  This paper proves sharp lower bounds on a resonance counting function for
obstacle scattering in even-dimensional Euclidean space without a need for
trapping assumptions. Similar lower bounds are proved for some other compactly
supported perturbations of the Laplacian on even-dimensional Euclidean space,
for example, for the Laplacian for certain metric perturbations. The proof uses
a Poisson formula for resonances, complementary to one proved by Zworski in
even dimensions.
