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 refined 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 that rigorously determines whether a leading-order Fredholm determinant asymptotic, with relative error 1/mu, suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion at K = pi, we derive the complete strong-coupling ground-state asymptotic expansion z_1^{pi,s}(mu) = -2mu + 6 + 8/mu + O(mu^{-2}), cross-validated against the exact benchmark and two independent high-precision numerical schemes. We rigorously establish the corresponding spectral gap and the Fredholm-determinant underestimation factor 2 + O(mu^{-2}). We independently confirm that the known K = 0 constant requires no analogous refinement. Finally, we contrast 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.