$d$ is even the resonances lie on $\Lambda$ makes them generally more difficult to study in the even-dimensional case than in the odd-dimensional case.

A point on $\Lambda$ can be described by its modulus and argument, where we do not identify points which have arguments differing by nonzero integral multiples of $2\pi$. Thus the physical half plane corresponds to $\Lambda_0
{\stackrel{\rm{def}}{=}}\{ \lambda \in \Lambda:
0<\arg \lambda <\pi\}$. Likewise, for $m\in \Integers$ we may define the $m$th sheet to be $$\Lambda_m {\stackrel{\rm{def}}{=}}\{ \lambda \in \Lambda: m\pi <\arg \lambda<(m+1)\pi\}$$ which is homeomorphic with the physical region and can be identified with the upper half plane when convenient.

Vodev [@vodeveven; @vodev2], following earlier work of Intissar [@intissar] studied the resonance counting function $n_V(r,a)$, defined to be the number of resonances (counted with multiplicity, here and everywhere) with norm at most $r$ and argument between $-a$ and $a$. He showed that there is a constant $C$ which depends on $V$ but not on $r$ or $a$ so that $$n_V(r,a) \leq C a( r^d + (\log a)^d)),\; \text{for}\; r,\; a>1.$$

The most general lower bound known is due to SáBarreto ([@SaB1], $d\geq 4$) and Chen ([@chen], d=2): $$\lim \sup_{r \rightarrow \infty} \frac{\# \{ \lambda_j:\; \text{pole 