Second-order dissociation of a Hubbard bound pair in a common spatially correlated lattice potential

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.