Selberg–Residue Analysis of the Equally Spaced Richardson–Gaudin Pairing Model

Abstract

Authorship and verification statement. The scientific problem, the original methodological direction, and the research objectives were formulated by W. V. Pogosov. The analytical derivations and numerical computations in this project were carried out by an advanced large language model (LLM), through an iterative interaction with the author. The author has not exhaustively checked all calculations and proofs. This is a working manuscript with explicit unresolved assumptions, supplied for inspection, reproduction, and further development. The manuscript develops a Selberg–residue approach to the equally spaced Richardson–Gaudin pairing model. Meromorphic contour periods are reduced to finite determinant expressions. Complementary kernels and sign-corrected minor expansions organize ground-state, marked, blocked, and nested-contour sectors. The text studies their thermodynamic asymptotics and candidate finite-size energy corrections while distinguishing exact period identities from physical eigenenergy interpretations. The method originated in W. V. Pogosov, “'Probabilistic' approach to Richardson equations,” Journal of Physics: Condensed Matter 24, 075701 (2012), doi:10.1088/0953-8984/24/7/075701 (https://doi.org/10.1088/0953-8984/24/7/075701), arXiv:1111.2907 (https://arxiv.org/abs/1111.2907). That work treated the ground state and obtained the required coupling dependence without a full direct evaluation of the determinant normalization. Within this line of work, it also first formulated explicitly the Hamiltonian origin of electron–hole symmetry in the exact Richardson analysis; its Appendix A acknowledges earlier observations of spectral duality. The subsequent paper, “Excited states in Richardson pairing model: 'probabilistic' approach,” Progress of Theoretical Physics 128, 1–14 (2012), doi:10.1143/PTP.128.1 (https://doi.org/10.1143/PTP.128.1), arXiv:1208.1088 (https://arxiv.org/abs/1208.1088), extended the approach to excited states, again leaving a common determinant normalization unevaluated directly. The author's 2014 doctoral dissertation, Superconductors and dilute superfluid Bose systems: From the microscopic to the macroscopic scale (in Russian), developed direct determinant evaluations by successive row differences and triangular reduction. See the Lebedev Physical Institute dissertation-council record (https://www.lebedev.ru/ru/fian-dissertation-councils/posts.html?id=188) and the complete thesis on ResearchGate (https://www.researchgate.net/publication/271840408_Sverhprovodniki_i_razrezennye_sverhtekucie_boze-sistemy_ot_mikro-_k_makrourovnu) (doi:10.13140/2.1.1066.5766 (https://doi.org/10.13140/2.1.1066.5766)). From the perspective of the broader program pursued here, these earlier results constituted a partial solution. The present project attempts to carry that program through; completion is not claimed. Version 1.0 incorporates a repair of the confluent lattice-minor argument, using a saddle-localized continuation and an exponentially small error controlled on integer indices. Important open inputs remain: contour selection, uniform Gaussian energy control, the proposed Hessian-to-spectral-shift identification, and finite-defect comparisons. The correspondence between the period endpoint and the physical real-root density is also conjectural. The deposit contains the full manuscript, LaTeX sources, bibliography, fifteen supporting scientific programs, fourteen labelled historical reports, and a current separately implemented lattice-check report. The latter records 96 small exact-arithmetic comparisons and 24 larger minor and derivative checks. Independence here concerns implementations within the LLM-assisted workflow, not independent human verification. The earlier program collection has not been rerun in full for this release, and finite tests do not prove the unresolved analytical assumptions.