The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of manybody bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in threebody problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of manybody problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semiinfinite line and interact with each other through a pairwise δfunction potential with a particular positiondependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of twobody bound states. We also study the twobody scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.
I. INTRODUCTION
In his seminal paper in 1970, Efimov considered three identical bosons with shortrange pairwise interactions.^{1} He pointed out that, when the twobody scattering length diverges, an infinite number of threebody bound states appear with energy levels $ { E n} n \u2208 \mathbb{Z}$ forming a geometric sequence. This phenomenon—generally known as the Efimov effect—has attracted much attention because the ratio $ E n + 1 / E n \u2248 1 / ( 22.7 ) 2$ is independent of the details of the interactions as well as of the nature of the particles: It is universal. More than 35 years after its prediction, this effect was finally observed in cold atom experiments,^{2–6} which has triggered an explosion of research on the Efimov effect. For more details, see the reviews in Refs. 7–11 (see also Refs. 12–14 for a more elementary exposition).
Aside from its universal eigenvalues ratio, the Efimov effect takes its place among the greatest theoretical discoveries in modern physics, because it was the first quantum manybody phenomenon to demonstrate discrete scale invariance—an invariance under enlargement or reduction in the system size by a single scale factor.^{15} It is now known that the emergence of a geometric sequence in the bound states' discrete energies is associated with the breakdown of continuous scale invariance to discrete scale invariance^{16} and can be found in a wide spectrum of quantum manybody problems in various dimensions.^{17–23} The notion of the Efimov effect has, therefore, now been broadened to include those generalizations, so that its precise meaning varies in the literature. In the present paper, we will use the term “Efimov effect” to simply refer to the onset of a geometric sequence in the energies of manybody bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance.
To date, there exist several theoretical approaches to study the Efimov effect. The most common approach is to directly analyze the manybody Schrödinger equation, which normally involves the use of Jacobi coordinates, hyperspherical coordinates, the adiabatic approximation, and the Faddeev equation.^{7} Another popular approach is to use second quantization or quantum field theory.^{8} Though the problem itself is conceptually simple, it is hard for students and nonspecialists to master these techniques and to work out the physics of the Efimov effect. The essential part of this phenomenon, however, can be understood from undergraduatelevel quantum mechanics without using any fancy techniques.
The rest of this paper is devoted to the detailed analysis of the eigenvalue problem of H. Before going into details, however, it is worth summarizing the symmetry properties of the model. Of particular importance are the following:

Permutation invariance. Thanks to the relation $ g ( x 1 ) \delta ( x 1 \u2212 x 2 ) = g ( x 2 ) \delta ( x 2 \u2212 x 1 )$, the Hamiltonian (1) is invariant under the permutation of coordinates, $ ( x 1 , x 2 ) \u21a6 ( x 2 , x 1 )$. Note that this permutation invariance is necessary for Eq. (1) to be a Hamiltonian of indistinguishable particles, where, for bosons, the twobody wavefunction should satisfy $ \psi ( x 1 , x 2 ) = \psi ( x 2 , x 1 )$. We will see in Sec. III A that this invariance greatly simplifies the analysis.
 Scale invariance. Thanks to the relations $ g ( e t x 1 ) = e \u2212 t g ( x 1 )$ and $ \delta ( e t x 1 \u2212 e t x 2 ) = e \u2212 t \delta ( x 1 \u2212 x 2 )$, the Hamiltonian (1) transforms as $ H \u21a6 e \u2212 2 t H$ under the scale transformation $ ( x 1 , x 2 ) \u21a6 ( e t x 1 , e t x 2 )$. This transformation law has significant implications for the spectrum of H. Let $ \psi E ( x 1 , x 2 )$ be a solution to the eigenvalue equation $ H \psi E ( x 1 , x 2 ) = E \psi E ( x 1 , x 2 )$. Then, it follows from $ e \u2212 2 t H \psi E ( e t x 1 , e t x 2 ) = E \psi E ( e t x 1 , e t x 2 )$ that $ \psi E ( e t x 1 , e t x 2 )$ satisfies $ H \psi E ( e t x 1 , e t x 2 ) = e 2 t E \psi E ( e t x 1 , e t x 2 )$; that is, $ \psi E ( e t x 1 , e t x 2 )$ is proportional to the eigenfunction $ \psi e 2 t E ( x 1 , x 2 )$ corresponding to the eigenvalue $ e 2 t E$. The proportionality coefficient can be determined by requiring that both ψ_{E} and $ \psi e 2 t E$ be normalized. The result is the following scaling law:$ \psi e 2 t E ( x 1 , x 2 ) = e t \psi E ( e t x 1 , e t x 2 ) .$
If this indeed holds for any $ t \u2208 \mathbb{R} , \u2009 e 2 t E$ can take any arbitrary (positive) value so that the spectrum of H is continuous. As we will see in Sec. III B, however, if g_{0} is smaller than a critical value $ g \u2217$, Eq. (3) holds only for some discrete $ t \u2208 t \u2217 \mathbb{Z} = { 0 , \xb1 t \u2217 , \xb1 2 t \u2217 , \u2026}$; that is, continuous scale invariance is broken to discrete scale invariance, defined by a characteristic scale $ t \u2217$. As a consequence, there appears a geometric sequence of (negative) energy eigenvalues, $ { E 0 , E 0 e \xb1 2 t \u2217 , E 0 e \xb1 4 t \u2217 , \u2026}$, where $ E 0 ( < 0 )$ is a newly emergent energy scale. One of the goals of this paper is to show this result using only undergraduatelevel calculus.
