Spectral properties of the diffraction problem for a thin screen (Q1069068)

The paper is a continuation of an earlier work of the author [ibid. 25, 31-42 (1984); translation from Sib. Mat. Zh. 25, No.1(143), 39-52 (1984; Zbl 0556.35117)] and is concerned with spectral properties of the following diffraction problem: \[ \Delta U+k^ 2U=0\quad in\quad {\mathcal D}/S,\quad \Lambda U=\mp (1/\sigma)\partial U/\partial N+H\quad on\quad S_{\pm},\quad \partial U/\partial \omega +\beta U=0\quad on\quad \partial {\mathcal D}. \] Here, \(k\in C\), \({\mathcal D}\) is a bounded domain in \(R^ 3\) with a smooth boundary \(\partial {\mathcal D}\), \(S\subset {\mathcal D}\) is a smooth two-sided surface with the (edge) boundary \(\Gamma\) and normal N, \(\lambda\) is a spectral parameter, \(\sigma >0\) is a function given on \(S_+\) and \(S_-\), \(\omega\) is a smooth vector field transverse to \(\partial {\mathcal D}\) and \(\beta\) is a smooth function given on \(\partial {\mathcal D}\). The functions U and H belong to suitable spaces defined in the paper. The main results of the paper are following: (i) The distribution of eigenvalues \(\lambda_ n\) of the problem is determined; (ii) It is proved that the restrictions of the root functions (simple and generalized eigenfunctions) to \(S_- \cup S_+\) form a basis with respect to Abel's summation in a suitably chosen space over \(S_- \cup S_+\). If \(k^ 2\in R\) then the basis is a Bari basis with brackets. As a consequence, the solution to the problem for \(\lambda \not\in \{\lambda_ n\}\) can be sought using the Fourier method. In addition, two related problems are also considered: One with the boundary condition on \(\partial {\mathcal D}\) replaced by a radiation condition and the other formulated in a two dimensional space.