We study the three-boson Schrodinger operator on the two-dimensional integer lattice with pairwise contact interactions. First, we obtain precise asymptotics of the two bound states below the essential spectrum at zero total quasimomentum in the strong-coupling limit mu to infinity: the ground state energy is -3 mu + 6 + O(1/mu), and the first excited state energy is - mu + C + O(1/mu), where C = 4 - delta approximately 3.96458, with delta determined by the transcendental equation b0(delta) = 1/(1+delta) involving the lattice Green's function. The associated spectral gap is 2 mu + O(1). Second, using the discrete Agmon comparison method and the Paley-Wiener theorem, we establish exponential localization of the ground-state wavefunction with a logarithmic upper bound on the decay rate alpha(mu) at most ln(3 mu) + O(1/mu), reflecting the bounded nature of the lattice dispersion; the sharp asymptotic rate is conjectured. Third, at quasimomentum pi, the parity symmetry is preserved, but the odd quadratic form becomes positive definite with a unique eigenvalue of order 1/mu that never reaches the Birman-Schwinger threshold, so the odd subspace yields only a virtual level. The even subspace supports exactly one bound state with energy -2 mu + 6 + 8/mu + O(1/mu^2), and the spectral gap to the essential spectrum is mu - 2 - 8/mu + O(1/mu^2). The reduction from two bound states at zero to one at pi preserves the total spectral flow and is a lattice-specific phenomenon. Our approach uses an invariant subspace decomposition of the Birman-Schwinger operator and the Krein-Rutman theorem to guarantee uniqueness and strict positivity of the ground state. These results have direct implications for quantum simulation of few-boson systems in optical lattices.
We study the three-boson Schrodinger operator on the two-dimensional integer lattice with pairwise contact interactions. First, we obtain precise asymptotics of the two bound states below the essential spectrum at zero total quasimomentum in the strong-coupling limit mu to infinity: the ground state energy is -3 mu + 6 + O(1/mu), and the first excited state energy is - mu + C + O(1/mu), where C = 4 - delta approximately 3.96458, with delta determined by the transcendental equation b0(delta) = 1/(1+delta) involving the lattice Green's function. The associated spectral gap is 2 mu + O(1). Second, using the discrete Agmon comparison method and the Paley-Wiener theorem, we establish exponential localization of the ground-state wavefunction with a logarithmic upper bound on the decay rate alpha(mu) at most ln(3 mu) + O(1/mu), reflecting the bounded nature of the lattice dispersion; the sharp asymptotic rate is conjectured. Third, at quasimomentum pi, the parity symmetry is preserved, but the odd quadratic form becomes positive definite with a unique eigenvalue of order 1/mu that never reaches the Birman-Schwinger threshold, so the odd subspace yields only a virtual level. The even subspace supports exactly one bound state with energy -2 mu + 6 + 8/mu + O(1/mu^2), and the spectral gap to the essential spectrum is mu - 2 - 8/mu + O(1/mu^2). The reduction from two bound states at zero to one at pi preserves the total spectral flow and is a lattice-specific phenomenon. Our approach uses an invariant subspace decomposition of the Birman-Schwinger operator and the Krein-Rutman theorem to guarantee uniqueness and strict positivity of the ground state. These results have direct implications for quantum simulation of few-boson systems in optical lattices.
We compare two three-particle lattice models on the square lattice at total quasimomentum zero: three identical bosons with attractive contact interactions, and a two-fermion plus one distinguishable particle system with attractive interspecies contact. Using the Birman--Schwinger principle combined with symmetry reduction and strong-coupling asymptotics, we distinguish shared methodology from model-dependent spectral conclusions. For bosons, the ground-state energy behaves as minus three times the coupling plus six minus three over twice the coupling plus higher-order corrections, and exactly two bound states exist below the threshold for large coupling; the second lies at a finite distance 0.03542 below the threshold, a genuine eigenvalue rather than a threshold resonance. For fermions, the critical mass ratio is pi divided by pi minus two, approximately 2.75194; above it, a doubly degenerate level appears at strong coupling. Near onset, the binding energy follows a linear-over-logarithmic law. The results are compared with numerical finite-grid Faddeev computations and with recent three-dimensional classifications of threshold states.
We present a rigorous 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 determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but 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, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare 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.
