Maximal order of growth for the resonance counting functions for generic potentials in even dimensions T. ChristiansenPeter Hislop 81U0532U0535P25resonancescounting functionSchroedinger operators We prove that the resonance counting functions for Schroedinger operators $H_{V} = - \Delta + V$ on $L^{2}(\mathbb{R}^{d})$, for $d \geq 2$ \textit{even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet $\Lambda_{m}$, $m \in \mathbb{Z} \setminus \{0\}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m \in \mathbb{Z} \setminus \{0\}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet $\Lambda_{0}$ determine the poles on $\Lambda_m$. 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 $\Lambda_{m}$, $m \in \mathbb{Z} \setminus \{0\}$. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4007 10.1512/iumj.2010.59.4007 en Indiana Univ. Math. J. 59 (2010) 621 - 660 state-of-the-art mathematics http://iumj.org/access/