Closed guides
The spectral theory and its applications. Generalities and compact operators

The study of waveguides is one of the fields of application of spectral theory. We will restrict ourselves to the case of closed waveguides, so as to remain within the framework of bounded operators, and we will content ourselves with the two-dimensional case, which allows explicit calculations, but whose conclusions remain essentially valid in higher dimensions.

Let's consider the case of an acoustic guide: the edges of the guide σ₀ and σ_h are formed by horizontal straight lines with ordinates y = 0 and y = h respectively. A bounded obstacle of boundary Γ located between these two straight lines diffracts the acoustic waves propagating in the guide. The domain Ω lies between the walls of the guide and the obstacle; the perturbation potential verifies Hemlholtz's equation, as well as natural boundary conditions, homogeneous at the edges of the guide.