Relative tail payment and spectral enclosures
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Abstract
We sharpen the relative tail-payment estimate for a self-adjoint diagonal operator perturbed by a bounded self-adjoint operator with bounded first commutator. A smooth spectral cutoff replaces the growing-band argument and gives complete relative tail payment at the threshold A ≥ max{16P, (1152K²/δ)^(1/3)}, where P bounds the perturbation, K bounds its first commutator, and δ is the positive finite-head margin. The estimate retains the original finite head and the complete receiving tail. A shifted version yields indexed lower enclosures for every low Ritz eigenvalue, including an indefinite finite head. A fixed rank-two perturbation of a compact-resolvent diagonal operator shows that the cubic truncation power is optimal in this first-commutator class. Jacobi examples illustrate the resulting enclosures with explicit commutator budgets. This supplement develops the relative tail-payment method of the arithmetic matrix criteria predecessor.
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- Relative_tail_payment_and_spectral_enclosures.pdf90.1 KiB · pdf
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Available editions
- Edition 1 · 2026-10-29 · public
Research details
- Subjects
- Mathematics · Spectral Theory · Numerical Analysis
- Keywords
- bounded commutator, spectral truncation, Schur complement, Ritz eigenvalue, relative tail estimate
- Manuscript date
- 2026-10-29
- Reader reuse
- none
Scope
First-commutator relative tail estimates for bounded self-adjoint perturbations preserving the diagonal operator domain, indexed Ritz eigenvalue enclosures, and a fixed-operator optimal-power example. Mathematical supplement to the cited arithmetic matrix criteria paper.
Disclosures
The author used OpenAI ChatGPT to assist with mathematical exploration, proof checking, literature comparison, and preparation of the text. The author takes responsibility for the mathematical statements and presentation.
Reproduction
The PDF contains the complete mathematical argument. The accompanying package includes editable LaTeX source, build_pdf.py, reader instructions, citation metadata and a SHA-256 member manifest.
References and related resources
- S. Schubr, The Riemann Hypothesis: Arithmetic Matrix Criteria and Certified Weil Bounds, edition 7 (2026). · isSupplementTo
- J. Zhang, Spectral localization estimates for abstract linear Schrödinger equations (2024). · references
- H. Yu and T.-C. Wei, Universal energy-space localization and stable quantum phases against time-dependent perturbations (2026). · references