It should be noted that there is no translation invariance in our model: It is explicitly broken by the boundary at x = 0 as well as by the positiondependent coupling strength (2). This noninvariance means that the total momentum—the canonical conjugate of the centerofmass coordinate—is not a welldefined conserved quantity. In other words, the twobody wavefunction cannot be of the separationofvariable form $ \psi ( x 1 , x 2 ) = e iPX / \u210f \varphi ( x )$, where $ X = ( x 1 + x 2 ) / 2$ is the centerofmass coordinate, P is the total momentum, $ x = x 1 \u2212 x 2$ is the relative coordinate, and $\varphi $ is the wavefunction of relative motion. In Sec. II, we will first introduce an alternative coordinate system that is more suitable for the twobody problem on the halfline $ \mathbb{R} +$, before solving the problem in Sec. III.
II. TWOBODY PROBLEM WITHOUT TRANSLATION INVARIANCE
III. TWOBODY EFIMOV EFFECT WITH BOUNDARY
A. Solution to the angular equation
Let us first solve the angular equation (11a). To this end, we need to specify the connection conditions at θ = 0 and the boundary conditions at $ \theta = \xb1 \pi / 4$. We start with the connection conditions equivalent to the δfunction potential.
B. Boundarylocalized twobody Efimov states
C. Twobody scattering off the boundary
We note in closing that the reflection amplitude (30) can be regarded as the scattering matrix (Smatrix) element $ S k \lambda , k \u2032 \lambda \u2032 = ( \Psi k \lambda out , \Psi k \u2032 \lambda \u2032 in )$ for $ \lambda = \lambda \u2032 = \lambda 0 ( < 0 )$, where $ \Psi k \lambda in = r \u2212 1 / 2 R k \lambda \Theta \lambda $ is the instate, $ \Psi k \lambda out = \Psi k \lambda in \xaf$ is the outstate given by the complex conjugate (i.e., time reversal) of the instate, and $ ( \xb7 , \xb7 )$ is the inner product defined by $ ( f , g ) = \u222b 0 \u221e \u222b 0 \u221e f \xaf g \u2009 d x 1 d x 2 = \u222b 0 \u221e \u222b \u2212 \pi / 4 \pi / 4 f \xaf g \u2009 rdrd \theta $. In fact, it follows from the orthonormality $ \u222b \u2212 \pi / 4 \pi / 4 \Theta \lambda \Theta \lambda \u2032 d \theta = \delta \lambda \lambda \u2032$,^{41} the identity $ ( k 2 \u2212 k \u2032 2 ) R k \lambda R k \u2032 \lambda = ( d / d r ) ( R k \lambda R k \u2032 \lambda \u2032 \u2212 R k \lambda \u2032 R k \u2032 \lambda )$, and the asymptotic behavior $ R k \lambda \u2192 e \u2212 ikr + S \lambda ( k ) e ikr$ that this Smatrix element takes the form $ S k \lambda , k \u2032 \lambda \u2032 = 2 \pi \delta ( k \u2212 k \u2032 ) \delta \lambda \lambda \u2032 S \lambda ( k )$. One nice thing in this formulation is that it is obvious that there is no scattering between different channels λ and $ \lambda \u2032$.
IV. CONCLUSION
In this paper, we have introduced a toy scaleinvariant model of two identical bosons on the halfline $ \mathbb{R} +$, where interparticle interaction is described by the pairwise δfunction potential with the particular positiondependent coupling strength given in Eq. (2). We have seen that, if the twobody interaction is sufficiently attractive, continuous scale invariance is broken down to discrete scale invariance. In the boundstate problem where the bosons are bound together and localized to the boundary, this discrete scale invariance manifests itself in the onset of the geometric sequence of binding energies. In the scattering problem where the twobody bound state is scattered by the boundary, on the other hand, this discrete scale invariance manifests itself in the logperiodic behavior of the reflection amplitude. Hence, by breaking translation invariance of this onedimensional problem, we can construct a twobody model that exhibits the Efimov effect. In contrast to the ordinary Efimov effect in threebody problems in three dimensions, our model can be solved exactly by just using undergraduatelevel calculus.
Finally, it should be mentioned the stability issue of the model and its cure. As is evident from Eq. (27), there is no lower bound in the energy spectrum $ { E n}$ for $ g 0 < g \u2217$. This absence of ground state is inevitable if the system is invariant under the full discrete scale invariance that forms the group $\mathbb{Z}$. (As discussed in the introduction, the full discrete scale invariance leads to the geometric sequence $ { E 0 , E 0 e \xb1 2 t \u2217 , E 0 e \xb1 4 t \u2217 , \u2026}$, which cannot be bounded from below if $ E 0 < 0$.) In order to make the spectrum lowerbounded, we, therefore, have to break this invariance under $\mathbb{Z}$. The easiest way to do this is to replace the shortdistance singularity of the inversesquare potential by, e.g., a squarewell potential. Such regularization procedures have been widely studied over the years in the context of renormalization of the inversesquare potential. For more details, we refer to Refs. 24–31.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
APPENDIX A: MODIFIED BESSEL FUNCTIONS OF IMAGINARY ORDER
In this section, we summarize the short and longdistance behaviors of the modified Bessel functions. For details, we refer to Ref. 42.
APPENDIX B: BOUNDARY AS THE INFINITELY HEAVY THIRD PARTICLE
In this section, we show that the twobody problem on the halfline $ \mathbb{R} +$ discussed in the main text is equivalent to a threebody problem on the whole line $\mathbb{R}$ with an infinitely heavy third particle. We note that this section is not necessary for understanding the main text.