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Polynomial bound on the number of scattering poles. (English) Zbl 0535.35067
In this paper it is shown that the number of the poles of the scattering matrix associated to the operator \(\Delta +V\) is of polynomial growth, where V is an infinitely differentiable potential with compact support. More precisely, let \(\{\lambda_ j\}\) be the set of poles, and let \(N(T)=\max \{j| | \lambda_ j|<T\}.\)- It is shown that there exist constants C, \(p>0\) with \(N(T)\leq C(1+T)^ p.\) To prove this result, the integral operator \(H(\lambda)u(x)=\int \rho(x)e_{\lambda}(x,y)V(y)u(y)dy\) is considered, where \(e_{\lambda}\) is the fundamental solution of the Helmholtz equation, and where \(\rho\) is a cut-off function. It is shown that \(H(\lambda)^ k\) is trace class for k sufficiently large. The result is obtained from growth estimates for the entire function \(h(\lambda)=\det(I-H(\lambda))^ k.\)- Also, Poisson-type formulas for the fundamental solution of the wave equation and the trace of the solution operator to the wave equation are discussed.
Reviewer: H.D.Alber

MSC:
35P25 Scattering theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81U20 \(S\)-matrix theory, etc. in quantum theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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