Citation
Lyford, William Carl (1973) Scattering Theory for the Laplacian in Perturbed Cylindrical Domains. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CY60CP94. https://resolver.caltech.edu/CaltechTHESIS:12062017145856672
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 Nspace, 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 H_{0}, is defined in the Hilbert space H_{0} = L_{2}(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 H_{0} has only absolutely continuous spectrum. The eigenfunction expansion is used to construct the resolvent operator, the spectral measure, and a spectral representation for H_{0}.
The perturbed operator, denoted by H, is defined in the Hilbert space H = L_{2}(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 H_{0} onto H which has the additional property that JD(H_{0}) = 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
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and
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exist. The operators W_{±}(H, H_{0}; J) map H_{0} isometrically onto H^{ac} = π^{ac}H and provide a unitary equivalence between H_{0} and H^{ac}, the part of H in H^{ac}. Furthermore,
[W_{±}(H, H_{0}; J)]^{*} = W_{±}(H, H_{0}; 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, H_{0}; 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 H_{0} 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, H_{0}; J) = W_{+}(H_{0}, H; J^{*})(W_H_{0}, H; J)
is constructed.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  2 April 1973  
Funders: 
 
Record Number:  CaltechTHESIS:12062017145856672  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12062017145856672  
DOI:  10.7907/CY60CP94  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10594  
Collection:  CaltechTHESIS  
Deposited By:  Benjamin Perez  
Deposited On:  07 Dec 2017 00:25  
Last Modified:  21 Dec 2019 02:07 
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