Sharp polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in \(\mathbb{R}{}^ n\) (Q1191419)

In the cases of Laplacian in exterior domains and the Schrödinger operator \(-\Delta+V(x)\) with a potential \(V\in L_ 0^ \infty(\mathbb{R}^ n)\), \(n\geq 3\), odd, \textit{R. Melrose} [Journ. ``Equations Deriv. Partielles'', St. Jean-De-Monts 1984, Conf. No. 3, 8 p. (1984; Zbl 0621.35073)] and \textit{M. Zworski} [Duke Math. J. 59, No. 2, 311-323 (1989; Zbl 0705.35099)], respectively, proved the following sharp polynomial bound on the number \(N(r)\) of the scattering poles in the disk of radius \(r\): \(N(r)\leq Cr^ n+C\). The purpose of this work is to prove this sharp polynomial bound on the number of the scattering poles associated to the Laplace-Beltrami operator \(c(x)^{-1}\sum_{i,j=1}^ n \partial_{x_ i}(g_{ij}(x)\partial_{x_ j})\) in \(\mathbb{R}^ n\), where \(n\geq 3\), odd, and the metric becomes Euclidean outside a bounded domain in \(\mathbb{R}^ n\).