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Sharp polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in \(\mathbb{R}{}^ n\). (English) Zbl 0754.35105

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, R. Melrose [Journ. “Equations Deriv. Partielles”, St. Jean-De-Monts 1984, Conf. No. 3, 8 p. (1984; Zbl 0621.35073)] and 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 \(\mathbb{R}^ n\).
Reviewer: G.Vodev (Sofia)

MSC:

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
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