DEF0244canonicalv1Spectral Parameter Λ(s)
The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) holds, then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L.
Payload
Spectral Parameter Λ(s)
The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) holds, then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L.
Spectral Parameter Λ(s)
Summary
The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) holds, then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L.
Statement
\label{def:spectral-parameter}
The \textbf{spectral parameter} is the quadratic map
\[
\Lambda(s) = \iota_\tau^2\left(s(1-s) - \tfrac{1}{4}\right),
\]
where $\iota_\tau = \tfrac{2}{\pi + e} \approx 0.341304$ is the master constant.
Proof / Justification
This item is definitional. No manuscript proof is required.
Source Context
- Registry source:
book-03.jsonlline 67 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch24-the-spectral-correspondence.texlines 33-39
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Doors.SpectralCorrespondence - Name:
spectral_parameter
Dependencies
- Canonical: III.D26, III.D28
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.D29spectral-parameter-sdef:spectral-parameterRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (2)
Appears in (1)
Downstream uses (computed) (4)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
Sources
Version & History
Status disclaimer
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