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Letters in Mathematical Physics

Absence of the confinement–induced Efimov effect: a direct proof in a specific geometry

Marvin R. Schulz, Sylvain Zalczer

Published Nov 17, 2025• 0 citations
Publisher

Abstract

Abstract We consider a system of three particles with identical mass interacting via short-range potentials, such that two of the particles are on parallel lines in a plane and the third one is on a line perpendicular to this plane. In this geometry, we prove that the corresponding Schrödinger operator only has a finite number of eigenvalues under physically reasonable assumptions on the decay of the interaction potentials. Our result disproves a recent prediction made in physics literature.

DOI
10.1007/s11005-025-02019-7

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Abstract We consider a system of three particles with identical mass interacting via short-range potentials, such that two of the particles are on parallel lines in a plane and the third one is on a line perpendicular to this plane. In this geometry, we prove that the corresponding Schrödinger operator only has a finite number of eigenvalues under physically reasonable assumptions on the decay of the interaction potentials. Our result disproves a recent prediction made in physics literature.
By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.
In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].
Two particles that are just shy of binding may develop an infinite number of shallow bound states when a third particle is added. This counterintuitive effect was first predicted by Efimov for identical bosons interacting with a short-range pairwise potential. The Efimov effect persists for nonidentical particles if at least two of the three bonds are almost bound. The Efimov effect has recently been verified experimentally using ultracold atoms. We explain the origin of this effect using elementary quantum mechanics and summarize the experimental evidence for it.