Scattering Theory for the Laplacian in Perturbed Cylindrical Domains

Abstract

In this week, the theory of scattering with two Hilbert spaces is applied to a certain selfadjoint elliptic operator acting in two different domains in Euclidean N-space, RN. The wave operators and scattering operator are then constructed. The selfadjoint operator is the negative Laplacian acting on functions which satisfy a Dirichlet boundary condition. The unperturbed operator, denoted by H0, is defined in the Hilbert space H0 = L2(S), where S is a uniform cylindrical domain in RN, S = G x R, G a bounded domain in RN-1 with smooth boundary. For this operator, an eigenfunction expansion is derived which shows that H0 has only absolutely continuous spectrum. The eigenfunction expansion 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 Hilbert space H = L2(Ω), where Ω is perturbed cylindrical domain in RN with the property that there is a smooth diffeomorphism ɸ : Ω ↔ S which is the identity map outside a bounded region. The mapping ɸ is used to construct a unitary operator J mapping H0 onto 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 subspace of absolute continuity of H. Then the wave operators Refer to PDF for formula and Refer to PDF for formula exist. The operators W±(H, H0; J) map H0 isometrically onto Hac = 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 in R. A limiting absorption principle is proved for H which shows that H has no singular continuous spectrum. The limiting absorption principle is used to construct two sets of generalized eigenfunctions for H. The wave operators W±(H, H0; J are constructed in terms of these two sets of eigenfunctions. This construction and the above theorem yield the usual completeness and orthogonality results for the two sets of generalized eigenfunctions. It is noted that the construction of the resolvent operator, spectral measure, and a spectral representation for H0 can be repeated for the operator Hac and yields similar results. Finally, the channel structure of the problem is noted and the scattering operator S(H, H0; J) = W+(H0, H; J*)(W_H0, H; J) is constructed.