In this week, the theory of scattering with two Hilbert spacesis applied to a certain selfadjoint elliptic operator acting in twodifferent domains in Euclidean N-space, RN. The wave operatorsand scattering operator are then constructed. The selfadjoint operator is the negative Laplacian acting on functions which satisfy aDirichlet boundary condition.
The unperturbed operator, denoted by H0, is defined in the Hilbert space H0 = L2(S), where S is a uniform cylindrical domainin RN, S = G x R, G a bounded domain in RN-1 with smooth boundary.For this operator, an eigenfunction expansion is derived whichshows that H0 has only absolutely continuous spectrum. The eigenfunctionexpansion is used to construct the resolvent operator, the spectral measure, and a spectral representation for H0.
The perturbed operator, denoted by H, is defined in the Hilbertspace H = L2(Ω), where Ω is perturbed cylindrical domainin RN with the property that there is a smooth diffeomorphismɸ : Ω ↔ S which is the identity map outside a bounded region. The mapping ɸ is used to construct a unitary operator J mapping H0onto H which has the additional property that JD(H0) = D(H).
The following theorem is proved:
Theorem: Let πac be the orthogonal projection onto the subspaceof absolute continuity of H. Then the wave operators
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and
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exist. The operators W±(H, H0; J) map H0 isometrically ontoHac = πacH and provide a unitary equivalence between H0 and Hac,the part of H in Hac. Furthermore,
[W±(H, H0; J)]* = W±(H, H0; J*). □
It is proved that the point spectrum of H is nowhere dense inR. A limiting absorption principle is proved for H which showsthat H has no singular continuous spectrum. The limiting absorptionprinciple is used to construct two sets of generalized eigenfunctions for H.The wave operators W±(H, H0; J are constructed interms of these two sets of eigenfunctions. This construction and theabove theorem yield the usual completeness and orthogonality resultsfor the two sets of generalized eigenfunctions. It is noted thatthe construction of the resolvent operator, spectral measure, and aspectral representation for H0 can be repeated for the operatorHac and yields similar results. Finally, the channel structure ofthe problem is noted and the scattering operator
S(H, H0; J) = W+(H0, H; J*)(W_H0, H; J)
is constructed.
