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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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[1] Bardos, C., Guilot, J., Ralston, J.: La relation de Poisson pour l’equation des ondes dans un ouvert non borne application la theorie de la diffusion. Commun. Partial Differ. Equations7, 905-958 (1982) · Zbl 0496.35067
[2] Intissar, A.: A polynomial bound on the number of scattering poles for a potential in even dimensional space ? n . Commun. Partial Differ. Equations11, 367-396 (1986) · Zbl 0607.35069
[3] Intissar, A.: On the value distribution of the scattering poles associated to the Schr?dinger operator \(H = ( - i\vec V + \vec b(x))^2 + a(x)\) in ? n ,n?3. Preprint
[4] Lax, P.D., Phillips, R.S.: Scattering theory. New York: Academic Press 1967 · Zbl 0214.12002
[5] Melrose, R.B.: Polynomial bounds on the number of scattering poles. J. Funct. Anal.53, 287-303 (1983). · Zbl 0535.35067
[6] Melrose, R.B.: Polynomial bounds on the distribution of poles in scattering by an obstacle. Journ?es Equations aux d?riv?es partielle. Saint-Jean-de-montes (1984)
[7] Melrose, R.B.: Weyl asymptotics for the phase in obstacle scattering. Commun. Partial Differ. Equations13, 1431-1439 (1988) · Zbl 0686.35089
[8] Petkov, V.M.: Phase de diffusion pour des perturbations captives. Journ?es Equations aux d?riv?es partielles, Saint-Jean-de-Montes. (1990)
[9] Titchmarsh, E.C.: The theory of functions. Oxford: Oxford University Press 1968 · Zbl 0005.21004
[10] Vainberg, B.: Asymptotic methods in equations of mathematical physics. New York: Gordon and Breach 1988 · Zbl 0907.35078
[11] Vodev, G.: Polynomial bounds on the number of scattering poles for symmetric systems. Ann. Inst. Henri Poincar? Phys. th?or.54, 199-208 (1991) · Zbl 0816.35101
[12] Vodev, G.: Polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in ? n ,n?3 odd. Osaka J. Math.28 (1991) (to appear) · Zbl 0754.35102
[13] Zworski, M.: Distribution, of poles for scattering on the real line J. Funct. Anal.73, 277-296 (1987) · Zbl 0662.34033
[14] Zworski, M.: Sharp polynomial bounds on the number of scattering poles of radial potentials. J. Funct. Anal.82, 370-403 (1989) · Zbl 0681.47002
[15] Zworski, M.: Sharp polynomial bounds on the number of scattering poles. Duke Math. J.59, 311-323 (1989) · Zbl 0705.35099
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