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 $-\mathrm{\Delta }+V$ on ${\mathbb{R}}^d$ with $V$ bounded and compactly supported are points on $\mathrm{\Lambda }$, 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 $\mathrm{\Lambda }$ has maximal order of growth.