Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators
Loading original first page…
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 construction pays 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 complete localization errors.
Read the full paper in the original PDF.
Edition files
- Joint_cutoffs.pdf141.4 KiB · pdf
SHA-256
f40c73048da1829042f6f818f7a4bda4bad7ea2a9cae8dc8127b93f9b0ab43d7 - Joint_cutoffs_package.zip156.8 KiB · package
SHA-256
f93a4dea0a82585e393f2c81efdc3146c70aa91a61f8512140d14585d139a9ef
Available editions
Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Weil operator, negative spectrum, Schatten classes, joint cutoffs, finite-rank approximation, theta kernel, prime-phase recurrence
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
The results use the specified two-row constrained minimal-form realization H=A+B. The joint approximation-rank law is at fixed absolute tolerance. Slow-cutoff signed saturation and the fast-cutoff high-block gap have their explicitly stated, distinct parameter regimes. The fast estimate bounds the retained high-frequency block, with its complete localization errors.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.
Reproduction
The complete publication package includes the searchable manuscript PDF, self-contained LaTeX source, citation data, reader instructions, SHA-256 manifest and a standard-library Python verifier for exact finite scalar controls. The operator and asymptotic theorems are proved analytically in the manuscript.