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Outgoing DtN coefficient at degree zero and the explicit PEC topology zero mode

Example

Assume the Axiom of Choice (The Axiom of Choice), as required by the cavity supplier (and hence its Countable-Choice harmonic supplier). The following are explicit mathematical verifications of clauses of Spherical Hankel functions: explicit solutions, constant Wronskian and exact Dirichlet-to-Neumann bounds and Spectral completeness of the curl form in a smooth PEC cavity; no measurement, physical identification or empirical claim is made.

(i) At degree $l = 0$ the outgoing radial function of the spherical Hankel lemma is $h_{0} (z) = - i e^{i z} / z$ , so for $k, R > 0$ the Dirichlet-to-Neumann coefficient of Spherical Hankel functions: explicit solutions, constant Wronskian and exact Dirichlet-to-Neumann bounds is

$t_{0} = \frac{k h_{0}^{'} (k R)}{h_{0} (k R)} = i k - \frac{1}{R},$

whose real part $- 1 / R$ is negative and whose imaginary part $k$ is positive, exactly as the exact bounds of that lemma require in degree zero.

(ii) Fix $0 < a < b$ . In the spherical shell $a < ∣ x ∣ < b$ , the field $u (x) = grad ∣ x ∣^{- 1} = - x / ∣ x ∣^{3}$ is smooth on the closure, nonzero, normal to both boundary spheres, curl-free and divergence-free: $u = \nabla Φ$ with $Φ (x) = ∣ x ∣^{- 1}$ harmonic off the origin, so $curl u = curl \nabla Φ = 0$ and $div u = Δ Φ = 0$ there. Thus $u$ lies in the PEC class $X$ of the cavity spectral lemma with zero curl, and it has curl eigenvalue zero: eliminating all topology zero modes would incorrectly discard this explicit admissible vector field.

Facts & Assumptions

Given: The Axiom of Choice, $k > 0$ , $R > 0$ , the shell $a < ∣ x ∣ < b$ with $0 < a < b$ , and the functions $h_{0} (z) = - i e^{i z} / z$ and $Φ (x) = ∣ x ∣^{- 1}$ .

[L1]

The degree-zero outgoing radial function is $h_{0} (z) = - i e^{i z} / z$ and the DtN coefficients of the spherical Hankel lemma satisfy $- (l + 1) / R \leq Re t_{l} \leq - 1 / R$ and $0 < Im t_{l} \leq k$ , with $t_{l} = k h_{l}^{'} (k R) / h_{l} (k R)$ (Spherical Hankel functions: explicit solutions, constant Wronskian and exact Dirichlet-to-Neumann bounds).

[L2]

The curl of a gradient vanishes on its domain (The curl of the gradient of a $C^{2}$ function vanishes), and $Φ (x) = ∣ x ∣^{- 1}$ is smooth and harmonic on $R^{3} ∖ {0}$ , so $Δ Φ = 0$ there (The Laplace fundamental solution is harmonic off its pole).

[L3]

The complex exponential is entire with derivative itself and the algebra of derivatives holds (The complex exponential is entire and its complex derivative is itself, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L4]

The graph completion $X$ of Spectral completeness of the curl form in a smooth PEC cavity contains every smooth field on the closure with $ν \times u = 0$ , and the curl form there has nonnegative eigenvalues with the topology zero modes retained.

Verification

technique · direct

1.1givenL1L3algebra

Logarithmic differentiation of $h_{0} (z) = - i e^{i z} / z$ by [L3] gives $h_{0}^{'} (z) = - i e^{i z} \cdot i / z + i e^{i z} / z^{2} = h_{0} (z) (i - 1 / z)$ ; hence $k h_{0}^{'} (k R) / h_{0} (k R) = k (i - 1 / (k R)) = i k - 1 / R$ , so $Re t_{0} = - 1 / R < 0$ and $Im t_{0} = k > 0$ , consistent with [L1].

1.2givenL2L4algebra

Since $Φ$ is smooth on the shell and $u = \nabla Φ$ , [L2] gives $curl u = 0$ and $div u = Δ Φ = 0$ on $a < ∣ x ∣ < b$ ; on the boundary sphere $∣ x ∣ = c$ the field is $u = - x / c^{3}$ , a nonzero multiple of the outward normal, so its tangential part vanishes and $u$ is in the PEC class $X$ by [L4].

2.1step 1.2L4algebra∎

Because $curl u = 0$ and $u \neq 0$ , the field $u$ belongs to the kernel of the curl form on $V = {w \in X : div w = 0}$ : the eigenidentity of [L4] holds with $λ = 0$ , so the zero eigenspace is nontrivial and must be retained; discarding all topology zero modes would delete this admissible vector field.

The computation in (i) is the degree-zero instance of the exact DtN bounds; the field in (ii) is the explicit topology zero mode of the spherical shell, and no claim is made about other eigenvalues, about lossy media or about maximal tangential-trace realisations beyond the class $X$ of Spectral completeness of the curl form in a smooth PEC cavity.

Depends on

Dependency tree · 2 levels

Sources

NIST Digital Library of Mathematical Functions, Section 10.47 (spherical Bessel and Hankel functions)