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

Non-scattering energies for acoustic-type equations on manifolds with a single flat end