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by Ikebe [2]. He studied the operators -A and -A +q(x) acting on L2(R'9), where q(x) is a potential function. This method was also used to construct a scattering operator for the wave equation in the exterior of a bounded obstacle by Shenk [5]. The exterior problem has been treated by Lax and Phillips [4], using a different approach. They constructed the scattering operator with the aid of certain abstract representation theorems. They then applied their theory to obtain a complete set of generalized eigenfunctions of the reduced wave equation in an exterior domain. The methods of [4] are not applicable in their present form to our problem. We begin in ?2 by defining certain Hilbert spaces X0 and X1 of initial data for the wave equation with zero boundary conditions in S and Q, respectively. (It may easily be shown that all of the arguments in this paper carry over to the case of a semi-infinite cylinder as well as to the case of the boundary conditions discussed in I.) In Theorem 2.1 we establish the existence of a group of unitary transformations to(t) [,i(t)] acting on X0 [X]. In ?3, we construct spectral representations for the operators t&0(t) [,&(t)]. These spectral representations are expressed in terms of the functions w, (x; e). In ?4, we define the wave operators ' +. We prove that the operators * A exist as unitary transformations. This follows from Theorem 4.1. We discuss the scattering operator 9'= (4' +) - 1* - in ?5. An immediate consequence of the existence of the scattering operator is a local energy decay theorem

Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. II. Applications to scattering theory