Decay properties of zero-energy resonances of multi-particle Schr\"odinger operators and why the Efimov effect does not exist for systems of $N\geq 4$ particles

TL;DR

This paper investigates the decay properties of zero-energy resonances in multi-particle Schrödinger operators, demonstrating that for systems with four or more particles in dimensions three and higher, only finitely many negative eigenvalues exist, and clarifying the non-existence of the Efimov effect in these cases.

Contribution

It establishes decay rates of eigenfunctions at zero energy and proves the finiteness of negative eigenvalues for multi-particle systems with four or more particles in higher dimensions.

Findings

Zero-energy resonances correspond to eigenvalues for N≥3 in dimensions n≥3.

Systems with N≥4 particles in n≥3 have only finitely many negative eigenvalues.

The Efimov effect does not occur for N≥4 particles in dimensions n≥3.

Abstract

We consider $N$-body Schr\"odinger operators with a virtual level at the threshold of the essential spectrum. We show that in the case of $N\geq 3$ particles in dimension $n\geq3$ virtual levels correspond to eigenvalues of the system and we obtain decay rates of the corresponding eigenfunctions in dependence on the dimension and the number of particles. We prove that in dimension $n\geq 3$ the Hamiltonian of $N\geq 4$ particles interacting via short-range potentials admits only a finite number of negative eigenvalues. We extend our results to dimension $n=1$ and $n=2$ in case of $N\geq 4$ fermions.