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

Abstract

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.