Monotonicity-based shape reconstruction for an inverse scattering problem in a waveguide

We consider an inverse medium scattering problem for the Helmholtz equation in a closed cylindrical waveguide with penetrable compactly supported scattering objects. We develop novel monotonicity relations for the eigenvalues of an associated modified near field operator, and we use them to establish linearized monotonicity tests that characterize the support of the scatterers in terms of near field observations of the corresponding scattered waves. The proofs of these shape characterizations rely on the existence of localized wave functions, which are solutions to the scattering problem in the waveguide that have arbitrarily large norm in some prescribed region, while at the same time having arbitrarily small norm in some other prescribed region. As a byproduct we obtain a uniqueness result for the inverse medium scattering problem in the waveguide with a simple proof. Some numerical examples are presented to document the potentials and limitations of this approach.