Precise spectral asymptotics, exponential localization, and spectral gap estimates for the three-boson lattice Schrödinger operator

Abstract

We present a corrected strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K=pi. First, we give an exact closed-form benchmark for the fiber Fredholm determinant at a flat momentum, valid for every K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion that decides whether a leading-order Fredholm determinant asymptotic suffices to fix the constant-order additive energy correction. Third, applying the criterion to the formal branch z=-2mu+d at K=pi, we identify an algebraic crossing -2mu+6+8/mu+O(mu^{-2}) from the finite-rank principal part. However, we 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, and so lies in the same leading branch -3mu+O(1) as at K=0. Direct finite-volume diagonalisation of the full three-particle Hamiltonian confirms the refined asymptotic -3mu+6+O(mu^{-1}) and the spectral gap 2mu-2+O(mu^{-1}) to the two-particle threshold. The reduction from two bound states at K=0 to at least one at K=pi (the trimer) preserves the total spectral flow, and the binding is not weakened at the corner of the Brillouin zone. We independently confirm that the K=0 constant C approximately 3.96458 requires no analogous refinement.