DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact modal Dirichlet-to-Neumann map on a sphere

Definition

For $k > 0$ , $R > 0$ , and $g \in C^{\infty} (S^{2}; C)$ , let $g_{ℓ}$ be its $L^{2} (S^{2})$ -orthogonal projection onto $H_{ℓ}$ and set $τ_{ℓ} (k, R) = k \frac{(h_{ℓ}^{(1)})^{'} (k R)}{h_{ℓ}^{(1)} (k R)}, T_{k, R} g = \sum_{ℓ = 0}^{\infty} τ_{ℓ} (k, R) g_{ℓ} .$ The associated outgoing extension for $r \geq R$ is $v (r, ω) = \sum_{ℓ = 0}^{\infty} \frac{h_{ℓ}^{(1)} (k r)}{h_{ℓ}^{(1)} (k R)} g_{ℓ} (ω) .$ The denominator is nonzero by Outgoing spherical Hankel modes have positive flux and no positive-real zeros. The convergence, smoothness, PDE, and radiation properties are asserted in The spherical modal DtN map is well-defined and exact for smooth data ↗.

Depends on

Used by

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 10: Bessel Functions (standard reference, not scraped)
Plamen Stefanov, Scattering and Inverse Scattering in R^n, Appendix 1, §§1.1–1.2 (standard reference, not scraped)