We consider two bosons in the one-dimensional Bose–Hubbard model with hopping amplitude J and constant on-site repulsion U, both subject to the same weak, static, stationary Gaussian lattice potential. For a repulsively bound pair of total quasimomentum K we derive in closed form the second-order (golden-rule) kernel for transitions into the interacting two-particle scattering continuum at transferred momentum q. The kernel is strictly positive throughout the kinematically open region and vanishes as a square root at its edges. Its continuum factor coincides with that of the known dissociation rate under hopping modulation, while the static density coupling contributes a momentum-dependent form factor and a folded on-shell support. For flat high-pass disorder spectra the kernel gives an exact classification of when dissociation and elastic backscattering vanish at order W² in the disorder strength. For π/2 < |K| < π and U < 4J sin(|K|/2), a critical interaction U* = 4J√(−cos K) separates pairs whose second-order dissociation can be removed by a finite spectral gap from those for which it cannot, and there is a window in which backscattering is closed while dissociation stays open. As the gap edge q0 approaches the largest open transfer qc, the coefficient vanishes as Δ^(3/2), Δ^(1/2) or Δ, with Δ = qc − q0, or tends to a nonzero plateau, depending on the geometry of the open region and on whether the pass-band height or the on-site variance is held fixed. Exact finite-ring simulations of the bound-band population reproduce the predicted enhancement and suppression for the two normalizations. This preprint includes the complete analytic proofs and numerical-check appendices. It is not an accepted journal publication. The PDF is unchanged from the author’s final manuscript dated 15 October 2026. Supporting data and code: 10.5281/zenodo.23390681. Public repository with the manuscript, source, figures and reproduction instructions: https://github.com/SHIN09-git/hubbard-pair-correlated-disorder. Generative AI was used extensively in the research and manuscript preparation. The manuscript and the repository’s AI-use record disclose the reported model roles, human verification scope and limitations. JIN HANZHE is the sole human author and takes responsibility for the work.
We consider two bosons in the one-dimensional Bose–Hubbard model with hopping amplitude J and constant on-site repulsion U, both subject to the same weak, static, stationary Gaussian lattice potential. For a repulsively bound pair of total quasimomentum K we derive in closed form the second-order (golden-rule) kernel for transitions into the interacting two-particle scattering continuum at transferred momentum q. The kernel is strictly positive throughout the kinematically open region and vanishes as a square root at its edges. Its continuum factor coincides with that of the known dissociation rate under hopping modulation, while the static density coupling contributes a momentum-dependent form factor and a folded on-shell support. For flat high-pass disorder spectra the kernel gives an exact classification of when dissociation and elastic backscattering vanish at order W² in the disorder strength. For π/2 < |K| < π and U < 4J sin(|K|/2), a critical interaction U* = 4J√(−cos K) separates pairs whose second-order dissociation can be removed by a finite spectral gap from those for which it cannot, and there is a window in which backscattering is closed while dissociation stays open. As the gap edge q0 approaches the largest open transfer qc, the coefficient vanishes as Δ^(3/2), Δ^(1/2) or Δ, with Δ = qc − q0, or tends to a nonzero plateau, depending on the geometry of the open region and on whether the pass-band height or the on-site variance is held fixed. Exact finite-ring simulations of the bound-band population reproduce the predicted enhancement and suppression for the two normalizations. This preprint includes the complete analytic proofs and numerical-check appendices. It is not an accepted journal publication. The PDF is unchanged from the author’s final manuscript dated 15 October 2026. Supporting data and code: 10.5281/zenodo.23390681. Public repository with the manuscript, source, figures and reproduction instructions: https://github.com/SHIN09-git/hubbard-pair-correlated-disorder. Generative AI was used extensively in the research and manuscript preparation. The manuscript and the repository’s AI-use record disclose the reported model roles, human verification scope and limitations. JIN HANZHE is the sole human author and takes responsibility for the work.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.
