Sharp upper bounds on the number of the scattering poles

Abstract

For various compactly supported perturbations of the Laplacian in odd dimensions $n$, we prove a sharp upper bound of the resonance counting function $N(r)$ of the type $N(r)\le A_n{r}^n(1+o(1))$ with an explicit constant $A_n$. In a few special cases, we show that this estimate turns into an asymptotic.