Scattering Theory for the Laplacian in Perturbed Cylindrical Domains
<p>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<sup>N</sup>. The wave operators and scattering operator are then constructed. The selfadjoint operato...
| Summary: | <p>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<sup>N</sup>. 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.</p>
<p>The unperturbed operator, denoted by H<sub>0</sub>, is defined in the Hilbert space H<sub>0</sub> = L<sub>2</sub>(S), where S is a uniform cylindrical domain
in R<sup>N</sup>, S = G x R, G a bounded domain in R<sup>N</sup>-1 with smooth boundary.
For this operator, an eigenfunction expansion is derived which
shows that H<sub>0</sub> has only absolutely continuous spectrum. The eigenfunction
expansion is used to construct the resolvent operator, the
spectral measure, and a spectral representation for H<sub>0</sub>.</p>
<p>The perturbed operator, denoted by H, is defined in the Hilbert
space H = L<sub>2</sub>(Ω), where Ω is perturbed cylindrical domain
in R<sup>N</sup> 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<sub>0</sub>
onto H which has the additional property that JD(H<sub>0</sub>) = D(H).</p>
<p>The following theorem is proved:</p>
<p>Theorem: Let π<sup>ac</sup> be the orthogonal projection onto the subspace
of absolute continuity of H. Then the wave operators</p>
<p>Refer to PDF for formula</p>
<p>and</p>
<p>Refer to PDF for formula</p>
<p>exist. The operators W<sub>±</sub>(H, H<sub>0</sub>; J) map H<sub>0</sub> isometrically onto
H<sup>ac</sup> = π<sup>ac</sup>H and provide a unitary equivalence between H<sub>0</sub> and H<sup>ac</sup>,
the part of H in H<sup>ac</sup>. Furthermore,</p>
<p>[W<sub>±</sub>(H, H<sub>0</sub>; J)]<sup>*</sup> = W<sub>±</sub>(H, H<sub>0</sub>; J<sup>*</sup>). □</p>
<p>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<sub>±</sub>(H, H<sub>0</sub>; 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<sub>0</sub> can be repeated for the operator
H<sup>ac</sup> and yields similar results. Finally, the channel structure of
the problem is noted and the scattering operator</p>
<p>S(H, H<sub>0</sub>; J) = W<sub>+</sub>(H<sub>0</sub>, H; J<sup>*</sup>)(W_H<sub>0</sub>, H; J)</p>
<p>is constructed.</p>
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