fixed-sign potentials. In the odd-dimensional case Lax-Phillips [@l-p] and Vasy [@vasy] proved lower bounds on the number of pure imaginary resonances for potentials of fixed sign. In [@ch-hi4] it is shown that in even dimensions there are no “pure imaginary” resonances for positive potentials, and on each sheet $\Lambda_m$ of $\Lambda$, only finitely many for negative potentials. Both [@l-p] and [@vasy] use a monotonicity property for potentials of fixed sign. This paper also uses a monotonicity property, though it is more closely related to one used in [@scv]. Also important here are some results from one-dimensional complex analysis, more delicate than the corresponding complex-analytic arguments from [@scv].

[**Acknowledgments.**]{} The author gratefully acknowledges the partial support of the NSF under grant DMS 1001156.

Some Complex Analysis
=====================

The main result of this section is Proposition \[p:complex\], which, roughly speaking, controls the growth of a function $f$ analytic in a half plane in terms of the growth of the counting function for the zeros of $f$ in the half plane and the behavior of $f$ on the boundary of the half plane.

Both the statement and the proof of the following lemma bear some resemblance to those for Carathéodory’s inequality for the 