Lower bounds for resonance counting functions for obstacle scattering in even dimensions

In even dimensional Euclidean scattering, the resonances lie on the logarithmic cover of the complex plane. This paper studies resonances for obstacle scattering in ${\Bbb R}^d$ with Dirichlet, Neumann, or admissible Robin boundary conditions, when $d$ is even. Set $n_m(r)$ to be the number of resonances with norm at most $r$ and argument between $m\pi$ and $(m+1)\pi$. Then $\lim\sup_{r\rightarrow\infty}{\log n_m(r)\over\log r}=d$ if $m\in{\Bbb Z}\setminus\{0\}$.