Repulsively interacting particles in a periodic potential can form bound composite objects with long lifetimes, as has been observed experimentally [Winkler et al., Nature (London) 441, 853 (2006)]. In this paper, a complete two-particle solution of one-dimensional periodical potential was derived in a strong interaction regime, where the on-site approximation of a two-body interaction in the Bose-Hubbard model (BHM) is extended to include the interaction of nearest neighbors, which results in atom-pair hopping. The energy spectrum of the bound pair is drastically reshaped due to the pair-hopping term, and complex eigenenergy corresponding to metastable states is also found that has not been discovered in the usual BHM. When the absolute value of a center-of-mass quasimomentum wave vector is greater than a critical value (∣K∣>Kc), two bound-pair solutions are found. Furthermore, the spatial and momentum distributions of the bound pair displays a crossover from dark to bright soliton-like solutions in the extended BHM. Our results reduce to that of the usual BHM in the weak interaction case.
The coherent mobility of doublons, arising from second-order virtual dissociation-recombination processes, fundamentally limits their use as information carriers in the strongly interacting Bose-Hubbard model. We propose a disorder-free suppression mechanism by introducing an optimized nearest-neighbor pair-hopping term that destructively interferes with the dominant virtual hopping channel. Using the third-order Schrieffer-Wolff transformation, we derive an analytical optimal condition that accounts for lattice geometry corrections. Exact numerical simulations demonstrate that this optimized scheme achieves near-complete dynamical arrest and entanglement preservation in one-dimensional chains, while in two-dimensional square lattices, it significantly suppresses ballistic spreading yet permits a slow residual expansion. Furthermore, in the many-body regime, finite-size scaling analysis identifies the observed long-lived density-wave order as a prethermal plateau emerging from the dramatic separation of microscopic and thermalization timescales.
Unrefereed AI-assisted preprint, version 0.2.2, 9 October 2026. No author affiliation is given. For each m >= 1, set q = m^2 + m + 1. Attach q intermediate vertices to a root and q - 1 leaves to each intermediate vertex. This produces infinitely many pairwise nonisomorphic non-star trees of order q^2 + 1, with root Laplacian spectral support {0, m^2 + 1, (m + 1)^2 + 1}. The symbolic argument establishes Laplacian peak state transfer at time pi from the root to an intermediate vertex for odd m and to a leaf for even m, addressing Open Problem 7.1 of Coutinho, Guo and Schmeits, arXiv:2505.11986v4. The ten-vertex m = 1 member is their known Figure 10 example. Phase alignment is an application of their Lemma 5.2 and Corollary 5.3; the general peak-time classification follows their Theorem 5.4 and Corollary 5.5. The contribution presented here is the explicit infinite family and its spectral coefficients, targets and gcd calculation. For the specified pairs, all positive peak times are odd multiples of pi/g, where g = 5 when m = 2 mod 5 and otherwise g = 1. Novelty and priority are not certified. Peak transfer means attaining the mutual-projector absolute-sum bound, not unit amplitude. The probabilities tend to zero along both parity subsequences. This does not establish perfect transfer, efficient long-distance quantum communication or a classification of all trees. Exact standard-library reruns check actual full-tree Laplacians for m = 1..5, quotients for m = 1..1000, and gcd/phase regressions for m = 1..10000. Those finite checks support, not replace, the symbolic all-parameter proof and the analytic all-real-time Bezout argument. No complete Lean or other proof-assistant verification is supplied. 本研究は、清水竣平が個人の自主研究として企画・進行した、AIを活用した数学探索です。清水は探索方針の指示、検証の手配、成果の取りまとめを担当しました。証明の構成、計算による検証、原稿作成の大部分にはAIを使用しています。検証方法とその範囲は本文に記載しており、人間の専門家による査読は未実施です。 