Universality of excited three-body bound states in one dimension

We consider a two-body system consisting of a heavy particle of mass M and a light one of mass m, both constrained to 1D and interacting via a potential of finite range ξ₀. We define the mass ratio α ≡ M/m of the two particles in the heavy–light system.

After eliminating the heavy–light center-of-mass coordinate, the system is governed by the stationary Schrödinger equation

$$\begin{equation}\left[-\frac{1}{2}\frac{{\mathrm{d}}^{2}}{\mathrm{d}{\xi }^{2}}+v\left(\xi \right)\right]{\tilde{\psi }}^{(2)}={\mathcal{E}}^{(2)}{\tilde{\psi }}^{(2)}\end{equation} \tag{ 1 }$$

for the two-body wave function ${\tilde{\psi }}^{(2)}={\tilde{\psi }}^{(2)}(\xi )$ of the relative motion presented in dimensionless units. Here, ξ denotes the relative coordinate of the light particle with respect to the heavy one in units of the characteristic length ξ₀.

The two-body binding energy ${\mathcal{E}}^{(2)}$ and the potential

$$\begin{equation}v\left(\xi \right)={v}_{0}f(\xi )\end{equation} \tag{ 2 }$$

are both given in units of ${\hslash }^{2}/\mu {\xi }_{0}^{2}$ with the Planck constant ℏ and the reduced mass μ ≡ M/(1 + α) of the heavy–light system. Here, v₀ denotes the magnitude and f the shape of the interaction potential.

We assume an attractive interaction with v₀ < 0 and ∫dξf(ξ) > 0, as well as a symmetric shape f, that is $f(\xi )=f(\left\vert \xi \right\vert )$. Moreover, we choose v such that it describes a short-range interaction, i.e. ${\left\vert \xi \right\vert }^{2}f(\vert \xi \vert )\to 0$ as $\left\vert \xi \right\vert \to \infty$.

In particular, we perform the analysis in this article for different two-body interactions of finite-range, namely a potential having the polynomially decaying shape

$$\begin{equation}{f}_{\mathrm{L}}(\xi )\equiv \frac{1}{{(1+{\xi }^{2})}^{3}}\end{equation} \tag{ 3 }$$

of the cube of a Lorentzian, and one with the shape

$$\begin{equation}{f}_{\mathrm{G}}(\xi )\equiv \mathrm{exp}(-{\xi }^{2})\end{equation} \tag{ 4 }$$

of a Gaussian decaying exponentially.

In contrast to the zero-range interaction of shape

$$\begin{equation}{f}_{\delta }(\xi )\equiv \delta (\xi ),\end{equation} \tag{ 5 }$$

the finite-range potentials whose shapes are determined by equations (3) and (4) can support more than a single bound state, depending on the magnitude v₀.

We label the two-body wave function ${\psi }_{r}^{(2)}$ and the binding energy ${\mathcal{E}}_{r}^{(2)}$ as solutions of equation (1) by the index r, where r = 0, 1, 2, ... denotes the number of nodes of ${\tilde{\psi }}_{r}^{(2)}$. Even values of r indicate symmetric two-body wave functions, whereas odd values indicate antisymmetric ones, that is

$$\begin{equation}{\tilde{\psi }}_{r}^{(2)}(-\xi )={(-1)}^{r}{\tilde{\psi }}_{r}^{(2)}(\xi ).\end{equation} \tag{ 6 }$$

We now add a third particle to the two-body system, also constrained to 1D and identical to the other heavy particle of mass M. We assume the same interaction between each heavy and the light particle as introduced in section 2.1, but no interaction between the two heavy ones. The resulting three-body system is depicted in figure 1. Here, y₂₃ denotes the relative coordinate between the two heavy particles (called here particles 2 and 3) and x₁ the relative coordinate of the light particle (particle 1) with respect to the center of mass C of the two heavy ones, both in units of the characteristic length ξ₀.

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By eliminating the center-of-mass motion of this heavy–heavy–light system, we arrive at the Schrödinger equation

$$\begin{equation}\left[-\frac{{\alpha }_{x}}{2}\frac{{\partial }^{2}}{\partial {x}_{1}^{2}} - \frac{{\alpha }_{y}}{2}\frac{{\partial }^{2}}{\partial {y}_{23}^{2}}+v\left({x}_{1} + \frac{{y}_{23}}{2}\right) + v\left({x}_{1} - \frac{{y}_{23}}{2}\right)\right]\tilde{\psi }=\mathcal{E}\tilde{\psi }\end{equation} \tag{ 7 }$$

for the three-body wave function $\tilde{\psi }=\tilde{\psi }({y}_{23},{x}_{1})$ of the two relative motions in coordinate representation. The coefficients α_x ≡ (1 + 2α)/[2(1 + α)] and α_y ≡ 2/(1 + α) depend only on the mass ratio α, and $\mathcal{E}$ is the dimensionless three-body energy in units of ${\hslash }^{2}/\mu {\xi }_{0}^{2}$.

