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

radius.

Received by the editor Mar 30, 2005. ch

iv