improvement over the trivial Weyl bound and
exponential decay up to twice the Ehrenfest time.Comment: 36 pages, 5 figures; minor revision
