Capacity estimates for the scattering phase
Let \(H\) be the selfadjoint nonnegative definite operator \(H:=-\frac 12\Delta\) acting in \(L^6({\mathbb R}^n)\) where \(n\geq 3\). The authors perturb \(H\) by imposing Dirichlet boundary conditions on a given compact subset \(K\) of \({\mathbb R}^N\). This gives a selfadjoint nonnegative definite operator \(H_Y\) acting in \(L^2(Y)\), where \(Y\) is the complement of \(K\). Let \(\theta_Y(\lambda)\) be the scattering phase for the scattering system \(\{H_Y, H;J\}\), where \(J\) denotes the restriction to \(Y\). The authors show that the variation in \(\theta_Y(\lambda)\) (modulo integer multiples of \(\pi /2\)) as \(K\) varies may be estimated in terms of the capacity of the symmetric difference of the obstacles. Furthermore, for the case where \(K\) has smooth boundary, \(H_{\beta}:=H+\beta 1_{k}\) (\(\beta >0\)) and \(\theta_{\beta}(\lambda)\) is the scattering phase for the scattering system \(\{ H_{\beta}, H\}\), the authors prove that \(\theta_{\beta}(\lambda)\) converges to \(\theta_{Y}(\lambda)\) (again modulo integer multiples of \(\pi /2\)) roughly at the rate \(\beta^{-1/2}\) as \(\beta \to \infty.\) A proof that \(\theta_{\beta}(\lambda)\) converges to \(\theta_{Y}(\lambda)\) as \(\beta\to\infty\) was sketched in [\textit{T. Kato}, Hadronic J. 1, 134-154 (1978; Zbl 0426.47004)].
scientific article; zbMATH DE number 1348147
scientific article