The inverse acoustic obstacle scattering problem and its interior dual

This paper addresses possible connections between two classical inverse problems arising in wave propagation. The first is the problem of extracting geometrical information about an unknown bounded domain from a knowledge its eigen-frequencies. The chosen method of investigation being the high frequency asymptotics of the associated counting function. The second problem is the inverse obstacle scattering problem. That is the determination of an unknown obstacle from far field data. This problem is investigated through the high frequency asymptotics of the associated scattering phase. It turns out that there is a remarkable similarity between the asymptotic expansions in each of these problems. We discuss a number of ideas and techniques along the way including representations of the scattering matrix and the Kirchoff approximation. We also show how to solve scattering problems for polygonal obstacles. Whether there is a deep physical connection between interior and exterior scattering problems remains a challenging area of research.