Lower bounds for resonance counting functions for Schr\"odinger operators with fixed sign potentials in even dimensions

If the dimension $d$ is even, the resonances of the Schr\"odinger operator $-\Delta +V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show that for fixed sign potentials $V$ and for nonzero integers $m$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth.