Efimov effect in an internal spectral gap of a three-particle lattice Hamiltonian

Résumé : 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 \(\Z^3\), with repulsive pair interactions. For each fixed fermion-to-third-particle mass ratio \(γ>γ_*\approx13.6069657\) and total quasi-momentum near \(π\mathbf1\), 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.