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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 |

### Keywords:

polynomial bound; scattering poles; number of the poles; scattering matrix; Poisson-type formulas; fundamental solution
Full Text:
DOI

### References:

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