In this article, we determine the energies and wave functions of bound states in the three-body system, provided the heavy–light subsystems support a weakly-bound excited state, that is ${\mathcal{E}}_{r}^{(2)}\to {0}^{-}$ for r = 1, 2, 3. We are mainly interested in the question of whether or not there exist three-body quantities that show universal behavior in these cases. Universality denotes the property of physical quantities to show a behavior that is independent of the short-range details of the interparticle interactions.

In reference [17], we have studied the situation when the heavy–light system is tuned to the ground-state threshold (r = 0). We have found universality in terms of both energies and corresponding wave functions of three-body bound states for ${\mathcal{E}}_{0}^{(2)}\to {0}^{-}$. In this limit, we have proven that for arbitrary attractive short-range heavy–light interactions obeying the conditions in section 2.1, the three-body energies and wave functions converge to the respective ones induced by a heavy–light contact interaction of shape f_δ , equation (5).

The differences between the ground- and an excited-state threshold are as follows. Tuning the heavy–light subsystems to an excited-state energy threshold immediately implies (i) a different number r of nodes and therefore a possibly different symmetry of the corresponding weakly-bound two-body bound state, and (ii) the existence of deeply-bound two-body states, whose binding energy does not approach zero. Depending on the symmetry of the weakly-bound two-body wave function, equation (6), we speak of even-numbered (r = 0, 2, ...) and odd-numbered (r = 1, 3, ...) thresholds.

We analyze the effect of both features on the universal behavior of energies and wave functions of three-body bound states close to an excited-state threshold in the heavy–light subsystems. We emphasize that there are two kinds of three-body bound states, namely (i) excited three-body bound states associated with the respective threshold, and (ii) deeply-lying three-body bound states. (i) The excited three-body bound states are embedded in a continuum that originates from deeply-bound heavy–light states in combination with an unbound heavy particle. For these states we expect a universal behavior. (ii) The deeply-bound three-body states keep a finite size when the excited-state threshold is approached, hence we expect them to be sensitive to the details of the interaction. For this reason, they do not show universal behavior and we refrain from analyzing them in this article.

The excited three-body states are so-called bound states in the continuum [24]. These states have a real-valued energy, but are positioned in the continuum of scattering states, that is $\mathcal{E} > {\mathcal{E}}_{0}^{(2)}$. Bound states in the continuum have been predicted [22, 23, 50] and realized [51–54] in a variety of physical systems. There are several mechanisms that can protect these states against decay into the continuum states, e.g. modulation of the interaction potential [22, 23], interference of resonances [50] and symmetry protection [51, 52]. For example the latter might be of relevance in our three-body system, as depending on the threshold r, the three-body bound states display a different symmetry than the continuum in which it is embedded. Nevertheless, we want to emphasize that in the current article the focus lies on obtaining such bound states in the continuum, and most importantly in analyzing their universality. We therefore refrain here from a deeper investigation of the underlying mechanism that allows their existence.

Apart from bound states, we want to mention that there can further exist three-body resonances [55–57] that are also located above the lowest two-body energy threshold,$\mathrm{R}\mathrm{e}\enspace \mathcal{E} > {\mathcal{E}}_{0}^{(2)}$. These resonances are however characterized by a non-vanishing imaginary part $\mathrm{I}\mathrm{m}\enspace \mathcal{E}\ne 0$ of the three-body energy and are not studied in the present article.

Similar to the results [17] found near the ground-state threshold $({\mathcal{E}}_{0}^{(2)}\to {0}^{-})$, we expect a set of universal three-body bound states associated with each heavy–light threshold $({\mathcal{E}}_{r}^{(2)}\to {0}^{-})$ characterized by the index r. To distinguish between these sets, we introduce the notation ${\mathcal{E}}_{r,n}$ for the energy of the nth three-body bound state |ψ_r,n ⟩ within the rth set. As shown in the following, we indicate with even and odd values of n the case of the two heavy particles being bosons or fermions, respectively.

