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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
##### Keywords:
Laplace-Beltrami operator
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##### References:
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