Scattering by Obstacles

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.

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

$$\text{ Tr }A = \sum \nolimits ({ dim }{V }_{j}){\lambda }_{j},$$

(A.1)

where {λ _j : j ≥ 1} = Spec A ∖{ 0} and V _j is the generalized λ _j -eigenspace of A.

We will make use of elementary results about trace class operators, established in §6 of Appendix A, Functional Analysis. In particular, if {u_j} is any orthonormal basis of H, then

$$\text{ Tr }A = \sum \nolimits (A{u}_{j},{u}_{j}),$$

(A.2)

the result being independent of the choice of orthonormal basis, provided A is trace class.

To begin the proof, let E_ℓ = ⊕_{j ≤ ℓ} V_j, and let P_ℓ = Q₁ + ⋯ + Q_ℓ denote the orthogonal projection of H onto E_ℓ. Thus

$$A{P}_{{\ell}} = {P}_{{\ell}}A{P}_{{\ell}},$$

restricted to E_ℓ, has spectrum {λ_j: 1 ≤ j ≤ ℓ}. We will choose an orthonormal set {u_j : j ≥ 1} according to the following prescription: {u_j : 1 + dim E_{ℓ − 1} ≤ j ≤ dim E_ℓ} will be an orthonormal basis of $\mathcal{R}({Q}_{{\ell}})$, with the property that Q_ℓ AQ_ℓ (restricted to $\mathcal{R}({Q}_{{\ell}})$) is upper triangular. That this can be done is proved in Theorem 4.7 of Chap. 1. Note that {u_j : 1 ≤ j ≤ dim E_ℓ} is then an orthonormal basis of E_ℓ, with respect to which AP_ℓ = P_ℓ AP_ℓ is upper triangular. It follows that the diagonal entries of ${P}_{{\ell}}A{P{}_{{\ell}}\bigr |}_{{E}_{{\ell}}}$ with respect to this basis are precisely λ_j, 1 ≤ j ≤ ℓ, counted with multiplicity dim V_j. Inductively, we conclude that the diagonal entries of each block Q_ℓ AQ_ℓ consist of dim V_ℓ copies of λ_ℓ.

Let H₀ denote the closed linear span of {u_j : j ≥ 1}, and H₁ the orthogonal complement of H₀ in H, and let R_ν be the orthogonal projection of H on H_ν. We can write A in block form

$$A = \left (\begin{array}{*{10}c} {A}_{0} & B \\ 0 & {A}_{1} \end{array} \right ),$$

(A.3)

where A_ν = R_ν AR_ν, restricted to H_ν. Clearly, A₀ and A₁ are trace class and, by the construction above plus (A.2), we have

$$\text{ Tr }{A}_{0} = \sum \nolimits (\text{ dim }{V }_{j}){\lambda }_{j}.$$

(A.4)

Thus (A.1) will follow if we can show that Tr A₁ = 0. If H₁ = 0, there is no problem.

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

$$Av = \mu v + w,\quad w \in {H}_{0}.$$

Hence $\mathcal{H}$ is invariant under A; let $\mathcal{A}$ denote A restricted to $\mathcal{H}$. Of course, H₀ is invariant under $\mathcal{A}$, and $\mathcal{A}$ restricted to H₀ is A₀.

Note that both T_μ = A₀ − μI (on H₀) and ${\mathcal{T}}_{\mu } = \mathcal{A}- \mu I$ (on $\mathcal{H}$) are Fredholm operators of index zero, and that

$$\text{ Codim }{\mathcal{T}}_{\mu }(\mathcal{H}) = 1 + \text{ Codim }{T}_{\mu }({H}_{0}).$$

Hence

$$\text{ Dim Ker}(\mathcal{A}- \mu I) = 1 + \text{ Dim Ker}({A}_{0} - \mu I).$$

It follows that the μ-eigenspace of $\mathcal{A}$ is bigger than the μ-eigenspace of A₀. But this is impossible, since by construction, for any μ ≠ 0, the μ-eigenspace of A₀ is the entire μ-eigenspace of A. Thus the lemma is proved.

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

$${\sup \limits_{j}}\ \vert z{\vert }^{j}\|{K}^{j}\| < \infty, \quad \forall \ z \in \mathbb{C},$$

(A.5)

a condition that is in fact equivalent to Spec K = { 0}. To prove Theorem A.1, it suffices to demonstrate the following.

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

$$\varphi (z) = \text{ det}(I + zK),$$

(A.6)

which is well defined for trace class K. By (6.45) of Appendix A,

$$\vert \varphi (z)\vert \leq {C}_{\epsilon }{e}^{\epsilon \vert z\vert },\quad \forall \ \epsilon > 0.$$

(A.7)

Also, by Proposition 6.16 of Appendix A, φ(z) ≠ 0 whenever I + zK is invertible. Now, if K is quasi-nilpotent, then, as remarked above, I + zK is invertible for all $z \in \mathbb{C}$. Hence φ(z) is nowhere vanishing, so we can write

$$\varphi (z) = {e}^{f(z)},$$

(A.8)

with f(z) holomorphic on $\mathbb{C}$. Now (A.7) implies Re f(z) ≤ C_ε + ε |z| for all ε > 0, and a Harnack inequality argument applied to this gives

$$\vert \text{ Re }f(z)\vert \leq {C^\prime}_{\epsilon } + \epsilon \vert z\vert, \quad \forall \ \epsilon > 0,$$

(A.9)

See Chap. 3, §2, Exercises 13–16. The estimate (A.9) in turn (e.g., by Proposition 4.6 of Chap. 3) implies that Re f is constant, so f is constant, and hence φ is constant. But, by (6.41) of Appendix A, we have

$$\text{ Tr }K = \varphi ^\prime(0),$$

(A.10)

so the lemma is proved. Hence the proof of Theorem A.1 is complete.

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].