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

Authors: T. J. Christiansen and P. D. Hislop

Issue: Volume 59 (2010), Issue 2, 621-660

Abstract: 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 $\Lambda_{m}$, $m \in \mathbb{Z} \setminus \{0\}$.