Spectral transitions for concentrated zero-mean energy forms
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Abstract
We study the Riesz operators of concentrating signed potential forms in the derivative-energy space H₀¹(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 L² 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.
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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
Bounded-interval derivative-energy Riesz operators with finitely many fixed separated centers and the stated zero-mean inverse-square-tail hypotheses. Includes full mixed Green geometry, bounded residual spectrum, equal-tail logarithmic drift, critical mean crossover and 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.
Reproduction
The supplementary ZIP contains the reviewed manuscript PDF, complete editable LaTeX sources, portable build instructions, citation metadata, a checksum manifest and a Python standard-library verifier. Normal and optimized runs check 37 exact Green and mixed-block configurations. These bounded algebraic checks support the normalization and finite geometry; the analytic spectral claims are proved in the manuscript.