Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators
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Abstract
We obtain a joint finite-rank approximation law for the complete Birman–Schwinger family of a constrained theta-weighted Weil operator. At each fixed absolute error tolerance, the required rank is at most C exp(C√(log(1/ε))) as the positive resolvent parameter ε tends to zero. The approximation controls the complete paired arithmetic tail, high-frequency block and mixed localization errors. It gives a corresponding bound on the entire negative spectral projection and implies that the negative eigenvalue magnitudes decay as C exp[−c log²(n+1)]. Thus the full negative part belongs to every Schatten class. We separate two arithmetic cutoff regimes. Effective simultaneous prime-phase returns give signed high-frequency saturation when the retained cutoff grows sufficiently slowly. In the faster regime of the rank construction, theta confinement and compressed-shift estimates instead give a strict high-block norm gap: its normalized upper bound is at most three quarters for every positive limiting arithmetic ratio, with a stronger branch above the theta-overlap threshold. Both arguments retain the original constraints, weighted norms and all localization errors.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Weil operators, finite-rank approximation, negative eigenvalues, Schatten classes, theta kernel
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
At fixed absolute tolerance the specified complete Birman–Schwinger family admits approximants of rank at most C exp(C√log(1/ε)); the whole negative part has stretched-log eigenvalue decay and belongs to every Schatten class. Slow-cutoff signed saturation and a strict fast-cutoff high-block gap hold in their stated joint regimes.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author owns Luminary Oddities and its Research Archive, where an author preprint of this work is available. No other relevant competing interests are declared.
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References and related resources
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