A model of zero width slits for an orifice in a semitransparent boundary (Q1204776)

The scattering problem for an obstacle with a small aperture is solved. The model constructed is based on extension theory. The structure of the model is as follows. Let \(\Omega^{in}\) be a domain in \(\mathbb{R}^ 3\) with a smooth boundary \(\partial \Omega\), \(\Omega^{ex} = \mathbb{R}^ 3 \backslash \overline{\Omega^{in}}\), \(x_ 0 \in \partial \Omega\). Let \(-\Delta = -(\Delta^{in} \oplus \Delta^{ex})\). Here \(\Delta^{in}\) and \(\Delta^{ex}\) are Laplace operators on \(\Omega^{in}\) and \(\Omega^{ex}\) respectively with boundary conditions of special type. The operator \(-\Delta\) restricted on smooth functions \(u\) \((u(x_ 0)=0)\) may be extended as selfadjoint operator. The parameters of the extension may be choosen such that the model Green's function coincides with the main component of the asymptotics for the respective ``real'' Green's function where the radius of the aperture \(d\) tends to zero. The main peculiarity of the paper is the change of Dirichlet or Neumann boundary conditions to a specific trapping condition with semitransparent boundary. The \(S\)-matrix for the scattering model is constructed, too.