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

TL;DR

This paper proves that in certain low-dimensional multi-particle quantum systems, virtual levels at the spectrum's bottom are actual eigenvalues, leading to the conclusion that the Efimov effect does not occur in these cases.

Contribution

The paper establishes the absence of the Efimov effect in multi-particle systems of one or two dimensions with specific particle counts, extending understanding of spectral properties.

Findings

Virtual levels at the spectrum's bottom are eigenvalues for N≥3 in 1D and N≥4 in 2D.

The Efimov effect does not occur in systems with N≥4 in 1D or N≥5 in 2D.

Results clarify spectral behavior of multi-particle low-dimensional quantum systems.

Abstract

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