there are many potentials with resonance counting functions for the $m$th sheet having maximal order of growth, the technique of [@ch-hi2] does not give a way of identifying them other than those which are scalar multiples of the characteristic function of a ball. In comparison, in odd dimensions $d\geq 3$ the only specific real-valued potentials $V\in L^\infty_{\comp}(\Real^d)$ which are known to have resonance-counting function with optimal order of growth are certain radial potentials [@zwradpot], though in that case asymptotics are known (see [@zwradpot] and [@dinh-vu]). Let $V\in L^\infty_{\comp}(\Real^d)$ and let $\Delta\leq 0$ denote the Laplacian on $\Real^d$. We define the resolvent $R_V(\lambda)=(-\Delta +V-\lambda^2)^{-1}$ for $\lambda$ in the “physical space”, $0<\arg \lambda <\pi$. With at most a finite number of exceptional values of $\lambda$, $R_V(\lambda)$ is bounded on $L^2(\Real^d)$ for $\lambda$ in this region. It is well known that for $\chi \in L^{\infty}_{\comp}(\Real^d)$, $\chi R_V(\lambda)\chi$ has a meromorphic continuation to $\Complex$ when $d$ is odd and to $\Lambda$, the logarithmic cover of $\Complex \setminus \{ 0\}$, when $d$ is even (e.g. [@lrb Chapter 2]). In either case, the resonances are defined to be the poles of $\chi R_V(\lambda)\chi$ when $\chi$ is chosen to satisfy $\chi V\equiv V$. The fact that when 