Abstract mixed boundary-value and spectral conjugation problems and their applications

Basing on the abstract Green formula, we study general approach to abstract boundary-value conjugation problems. We consider examples of some configurations of docked domains for conjugation problems using generalized Green formula for the Laplace operator. Also we consider spectral problems with two complex parameters: one of them can be treated as fixed and the other one as spectral. By means of the proposed general approach, we reduce these problems to the spectral problem for operator pencil with self-adjoint operator coefficients acting in Hilbert space and depending on two parameters.