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. We determine the logarithmic correction for every mode with unequal critical tails and prove that all escaping branches are simple. Mean perturbations at the square-root scale yield a complete finite Hermitian crossover model, its two-term eigenvalue expansion, and a quantified operator-norm remainder under 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.
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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
- none
Scope
Two-term spectral escape and critical mean-crossover laws for finitely many fixed separated concentrating signed potential forms in bounded-interval derivative energy, including unequal critical tails, simple branches, full Green interactions and a uniformly bounded spectral reserve. Log-curvature applications and real-root Hardy spectral classification with a fusion norm obstruction.
Disclosures
ChatGPT (OpenAI) assisted mathematical exploration and strengthening, symbolic reasoning, comparison of source arguments, internal proof review, verification-code development and manuscript preparation. The author is responsible for the mathematical content and presentation.
Reproduction
The 15-page manuscript and complete 12-member source package contain the proofs, five TeX components, standard-library verifier, build instructions, citation metadata and checksum manifest. The introduction develops the log-curvature energy quotient, an interacting two-center example and the trace-decomposition mechanism. Appendix A supplies the supporting real-root Hardy and fusion comparison. 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.