Capacitary Estimates for the Scattering Phase

Let H[y 12 D acting in L2 Rn., nG3. 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 uY l. be the scattering phase for the scattering system HY , H; J4, where J denotes restriction to Y. We show that the variation inuY l. modulo integer multiples of pr2. 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 H [Hqb 1 b )0.andu l.be the scattering b K b phase for the scattering system H , H4. We show furthermore that u l.con- b b verges touY l. again modulo integer multiples of pr2. roughly at the rate by1r2 as bª`.