Spectral escape and mean crossover for concentrating signed potential forms
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Abstract
We study the Riesz operators of concentrating signed potential forms in the derivative-energy space H_0^1(a,b). For finitely many fixed separated centers and nonzero even zero-mean profiles with inverse-square tails, exactly one positive and one negative eigenvalue per center escape to infinity. Their leading coefficients are the square roots of the eigenvalues of a full Dirichlet Green matrix weighted by the squared L^2 norms of the profile primitives. The remaining spectrum is uniformly bounded. Equal critical tails give a common logarithmic correction, and mean perturbations at the square-root scale yield a complete finite Hermitian crossover model with an operator-norm remainder under explicit moving isometric embeddings. We apply the results to localized log-curvature forms, including a positive-Fourier-inverse example with different even and odd escape coefficients. For real zeros, a trace-class Hardy localization identifies the absolutely continuous multiplicities and explains strong convergence with an unavoidable norm gap under genuine root fusion. The results concern bounded-interval energy operators; they assert no full-line theta sign theorem or polynomial-degree rate.
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Research details
- Subjects
- Mathematics · Functional Analysis · Spectral Theory
- Keywords
- spectral theory, concentrating signed potentials, Dirichlet Green matrix, derivative energy, Hardy inequality, zero-mean profiles
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Complete leading spectral escape and mean-crossover models for specified concentrating signed energy forms on bounded intervals, plus Hardy spectral analysis for real-root fusion.
Disclosures
AI-assisted symbolic reasoning, mathematical literature comparison, proof checking and manuscript preparation were used. The author is responsible for the statements and presentation. This preprint does not claim external peer review or formal proof verification.
Reproduction
11-page manuscript and 12 source-package members. Use python build.py --tectonic tectonic to compile the included assembly. Standard-library python verify_green.py, also under -O, checks finite Green/stiffness, parity and noncommuting crossover block identities.