Spectral problem of Poincaré and Steklov

We consider a spectral problem with a pair of boundary influence operators (transforming the Dirichlet data of a harmonic function into its Neumann data) for a pair of planar domains with a common boundary. Similar problems arise when we justify and optimize the computational methods such as domain decomposition and fictitious domains. The problem reduces to studying a pencil of one-dimensional integral operators with Cauchy and Grunsky kernels. The possibility of finding the eigenvalues and functions of the simplest pencils in a closed analytical form is investigated.