The Efimov Effect

On this page I will continue to add more acessible informations and heuristics behind the Efimov effect.

New Preprint:

On the Efimov Effect for Four Particles in Dimension Two

J. Rau and M.R. Schulz

We prove that the Schrödinger operator describing four particles in two dimensions, interacting solely through short-range three-body forces, can possess infinitely many bound states. This holds under the assumption that each three-body subsystem has a virtual level at zero energy. Our result establishes an analog of the Efimov effect for such four-particle systems in two dimensions.

Preprint:

Why a System of Three Bosons on Separate Lines Can Not Exhibit the Confinement Induced Efimov Effect

D. Hundertmark, M.R. Schulz and S. Vugalter

arXiv:2411.19263 [math-ph]

We study a system of three bosons interacting with short-range potentials which can move along three different lines. Two of these lines are parallel to each other within one plane. The third line is constrained to a plane perpendicular to the first one. Recently it was predicted in physics literature that such a system exhibits the so-called confinement induced Efimov effect. We prove that this prediction is not correct by showing that this system has at most finitely many bound-states.

Why Quantum Tunneling Can Explain the Efimov Effect

Jacobi Coordinates

Considering a system of three particles and introducing Jacobian coordinates as in the image on the left hand side one directly can idendify the double well operator as part of the full operator.

The full operator of this system in new coordinats will read:

Quantum Tunneling

Fate of a groundstate if a nearby potential well appears

Classical particles would not be allowed to enter areas where the potential is higher than the "Probability amplitude".

Quantum States will have an exponentially fast decaying probability to penetrate this forbidden area

Formation of a new ground state

The potential landscape is symmetric and consequently we expect a symmetric groundstate. The probability of existing in one of the two wells equals out.

Farly separated potential wells

The probability densitiy of quantum states vanishes exponetially fast in the classical forbidden area. There is no tunneling on long distances.

Resonances are very good in tunneling

We consider solutions of the corresponding Schrödinger equation that exist at the binding threshold. A small increase in energy would push them out of the potential well. However, these solutions are not proper eigenstates but rather generalized eigenstates, often referred to as resonance functions.

Important fact: Resonances will have much slower decay in the classic forbidden area. They only decay by some potential law (compared to exponential decay)

There will be interaction even on far seperation of potential wells.

Resonances will show Far Range Quantum Tunneling

As in ordinary quantum tunneling, the probability density equalizes between potential wells.

Remark:

Adding a small amount of energy to this equalized probability density is not sufficient to push the particle out of the combined wells.

From a mathematical perspective, a proper (non-generalized) eigenstate emerges: while the original operator had only essential spectrum, we now obtain a ground state in the discrete spectrum.

Efimov Effect in Mixed Dimensions

The Efimov Effect is a surprising phenomenon with certain universal properties where three particles in three-dimensional space that interact with short-range interactions show binding on long ranges, although none of the two-particle subsystems can bind on its own.

Question: Can a similar effect occur in systems of confined particles

For the following systems I see changes that such an effect occurs. They are also in certain senses exemplary for larger classes of Systems and consequently most interesting to study:

Project A

Two particles in dimension 2 interact with a third unconstrained particle
System is isotropic in confinements space (invariant under translations in the plane)
Existence of virtual levels (almost binding) in subsystems of particles causes restoring forces through quantum tunneling

Project B

Four two-dimensional Particles that interact solely with three-particle forces.
System is isotropic (invariant under mutual translations).
Three Particle Subsystems: Effective dimension is 4. (3 times two-dimensional particles minus two dimensonal isotropie)
Dimension Four: Virtual Levels (almost binding) does produce resonance solutions as in dimension three. Binding through quantum tunneling may exist.

Project C

Purely two-dimensional particles in different sub-spaces.
Compared to Project A: We lose isotropie in one direction. Restoring Forces will also be more isotropic, and binding gets less likely.

Literature Concerning Efimov Effect

SELECTION

Efimov, V. N. (1970). Weakly bound states of three resonantly interacting particles. Yadernaya Fizika, 12, 1080–1091.

"The original article by Efimov reporting on his findings [in russian]"

Interest in Efimov Effect is rising, and due to advancements in experiments in ultracold gases, it is a hot topic. (compare to the figure)

Data taken from: https://inspirehep.net/literature/73259

Naidon, P., & Endo, S. (2017). Efimov physics: a review. Reports on Progress in Physics, 80(5), 056001. IOP Publishing. https://doi.org/10.1088/1361-6633/aa50e8

"A extensive review article giving a great overwiew on almost all areas related to Efimov Effect and modern Research related to it: In particular in Section 9 mixed dimensional settings important for Projecta A,B and C are discussed."

Jafaev, D. R. (1974). On the theory of the discrete spectrum of the three-particle Schrödinger operator. Matematicheskii Sbornik (N.S.), 94(136), 567–593, 655–656.

"The first mathematical proof of the Efimov effect. The proof uses so-called Faddeev equations and is hard to digest"

Tamura, H. (1991). The Efimov effect of three-body Schrödinger operators. Journal of Functional Analysis, 95(2), 433–459. https://doi.org/10.1016/0022-1236(91)90038-7

Summary: Tamura provides a rigorous variational proof of the Efimov effect for three-body Schrödinger operators. The work shows how an infinite sequence of bound states can emerge at the threshold of binding, even when no two-body subsystem is bound. A key tool in the proof is the idea of a “conspiracy of potential wells,” where several weak interactions combine to produce binding that none of them could generate alone.

Klaus, M., & Simon, B. (1979). Binding of Schrödinger particles through conspiracy of potential wells. Annales de l’Institut Henri Poincaré, Section A: Physique Théorique (N.S.), 30(2), 83–87.

Summary: This short but influential paper investigates how binding (the formation of a stable quantum state) can occur for particles in quantum mechanics, even when individual potential wells are too weak to cause binding on their own. Klaus and Simon show that a collection of such “non-binding” wells can, when combined, produce a bound state—a phenomenon sometimes called a “conspiracy of wells.”

To get an overview:

My Thesis and in particular Chapters 3.1 and 5.1:

Schulz, Marvin Raimund. Analytic Studies of Atomic Structures and the Efimov Effect. Doctoral Thesis, Karlsruher Institut für Technologie (KIT), 2025.

DOI: https://doi.org/10.5445/IR/1000183607.

Thesis of A. Bitter:

Bitter, Andreas. Virtual levels of multi-particle quantum systems and their implications for the Efimov effect. Doctoral Thesis, Faculty of Mathematics and Physics, University of Stuttgart, 2020.

Supervisor: Prof. Timo Weidl.

DOI: 10.18419/opus-11315.

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