We study three particles on the two-dimensional square lattice with contact interactions of infinite strength, in every sector of total quasimomentum, for three statistics: identical bosons, two identical fermions with a distinguishable third particle, and two identical bosons with a distinguishable third particle. The atom-dimer sector associated with the infinitely degenerate eigenvalue of the pair-contact operator is analysed on the whole Brillouin zone. A Feshbach-Schur reduction gives explicit effective Hamiltonians built from the Dirichlet Laplacian and a rank-four exchange operator, and a spectral transfer theorem with explicit error bounds identifies the atom-dimer threshold of the three-body operator with that of the effective one; the eigenvalue counting function below the threshold is squeezed between those of the effective operator, so that the number of atom-dimer states bound uniformly in the coupling equals the number of bound states of the effective Hamiltonian. The threshold problem reduces to a four-by-four Hermitian matrix whose zero-energy limit is built from the two-point resistances of the square lattice and hence to a quartic polynomial. This yields the complete atom-dimer zones in closed form; the critical hopping ratio as an explicit function on the whole Brillouin zone with two-sided bounds and closed forms at all symmetry points; at most two atom-dimer states per sector with an explicit second threshold; exact onset laws of infinite order at the corner point and linear-over-logarithmic in the pure p-wave channel; and two exact dualities relating a change of statistics to a shift of quasimomentum by half a reciprocal-lattice vector. The reduction of the three-body eigenvalue problem to the lattice Skornyakov-Ter-Martirosyan equation is proved with an explicit isomorphism of eigenspaces.
We study three-particle Schrödinger operators on the two-dimensional lattice Z 2 and show that a critical mass ratio γ c ≈ 2.75194 governs the existence of a bound trimer in the fermionic 2 + 1 configuration (two identical fermions and a third particle). For γ < γ c there is a topological prohibition (Pauli suppression) of a three-body bound state, whereas for γ > γ c a doubly degenerate eigenvalue emerges below the essential spectrum with the strong-coupling asymptotics z(γ, λ) = −λ+e 0 (γ)+O(λ −1 ). Within a unified framework based on the Birman–Schwinger principle and strong-coupling asymptotic analysis, we compare this behaviour with the bosonic case of three identical particles, where two bound states exist below the essential spectrum and the ground-state energy satisfies z 1 s (µ) = −3µ + C 2 + O(µ −1 ). The resulting second-order phase transition with respect to the mass ratio γ is relevant for the design of experiments on fermionic trimers in optical lattices and for modelling excitonic complexes and defect-bound states in two-dimensional nanomaterials, where the critical value γ c serves as a design guideline for the observability of three-body bound states. We also outline a modified three-particle lattice model with two competing interaction channels, for which the Birman–Schwinger analysis naturally leads to a Landau-type scenario of a first-order phase transition in the space of trimer bound states. In the bosonic case we prove a strong-coupling theorem describing the existence and asymptotics of trimer bound states, while in the fermionic 2+1 case we establish a spectral phase-transition theorem that identifies an explicit critical mass ratio γ c separating the trimer and non-trimer regimes.
Bound States of 2+1 Fermionic Trimers on Lattice at Strong Couplings
Abstract In this paper, we investigate the bound states of 2+1 2 + 1 fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator Hγ,λ(K), H γ , λ ( K ) , focusing on energies below the continuum and within its gap. Depending on the quasi-momentum K , we show that if the mass ratio γ>0 γ > 0 between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling λ>0. λ > 0 . Conversely, if γ γ exceeds this threshold, Hγ,λ(K) H γ , λ ( K ) admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as λ→+∞. λ → + ∞ . For K=0, K = 0 , the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.
We prove the existence of an Efimov sequence in an internal gap of the essential spectrum of a three-particle lattice Hamiltonian. The system consists of two identical spinless fermions and one distinguishable particle on Z3, with repulsive pair interactions. For each fixed fermion-to-third-particle mass ratio γ>γ∗≈13.6069657 and total quasi-momentum near π1, the mixed zero-range coupling is tuned to a simultaneous upper-threshold resonance. Sufficiently strong nearest-neighbour repulsion between the fermions separates the upper fermion-pair two-cluster spectrum from the free three-particle band. Infinitely many discrete eigenvalues accumulate at the lower endpoint of the resulting gap, which coincides with the free upper edge. We establish logarithmic eigenvalue-counting asymptotics with the same positive coefficient as at zero fermion--fermion coupling. This coefficient also governs accumulation above the essential spectrum throughout the closed weak-coupling interval, including its critical endpoint. We classify the channel spectrum near the free upper edge for all nonnegative fermion couplings.