Monotonicity in inverse medium scattering on unbounded domains

We discuss a time-harmonic inverse scattering problem for the Helmholtz equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. We utilize this monotonicity relation to establish novel characterizations of the support of the scattering objects in terms of the far field operator. These are related to and extend corresponding results known from factorization and linear sampling methods to determine the support of unknown scattering objects from far field observations of scattered fields. An attraction of the new characterizations is that they only require the refractive index of the scattering objects to be above or below the refractive index of the background medium locally and near the boundary of the scatterers. An important tool to prove these results are so-called localized wave functions that have arbitrarily large norm in some prescribed region while at the same time having arbitrarily small norm in some other prescribed region. We present numerical examples to illustrate our theoretical findings.