Capacitary Estimates for the Scattering Phase

AbstractLet H≔−12Δ acting in L2(Rn), n≥3. Given a compact obstacle K, let HY be the operator obtained from H by imposing Dirichlet boundary conditions on K; here Y is the complement of K. Let θY(λ) be the scattering phase for the scattering system {HY,H;J}, where J denotes restriction to Y. We show that the variation in θY(λ) (modulo integer multiples of π/2) as K varies may be estimated in terms of the capacity of the symmetric difference of the obstacles. Suppose that K has smooth boundary. Let Hβ≔H+β1K (β>0) and θβ(λ) be the scattering phase for the scattering system {Hβ,H}. We show furthermore that θβ(λ) converges to θY(λ) (again modulo integer multiples of π/2) roughly at the rate β−1/2 as β→∞