Counting the Resonances in High and Even Dimensional Obstacle Scattering

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In this paper, we give a polynomial lower bound for the resonances of -Δ perturbed by an obstacle in even-dimensional Euclidean spaces, n≥4. The proof is based on a Poisson Summation Formula which comes from the Hadamard factorization theorem in the open upper complex plane. We take advantage of the singularity of regularized wave trace to give the pole/resonance counting function over the principal branch of logarithmic plane a lower bound.