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 $V\in L^{infty}_{\mathrm {comp}}(\mathbb R^d)$ 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.