The inverse resonance problem for $\Z_2$-symmetric analytic obstacles in   the plane
  We prove that a two-component mirror-symmetric analytic obstacle in the plane is determined by its resonance poles among such obstacles. The proof is essentially the same as in the interior case (part II of the series). A so-called interior/exterior duality formula is used to simplify the proof. A fair amount of exposition is included for the sake of completeness. 
Steve Zelditch