Spectral escape and mean crossover for concentrating signed potential forms
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Abstract
We study the spectrum of concentrating signed potential forms relative to derivative energy on a bounded interval. Classical weak-binding analysis identifies the role of profile primitives; here the interaction of finitely many fixed separated centers is retained through the full Dirichlet Green matrix. For nonzero even zero-mean profiles with inverse-square tails, we prove that exactly one positive and one negative eigenvalue per center diverge, with Green-matrix leading coefficients and mode-dependent logarithmic corrections. The entire remaining spectrum is uniformly bounded. Perturbing the profile means at the square-root scale gives a finite Hermitian crossover model, a two-term expansion for every escaping eigenvalue, and a quantitative operator-norm approximation. After energy normalization and a choice of phase, escaping modes converge uniformly to trace functions while their derivative energy develops explicit atomic concentrations; in the unperturbed case the diffuse and atomic parts each carry one half of the energy. A two-center example shows that ignoring off-diagonal Green interactions gives incorrect leading coefficients, and unequal tails produce different logarithmic drifts. Applications to localized log-curvature forms include an entire function with a strictly positive inverse Fourier density.
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Research details
- Subjects
- Mathematics · Functional Analysis · Spectral Theory
- Keywords
- signed potential forms, spectral concentration, Dirichlet Green matrix, eigenvalue asymptotics, Hardy inequality
- Manuscript date
- 2026-10-25
- Reader reuse
- none
Scope
Two-term full-operator spectral laws for finitely many fixed separated concentrating signed potential forms in bounded-interval derivative energy. Full Green interactions determine paired escaping eigenvalues and unequal-tail logarithmic corrections, with a uniformly bounded complementary spectrum. Square-root-scale mean perturbations yield a quantitative crossover model. Explicit eigenfunction profiles describe diffuse and atomic derivative-energy limits, including the unperturbed half-energy split. Log-curvature examples illustrate the interaction coefficients.
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 17-page manuscript and complete 12-member editable source package contain the analytic proofs, five TeX components, standard-library finite-verification code, build instructions, citation metadata and checksum manifest. Compile main.tex with an ordinary LaTeX distribution, or use python build.py --tectonic tectonic. Run python verify_green.py and python -O verify_green.py; both report 39 exact Green and mixed-block configurations. The analytic derivations establish the spectral asymptotics and eigenfunction energy-concentration law.