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

Abstract

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 = ( µ 1 , µ 2 , µ 3 ), where µ 1 acts at the origin and µ 2 , µ 3 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 ∈ R 3 : C ± (µ) = 0} define three critical surfaces in R 3 . 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 ( µ 2 , µ 3 )-plane is partitioned into domains D ± , which strictly govern the range of the spectral counting functions. We prove that the parameter µ 3  acts as a global phase controller: the critical values µ 3  = ±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 | µ 3 | > 2.