Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum

Abstract

We consider a system of N pairwise interacting particles described by the Hamiltonian H, where σ ess (H) = [0, ∞) and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if N ≥ 4 and none of the Hamiltonians corresponding to the subsystems containing N − 2 or less particles has an eigenvalue equal to zero then H has a finite number of negative energy bound states. This result provides a positive proof to a long-standing conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the "true" four-body Efimov effect.

Proof of the Super Efimov Effect

“…The operator k(0) is compact and in the vicinity of z = 0 the operator k(z) is compact as well. Because the interaction is tuned to the p-wave zero energy resonance from the BS principle (see Theorem 9 in [12]) we infer that k(0) = 1. By standard results in quantum mechanics the ground state of h(λ) for λ > 1 is doubly degenerate with the angular momentum l = ±1.…”

“…By standard results in quantum mechanics the ground state of h(λ) for λ > 1 is doubly degenerate with the angular momentum l = ±1. By the BS principle [12] it follows immediately that k(0) = 1 is an eigenvalue of k(0) with multiplicity 2. Due to spherical symmetry the largest eigenvalue of k(z) for z > 0 is also doubly degenerate.…”

“…The basic idea behind the proof of (1) stems from [2], namely, one uses symmetrized Faddeev equations and the Birman-Schwinger principle [5][6][7]12] for counting eigenvalues.…”

The absence of the Efimov effect in systems of one- and two-dimensional particles

“…However, to prove that virtual levels in subsystems of N − 1 particles correspond to eigenvalues and not to resonances was a very challenging problem, because the sum of the pair potentials does not decay in all directions at infinity, which makes it difficult to use Green's functions. This problem was first solved by D. Gridnev [12] and recently a proof with simpler methods and less restrictions on the potentials was given in [5]. In addition, it was shown in [5] that the Efimov effect can not occur in systems of N ≥ 4 one-or two-dimensional spinless fermions.…”

Existence of bound states of N-body problem in an optical lattice

“…The universality of our existence results is that we claim the existence of bound states in every purely attractive or purely repulsive system of arbitrary (finite) number of particles as soon as pairwise interactions between particles are strong enough. Such universality is less obvious, for example, in the Efimov physics (see [30,Section 1.2]), since currently the Efimov effect is known only for three-particle systems [11] and not every system can allow it. Moreover, our conditions to have a bound state can be derived using only the controllable parameters of the system, e.g.…”