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abstract: 'If $d$ is even, the resonances of the Schrödinger operator $-\Delta +V$ on $\Real^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of $\Complex \setminus \{0\}$. We show that for fixed sign potentials $V$ and for $m\in \Integers \setminus \{0\}$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth.'
address: 'Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA'
author:
- 'T.J. Christiansen'
title: Lower bounds for resonance counting functions for Schrödinger operators with fixed sign potentials in even dimensions 
---

Introduction
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The purpose of this paper is to prove some optimal lower bounds on the growth rate of resonance-counting functions for certain Schrödinger operators in even-dimensional Euclidean space. The resonances associated to the Schrödinger operator $-\Delta +V$, with potential $V\in L^{\infty}_{\comp}(\Real^d)$, lie on $\Lambda$, the logarithmic cover of $\Complex \setminus \{ 0 \}$, if $d$ is even. The main result of this paper is that for scattering by a fixed sign, compactly supported potential $V$ the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth for any $m\in \Integers \setminus \{ 0\}$. Though the results of [@ch-hi2] show that 