Airy Bergman Kernel and Acoustic Horizons in One-Dimensional Bose–Einstein Condensates

This work builds a controlled local bridge between microscopic Bogoliubov theory and the universal Airy structures known from random-matrix models, topological recursion and two-dimensional quantum gravity. Within the Bogoliubov and local-density approximations, it derives an effective Airy operator for upstream phonons near the acoustic horizon of a one-dimensional Bose–Einstein condensate from the Gross–Pitaevskii equation, and shows that, in the associated Airy coordinate, the equal-time density–density correlator has a universal (X₁ − X₂)⁻² short-distance behaviour. Matching this singularity to the Airy Bergman kernel W₀,₂ yields a concrete two-point Airy topological-recursion dictionary for a microscopic quantum fluid in an analogue-gravity setting.