Lower bounds for the number of resonances in even dimensional potential scattering (Q1963865)

Let \(V \in {\mathbb C}_0^\infty({\mathbb R}^n,{\mathbb R})\), \(n \geq 4,\) be even, \(P=-\Delta+V\) and \(\{\lambda_j \}\) are resonances of P with multiplicity \(M(\lambda_j)\). It is proved that \[ \sum_j {M(\lambda_j) \over {|\log|\lambda_j|+i \arg\lambda_j |} } = \infty. \] As a direct consequence of this result a lower bound for the number of resonances is obtained.