We consider a system of three particles in dimension d ≥ 4 interacting via shortrange potentials, where the two-body Hamiltonians have a virtual level at the bottom of the essential spectrum. In dimensions d = 2 (in case of fermions) and d = 3 the corresponding three-body Hamiltonian admits an infinite number of bound states, which is known as the Efimov effect. In this work we prove that this is not the case in higher dimensions. We investigate how the dimension and symmetries of the system influence this effect and prove the finiteness of the discrete spectrum of the corresponding three-body Hamiltonian.

We study virtual levels of N-particle Schrödinger operators and prove that if the particles are one-dimensional and N ≥ 3, then virtual levels at the bottom of the essential spectrum correspond to eigenvalues. The same is true for two-dimensional particles if N ≥ 4. These results are applied to prove the non-existence of the Efimov effect in systems of N ≥ 4 one-dimensional or N ≥ 5 two-dimensional particles.

We consider N -body Schrödinger operators with N ≥ 3 quantum particles interacting via short-range potentials in dimension d ≥ 3, where the essential spectrum coincides with the half line [0, ∞). We give the asymptotic behaviour of eigenfunctions corresponding to the eigenvalue at the threshold of the essential spectrum under the condition that the eigenfunctions are not orthogonal to the sum of the pair interactions. This condition is fulfilled when zero is the smallest eigenvalue and the pair interactions are negative. We also give examples of systems when this condition is not met.