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A near-unit volume-operator mode amplifies forcing error

Example

Conditionally suppose a compact volume operator $K_{k}$ has a normalized eigenmode $ϕ$ with eigenvalue $λ \neq 1$ , and suppose $I - K_{k}$ is injective. The Fredholm branch then applies. Forcing along $ϕ$ is amplified by exactly $∣ 1 - λ ∣^{- 1}$ ; this example does not claim that every physical contrast has such an eigenvalue.

Verification

Given: AC, the compact volume operator $K_{k}$ , injectivity of $I - K_{k}$ , and $K_{k} ϕ = λ ϕ$ with $∥ ϕ ∥_{2} = 1$ and $λ \neq 1$ .

[F1] AC, compactness of $K_{k}$ , and injectivity of $I - K_{k}$ give a unique solution for each forcing and a bounded inverse (Fredholm solvability and residual stability for volume scattering).

[F2] The compactness hypothesis is supplied for the model by The volume integral scattering operator is compact.

1.1givenF1F2algebra

By [F2], the model operator is compact; together with the assumed injectivity and $λ \neq 1$ , the Fredholm result [F1] applies. The vector $u = (1 - λ)^{- 1} ϕ$ satisfies $(I - K_{k}) u = (1 - λ)^{- 1} (1 - λ) ϕ = ϕ$ . Uniqueness from [F1] identifies this with the solution for forcing $f = ϕ$ , so $∥ u ∥_{2} = ∣ 1 - λ ∣^{- 1}$ .

2.1givenstep 1.1algebra∎

For perturbed forcing $f_{δ} = (1 + δ) ϕ$ , linearity and the same eigenvector calculation give $u_{δ} = (1 + δ) (1 - λ)^{- 1} ϕ$ . Therefore $∥ u_{δ} - u ∥_{2} = ∣ δ ∣ / ∣ 1 - λ ∣$ , while $∥ f_{δ} - f ∥_{2} = ∣ δ ∣$ . As $λ$ approaches $1$ through values unequal to $1$ , this exact amplification factor grows without bound; the eigenpair remains a conditional hypothesis.

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Sources

Andreas Neubauer and Ronny Ramlau, Lecture Notes: Integral Equations and Boundary Value Problems, §3.3 (standard reference, not scraped)
Habib Ammari, Brian Fitzpatrick, and Sanghyeon Yu, Mathematical and Computational Methods in Photonics: Tutorial Notes, §1.3 (standard reference, not scraped)