## 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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### References:

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