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Sharp bounds on the number of scattering poles in the two dimensional case. (English) Zbl 0829.35091

The paper extends to the two-dimensional case the upper bounds on the number of scattering poles of a class of compactly supported perturbations of the Laplacian. The scattering poles (known also as resonances) can be defined as poles of the meromorphic continuation of the cutoff resolvent which in the case of even dimension admits a meromorphic continuation on the Riemann logarithmic surface \[ A = \{z : - \infty < \arg z < + \infty\}. \] The main result shows that the counting function \(N(r,a)\) of the scattering poles associated to an operator \(G\) is bounded. The scattering poles, with multiplicity, are characterized as zero of a holomorphic function.

MSC:

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
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