  In this paper, we give a polynomial lower bound for the resonances of
$-\Delta$ perturbed by an obstacle in even-dimensional Euclidean spaces,
$n\geq4$. The proof is based on a Poisson Summation Formula which comes from
the Hadamard factorization theorem in the open upper complex plane. We take
advantage of the singularity of regularized wave trace to give the
pole/resonance counting function over the principal branch of logarithmic plane
a lower bound.
