Journal article
Equidistribution of phase shifts in obstacle scattering
J Gell-Redman, M Ingremeau
Communications in Partial Differential Equations | TAYLOR & FRANCIS INC | Published : 2019
Abstract
For scattering off a smooth, strictly convex obstacle (Formula presented.) with positive curvature, we show that the eigenvalues of the scattering matrix–the phase shifts–equidistribute on the unit circle as the frequency (Formula presented.) at a rate proportional to (Formula presented.), under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector (Formula presented.) not containing 1, there are (Formula presented.) eigenvalues for k large, where c d is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckma..
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Funding Acknowledgements
J.G.R. acknowledges the support of the Australian Research Council through Discovery Grant DP180100589. M.I. was funded by the LabEx IRMIA, and partially supported by the Agence Nationale de la Recherche project GeRaSic (ANR-13-BS01-0007-01). Both authors wish to thank the Australian Mathematical Sciences Institute and the Mathematical Sciences Institute at the Australian National University for their partial funding of the workshop "Microlocal Analysis and its Applications in Spectral Theory, Dynamical Systems, Inverse Problems, and PDE" at which part this project was completed.