A uniform elliptic reduction, an order-matching criterion, and precision benchmarks for the strong-coupling Birman-Schwinger analysis of the lattice three-boson trimer
Abstract
We present a rigorous strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant, valid for every quasimomentum K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but demonstrate that this crossing does not correspond to a true eigenvalue of the full Hamiltonian. The actual ground state obeys the rigorous variational bounds -3mu <= z_1^{pi,s}(mu) <= -3mu + 6, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare the asymptotic precision levels achieved across recent lattice few-body models and draw a structural parallel with the parity-based classification of topological band insulators at time-reversal-invariant momenta.
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BibTeX
@misc{dolgopolov2026uniform,
title = {{A uniform elliptic reduction, an order-matching criterion, and precision benchmarks for the strong-coupling Birman-Schwinger analysis of the lattice three-boson trimer}},
author = {Mikhail Dolgopolov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11848},
eprint = {2608.11848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11848},
}
RIS
TY - UNPB
TI - A uniform elliptic reduction, an order-matching criterion, and precision benchmarks for the strong-coupling Birman-Schwinger analysis of the lattice three-boson trimer
AU - Mikhail Dolgopolov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.11848
UR - https://arxiv.org/abs/2608.11848
AB - We present a rigorous strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant, valid for every quasimomentum K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but demonstrate that this crossing does not correspond to a true eigenvalue of the full Hamiltonian. The actual ground state obeys the rigorous variational bounds -3mu <= z_1^{pi,s}(mu) <= -3mu + 6, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare the asymptotic precision levels achieved across recent lattice few-body models and draw a structural parallel with the parity-based classification of topological band insulators at time-reversal-invariant momenta.
ER -