not give a way of identifying the particular potentials for which (\[eq:limsup\]) holds.

The main result of this paper is the following theorem.

\[thm:lbd\] Let $d$ be even. Suppose $V\in L^{\infty}_{\comp}(\Real^d)$ with $V$ bounded below by $\epsilon \chi_B$, where $\epsilon>0$ and $\chi_B$ is the characteristic function of a nontrivial ball $B$. Then for any nonzero $m\in \Integers$, $$\lim \sup _{r \rightarrow \infty} \frac{\log n_{m, \pm V}(r)}{\log r} =d.$$

We note that by Vodev’s result $d$ is the maximum value this limit can obtain. When this limit is $d$, we say that the $m$th counting function has maximal order of growth.

Theorem \[thm:lbd\], when combined with [@ch-hi2 Theorem 3.8], has the following theorem as an immediate corollary.

Let $d$ be even, and $K\subset \Real^d$ be a compact set with nonempty interior. Let $F$ denote either $\Real $ or $\Complex$. Then for $m\in \Integers$, $m\not
= 0$, the set $$\{ V\in C^{\infty}(K;F): \lim\sup_{r\rightarrow \infty } \frac{ \log n_{m,V}(r)}{\log r}=d\}$$ is dense in $C^{\infty}(K;F)$.

For the case of [*odd*]{} dimension $d\geq 3$, the analog of this theorem was proved in [@scv]. A stronger result holds in dimension $d=1$, see [@zworski1d] or [@froese; @regge; @simon].

Theorem \[thm:lbd\] may be compared with other results for 