  To study the location of poles for the acoustic scattering matrix for two
strictly convex obstacles with smooth boundaries, one uses an approximation of
the quantized billiard operator $M$ along the trapped ray between the two
obstacles. Using this method Ikawa and G{\'e}rard established the existence of
parallel rows of poles in a strip $Im z\leq c$ as $Re z$ tends to infinity.
Assuming that the boundaries are analytic and the eigenvalues of Poincar{\'e}
map are non-resonant we use the Birkhoff normal form for $M$ to improve this
result and to get the complete asymptotic expansions for the poles in any
logarithmic neighborhood of real axis.
