Maximal order of growth for the resonance counting functions for generic potentials in even dimensions

We prove that the resonance counting functions for Schr"odinger operators $H_V=-\mathrm{\Delta }+V$ on $L^2({\mathbb{R}}^d)$, for $d\ge 2$ {\it even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet ${\mathrm{\Lambda }}_m$, $m\in \mathbb{Z}\mathrm{\backslash }\{0\}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m\in \mathbb{Z}\mathrm{\backslash }\{0\}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet ${\mathrm{\Lambda }}_0$ determine the poles on ${\mathrm{\Lambda }}_m$. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by $C_m{r}^d$ on each sheet ${\mathrm{\Lambda }}_m$, $m\in \mathbb{Z}\mathrm{\backslash }\{0\}$.