1991 Mathematics Subject Classification. Primary 58J50, 54C40, 14E20;
Secondary 37A60, 46E25, 20C20
Key words and phrases. scattering resonance, obstacle, Ruelle transfer operator,
zeta function, billiard trajectory
Abstract. This work deals with scattering by obstacles which are finite dis-
joint unions of strictly convex bodies with smooth boundaries in an odd dimen-
sional Euclidean space. The class of obstacles of this type is considered which
are contained in a given (large) ball and have some additional properties:
its connected components have bounded eccentricity, the distances between
different connected components are bounded from below, and a uniform ’no
eclipse condition’ is satisfied. It is shown that if an obstacle K in this class
has connected components of sufficiently small diameters, then there exists a
horizontal strip near the real axis in the complex upper half-plane containing
infinitely many scattering resonances (poles of the scattering matrix), i.e. the
Modified Lax-Phillips Conjecture holds for such K. This generalizes a well-
known result of M. Ikawa concerning balls with the same sufficiently small
Received by the editor Mar 30, 2005. ch
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