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Low energy scattering asymptotics for planar obstacles

T. J. Christiansen and K. Datchev

Vol. 5 (2023), No. 3, 767–794
DOI: 10.2140/paa.2023.5.767

We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle’s logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on 2, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero.

resolvent, scattering matrix, scattering phase, Dirichlet boundary condition, capacity
Mathematical Subject Classification
Primary: 35P25, 47A40
Secondary: 35J25
Received: 25 October 2022
Accepted: 2 February 2023
Published: 24 August 2023
T. J. Christiansen
Department of Mathematics
University of Missouri
Columbia, MO
United States
K. Datchev
Department of Mathematics
Purdue University
West Lafayette, IN
United States