of $R_V(\lambda)$}\; \text{with} \;\frac{1}{r}\leq |\lambda_j|\leq r,\; |\arg \lambda_j| \leq \log r \}}{(\log r)( \log \log r)^{-p}}
=\infty \; \forall p>1$$ for any nontrivial $V\in C_c^{\infty}(\Real^d;\Real)$. This follows the earlier work of [@s-t]. We note that the assumption that the potential is real-valued is crucial here. There are explicit examples of nontrivial complex-valued potentials $V\in L^{\infty}_{\comp}(\Real^d)$ which can be chosen to be smooth so that the corresponding Schrödinger operator $-\Delta +V$ has neither eigenvalues nor resonances [@autin; @chex; @iso].

For $m\in \Integers$, let $n_m(r)=n_{m,V}(r)$ be the number of resonances of $-\Delta + V$ which both lie on $\Lambda_m$ and have norm at most $r$. We call this the resonance counting function for the $m$th sheet. It follows from Vodev’s result that $n_m(r)= O(r^d)$ as $r\rightarrow \infty$. On the other hand, lower bounds have proved more elusive. The results of [@ch-hi2 Theorem 1.1] show that “generically” for potentials $V\in L^{\infty}_{\comp}(\Real^d)$, $m\in \Integers \setminus \{ 0\}$, $$\label{eq:limsup}
\lim \sup _{r\rightarrow \infty}\frac{\log n_{m,V}(r)}{\log r}=d.$$ However, the result of [@ch-hi2] is nonconstructive in the sense that other than potentials which are nonzero positive scalar multiples of the characteristic function of a ball and those complex-valued potentials which are isoresonant with them [@iso], that paper does 