Abstract
In this chapter we study the phenomenon of scattering by a compact obstacle in Euclidean space \({\mathbb{R}}^{3}\). We restrict attention to the three-dimensional case, though a similar analysis can be given for obstacles in \({\mathbb{R}}^{n}\) whenever n is odd. The Huygens principle plays an important role in part of the analysis, and for that part the situation for n even is a little more complicated, though a theory exists there also.
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1 A. Lidskii’s trace theorem
The purpose of this appendix is to prove the following result of V. Lidskii, which is used for (8.2):
Theorem A.1.
If A is a trace class operator on a Hilbert space H, then
We will make use of elementary results about trace class operators, established in §6 of Appendix A, Functional Analysis. In particular, if {uj} is any orthonormal basis of H, then
To begin the proof, let Eℓ = ⊕j ≤ ℓ Vj, and let Pℓ = Q1 + ⋯ + Qℓ denote the orthogonal projection of H onto Eℓ. Thus
Let H0 denote the closed linear span of {uj : j ≥ 1}, and H1 the orthogonal complement of H0 in H, and let Rν be the orthogonal projection of H on Hν. We can write A in block form
Lemma A.2.
If H 1 ≠ 0, then Spec A 1 = {0}.
Proof.
Suppose Spec A1 contains an element μ ≠ 0. Since A1 is compact on H1, there must exist a unit vector v ∈ H1 such that A1 v = μv. Let \(\mathcal{H} = {H}_{0} + (v)\). Note that
Note that both Tμ = A0 − μI (on H0) and \({\mathcal{T}}_{\mu } = \mathcal{A}- \mu I\) (on \(\mathcal{H}\)) are Fredholm operators of index zero, and that
A linear operator K is said to be quasi-nilpotent provided Spec K = { 0}. If this holds, then (I + zK)− 1 is an entire holomorphic function of z. The convergence of its power series implies
Lemma A.3.
If K is a trace-class operator on a Hilbert space and K is quasi-nilpotent, then Tr K = 0.
To prove Lemma A.3, we use results on the determinant established in §6 of Appendix A, Functional Analysis. Thus, we consider the entire holomorphic function
A proof of Lidskii’s theorem—avoiding the first part of the argument given above, and simply using determinants, but making heavier use of complex function theory—is given in [Si2].
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Taylor, M.E. (2011). Scattering by Obstacles. In: Partial Differential Equations II. Applied Mathematical Sciences, vol 116. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7052-7_3
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