Spectral escape for interacting and coalescing signed potential forms
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Abstract
We determine how the spectrum of a concentrating signed potential form splits between escaping eigenvalues and a bounded remainder as its centers coalesce. Zero profile integrals alone do not rule out escape, and relative leading-order estimates need not determine logarithmic shifts to bounded error. For two unequal asymmetric zero-mean profiles separated by more than their width, we obtain logarithmic formulas with bounded additive remainders on both escaping scales throughout the mesoscopic collision regime. Exact primitive correlations and Green geometry, together with a cancellation in a quadratic spectral pencil, control mixed trace terms that can exceed the slower scale. For a noncancelling cluster at bounded separation-to-width ratios and zero background, we identify the operator remaining after removal of the actual escaping pair. Aggregate inverse-square tails determine its essential and absolutely continuous channels; vanishing aggregate tails give trace-norm convergence. Fixed-threshold eigenvalue counts and signed spectral masses quantify how a bounded bulk balances the escaping pair's logarithmic trace drift. At exact cancellation, a primitive estimate controls the complete operator for measurable profiles. An anchored cusp family exhibits regularity-dependent transition scales and a logarithmic endpoint, with a vanishing rank-two approximation error along every joint path. Identical even profiles supply explicit scalar matching through all collision ratios.
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Research details
- Subjects
- Mathematics · Functional Analysis · Spectral Theory
- Keywords
- signed potential forms, spectral concentration, Dirichlet Green matrix, coalescing centers, eigenvalue asymptotics, Hardy inequality
- Manuscript date
- 2026-10-25
- Reader reuse
- none
Scope
Unequal asymmetric zero-mean profiles in bounded-interval derivative energy admit two-scale mesoscopic logarithmic laws with bounded additive errors. For noncancelling bounded-ratio clusters with zero background, actual spectral deflation identifies the surviving compressed operator; aggregate inverse-square tails determine its channels, and vanishing aggregate tails give trace-norm convergence. Signed fixed-threshold counts and spectral masses explain trace compensation. A full primitive estimate treats measurable cancellation, and an anchored cusp family gives regularity-dependent all-path transition scales. Identical even profiles admit explicit scalar matching through all collision ratios.
Disclosures
ChatGPT (OpenAI) was used to generate and revise manuscript drafts, propose proof developments and strengthenings, compare literature arguments, develop verification code, and assist with internal proof review. The author directed the research and revision decisions and is responsible for the mathematical content and presentation. This work received no external funding. The author’s organization, Luminary Oddities, operates the Luminary Research Archive hosting the preprint. The author declares no other relevant competing interests.
Reproduction
The 30-page manuscript and 14-member source package contain complete analytic proofs in seven TeX files, standard-library finite-verification code, build instructions, citation metadata and checksums. Compile main.tex with a LaTeX distribution or run python build.py --tectonic tectonic. The code python verify_green.py, also under python -O, checks 39 exact finite Green and mixed-block configurations; the analytic proofs establish the spectral asymptotics.