This work is an AI-assisted mathematical exploration initiated and coordinated by Shunpei Shimizu as a personal independent research project. Shimizu directed the exploration, arranged verification, and compiled the results. AI systems performed most proof construction, computational checking, and manuscript preparation. The verification methods and their scope are documented in the manuscript. The work has not received specialist human peer review. The submitter has not independently certified every mathematical step. Separate AI analyses and implementations are not human refereeing. No cited-author or institutional endorsement is implied. Read peak_trees_en_v0.2.2.tex first. The current package is Quantum_peak_trees_publication_v0.2.2_20261009.zip; QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md explains its contents. Version 0.2.2 removes author affiliation; mathematical text, equations, statements, proofs and verification programs are unchanged from version 0.2.1. The current package omits old manuscript sources and old full ZIPs. Earlier verification records retain their original dates; they are not human peer review. The built-in LaTeX compiler failed at Windows environment setup before TeX diagnostics. No PDF has been generated or visually verified; editable TeX and exact-check code are supplied. Documentation clarification, 10 October 2026. The retained README_EN.txt and README_JA.md are historical documentation; the corresponding preserved v0.1.0 copies describe the 8 October 2026 working draft. QUANTUM_TREES_README_PUBLICATION_v0.2.1_EN.md describes the superseded v0.2.1 package. Their source names, payload inventories, draft-status statements and no-Publish instructions are dated history, not the current record state or operative instructions. Independent checks in those documents mean separate AI analyses or implementations, not human refereeing. The current entrypoints are peak_trees_en_v0.2.2.tex, Quantum_peak_trees_publication_v0.2.2_20261009.zip and QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md. Refer to the current TeX for the argument where historical checker comments mention an absent review.md. This notice makes no change to the mathematical manuscript or verification programs. Publication packaging clarification, 10 October 2026. In addition to the three historical READMEs identified above, the two QUANTUM_TREES_ATTRIBUTION_AI_ROLE_ADDENDUM files and novelty_attribution_math_monitor_review_20261009.zip are dated supplementary review records. The r1 addendum and the current v0.2.2 manuscript give the operative author and AI contribution disclosure. The novelty archive also compares other manuscripts; those comparisons are context, not additional results of this Quantum record. Earlier private-save and no-publication instructions describe past workflow only. Current scientific claims are the explicitly stated tree-family construction and specified-pair peak times. No mathematical content, verification program or frozen historical attachment is changed by this clarification.
We explicitly carry through to the Hubbard endpoint the product-state (Hartree-Fock-type) case that Waite et al. state is available within their framework. The closest route is their Theorem 7 with a subset of cardinality one. We give explicit constants, track each guiding spinor through the reduction, and provide a finite description of the resulting site-local Slater determinant. The target is a repulsive Fermi-Hubbard Hamiltonian on a full open rectangular square grid, with spatially uniform positive hopping and repulsion, local vector fields, and the entire fixed-number sector Nf = M. With inverse-polynomial squared overlap and normalized decision precision, the resulting promise problem is PromiseBQP-complete; hardness holds on the subclass with a unique sector ground state and an inverse-polynomial spectral gap. The guiding spinors are retained from their creation-time penalty ground states and rounded to normalized dyadic-pair descriptions. The proof includes explicit global perturbation bounds and preparation and energy-estimation costs. MSC2020: Primary 68Q12; Secondary 81P68, 81V70. The archive contains the complete LaTeX source, vector figures, numerical test data, build instructions, a manifest, and SHA-256 digests. AI-use disclosure: The natural-language proofs, diagnostic code, partial formal proofs, and manuscript were generated using GPT-6 Astra and Claude Fable 5.1 under the author's direction on the research question and acceptance criteria. AI review comprised four independent audits and one heterogeneous adversarial audit across the two model families; completed human mathematical proof reviews number zero. Formal checking is peripheral: five source-side and five finite-endpoint declarations have Lean kernel proofs, Schrieffer-Wolff composition is zero of eight original targets, and one overgeneral transfer signature has a singleton counterexample. The full uniform completeness theorem is not formally proved. The author supplied the presentation revisions and remains responsible for final acceptance of the work. Appendix C distinguishes the earlier high-precision numerical example from the new certified Arb interval calculation.