  In this paper, we consider the scattering theory for acoustic-type equations
on non-compact manifolds with a single flat end. Our main purpose is to show an
existence result of non-scattering energies. Precisely, we show a Weyl-type
lower bound for the number of non-scattering energies. Usually a scattered wave
occurs for every incident wave by the inhomogeneity of the media. However,
there may exist suitable wavenumbers and patterns of incident waves such that
the corresponding scattered wave vanishes. We call (the square of) this
wavenumber a non-scattering energy in this paper. The problem of non-scattering
energies can be reduced to a well-known interior transmission eigenvalues
problem.
