Alphabeta MathHow statement and proof provenance work
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Spherical Bessel and outgoing Hankel function conventions
Definition
For real , let and define, for integers , The definition fixes the outgoing convention. The functions solve the radial Helmholtz equation and have the recurrence, origin, Wronskian, and infinity behavior by Spherical Bessel and outgoing Hankel recurrences solve the radial Helmholtz equation ↗.
These symbolic definitions assert no stable or certified numerical evaluation algorithm for large or .
Depends on
Used by
- Exact modal Dirichlet-to-Neumann map on a sphere Definition
- Scaled spherical Bessel evaluation on positive real arguments Definition
- Exact matching for the radial square well in the zeroth partial wave Example
- The monopole symbol of the exact spherical DtN map Example
- The outgoing monopole field of a sound-soft sphere Example
- Every outgoing spherical Helmholtz mode is an outgoing Hankel mode Lemma
- Outgoing spherical Hankel modes have positive flux and no positive-real zeros Lemma
- Spherical Bessel and outgoing Hankel recurrences solve the radial Helmholtz equation Lemma
- Spherical Hankel modulus and uniform radial bounds Lemma
- Radial volume scattering separates into matched partial-wave boundary problems Theorem
- Soundness and eventual absolute precision for spherical Bessel enclosures Theorem
Dependency tree · two levels
12 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
- Maximilian Kreuzer, Quantum Theory: Lecture Notes, Chapter 8 (standard reference, not scraped)
- Pierre Capel, Introduction to Scattering Theory, TALENT lecture notes (standard reference, not scraped)