Scattering Theory for the Laplacian in Perturbed Cylindrical Domains

Citation

Lyford, William Carl (1973) Scattering Theory for the Laplacian in Perturbed Cylindrical Domains. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CY60-CP94. https://resolver.caltech.edu/CaltechTHESIS:12062017-145856672

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, R^N. 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 R^N, S = G x R, G a bounded domain in R^N-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 R^N 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 π^ac be the orthogonal projection onto the subspace of absolute continuity of H. Then the wave operators

Refer to PDF for formula

and

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exist. The operators W_±(H, H₀; J) map H₀ isometrically onto H^ac = π^acH and provide a unitary equivalence between H₀ and H^ac, the part of H in H^ac. 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 H^ac 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.

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