Lower bounds for resonance counting functions for Schrödinger operators with fixed sign potentials in even dimensions

Abstract

If $d$ is even, the resonances of the Schrödinger operator $-\Delta +V$ on ${\mathbb{R}}^d$ with $V\in L_{\mathrm{comp}}^{infty}({\mathbb{R}}^d)$ are points on $\Lambda$, the logarithmic cover of $\mathbb{C}\setminus \{0\}$. We show that for fixed sign potentials $V$ and for $m\in \mathbb{Z}\setminus \{0\}$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth.