A complete geometric classification of spectral configurations for two-particle Schrödinger operators on Z with a rank-three interaction

We study a family of lattice Schrödinger operators H_µ(K), describing two identical bosons on the one-dimensional lattice Z, where K ∈ T is the quasi-momentum. The interaction is described by the coupling vector µ = (µ₁, µ₂, µ₃), where µ₁ acts at the origin and µ₂, µ₃ at the sites |x| = 1.2. In this paper, we focus on the zero quasi-momentum fiber H_µ(0), whose essential spectrum is the interval [0, 8], while discrete eigenvalues may occur both below 0 and above 8.

Using the Fredholm determinant, we derive explicit threshold polynomials whose zero sets:

Γ^± := {µ ∈ R³: C^±(µ) = 0}

define three critical surfaces in R³. These surfaces partition the parameter space into regions with constant eigenvalue counts and identify exactly the parameter values at which eigenvalues emerge from, or are absorbed into, the thresholds 0 and 8.

A central result of this work is the complete geometric classification of the spectral configuration sets S = (n₋, n₊).  We show that the (µ₂, µ₃)-plane is partitioned into domains D^±, which strictly govern the range of the spectral counting functions. We prove that the parameter µ₃ acts as a global phase controller: the critical values µ₃  = ±2 separate qualitatively different spectral regimes, determining the set of all attainable configurations under the global rank-three constraint.

Finally, we extend our analysis to K  ≠ 0 and demonstrate that the discrete spectrum is preserved for  all quasi-momenta whenever |µ₃| > 2.

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