In order to reveal the dependency of three-body universality on (i) the symmetry of the weakly-bound state and (ii) the existence of deeply-bound states in the heavy–light subsystems, we compare the energies and wave functions of three-body bound states for r > 0 with the corresponding ones for r = 0. Due to the reported [17] universality for r = 0, we can use the scale-invariant three-body binding energies and wave functions for the heavy–light contact interaction as representatives for this set of states. We therefore base our analysis on the following two quantities:

(a) The relative deviation

$$\begin{equation}{\Delta}{{\epsilon}}_{r,n}\equiv \left\vert \frac{{{\epsilon}}_{r,n}-{{\epsilon}}_{n}^{\star }}{{{\epsilon}}_{n}^{\star }}\right\vert \end{equation} \tag{ 8 }$$

between the energy ratios _r,n and ${{\epsilon}}_{n}^{\star }$. Here,

$$\begin{equation}{{\epsilon}}_{r,n}\equiv \frac{{\mathcal{E}}_{r,n}}{\left\vert {\mathcal{E}}_{r}^{(2)}\right\vert }\end{equation} \tag{ 9 }$$

denotes the ratio of the nth three-body bound state energy ${\mathcal{E}}_{r,n}$ within the rth set to the energy ${\mathcal{E}}_{r}^{(2)}$ of the weakly-bound heavy–light bound state. Moreover, we define the ratio

$$\begin{equation}{{\epsilon}}_{n}^{\star }\equiv \frac{{\mathcal{E}}_{0,n}}{\left\vert {\mathcal{E}}_{0}^{(2)}\right\vert }\end{equation} \tag{ 10 }$$

between the corresponding three- and two-body energies found for the special case of a contact heavy–light interaction of shape f_δ , equation (5).

(b) The fidelity

$$\begin{equation}{F}_{r,n}\equiv {\left\vert \langle {\psi }_{r,n}\vert {\psi }_{n}^{\star }\rangle \right\vert }^{2}\end{equation} \tag{ 11 }$$

defined as the overlap between a three-body bound-state |ψ_r,n ⟩ for a finite-range heavy–light potential, and the corresponding state $\vert {\psi }_{n}^{\star }\rangle$ for the contact interaction in arbitrary representation.

We solve equation (7) within the framework of the Faddeev equations [20, 58]. There we can easily incorporate the necessary boundary condition

$$\begin{equation}\tilde{\psi }({y}_{23},{x}_{1})\to 0,\quad \mathrm{a}\mathrm{s}\enspace \left\vert {x}_{1}\right\vert \to \infty \enspace \mathrm{a}\mathrm{n}\mathrm{d}\enspace \left\vert {y}_{23}\right\vert \to \infty ,\end{equation} \tag{ 12 }$$

to obtain three-body bound states embedded in a continuum. For this reason, we consider the homogeneous form of the Faddeev equations. In order to select other types of states, e.g. resonance or scattering states, a different form of the boundary condition and therefore of the Faddeev equations is required. However, in this article, we restrict ourselves to bound states in the continuum which obey equation (12).

We show in appendix A that the wave function ψ(k₂₃, p₁) ≡ ⟨k₂₃, p₁|ψ⟩ of a three-body bound state in momentum representation is given by the superposition

$$\begin{align}\hfill \psi ({k}_{23},{p}_{1})& ={\phi }^{(2)}\left(-\frac{{\alpha }_{y}}{2}{k}_{23}-{\alpha }_{x}{p}_{1},{k}_{23}-\frac{1}{2}{p}_{1}\right)\hfill \\ \hfill & \quad \pm {\phi }^{(2)}\left(\frac{{\alpha }_{y}}{2}{k}_{23}-{\alpha }_{x}{p}_{1},-{k}_{23}-\frac{1}{2}{p}_{1}\right)\hfill \end{align} \tag{ 13 }$$

of the Faddeev component ϕ⁽²⁾ evaluated at different arguments. Here, k₂₃ denotes the relative momentum between particles 2 and 3, whereas p₁ is the momentum of particle 1 relative to the center of mass of particles 2 and 3.

Moreover, the plus and minus signs in equation (13) distinguish the case of the two heavy particles being bosons or fermions. This is evident from considering the exchange of the two heavy particles being described by k₂₃ → −k₂₃ which leads to the exchange symmetry

$$\begin{equation}\psi (-{k}_{23},{p}_{1})=\pm \psi ({k}_{23},{p}_{1})\end{equation} \tag{ 14 }$$

for the total three-body wave function. As k₂₃ is the momentum corresponding to the relative distance y₂₃ between the two heavy particles, the symmetry (antisymmetry) of ψ(k₂₃, p₁) with respect to the line k₂₃ = 0 directly relates via the Fourier transform to the symmetry (antisymmetry) of the wave function ψ(y₂₃, x₁) with respect to the line y₂₃ = 0 in coordinate space.

The separable expansion [19–21] simplifies the Faddeev equations (see appendix A.3) and allows to analyze the influence of deeply-bound two-body states on three-body universality. We therefore expand the Faddeev component

$$\begin{equation}{\phi }^{(2)}(k,p)\equiv \pm \frac{1}{{\mathcal{E}}_{p}-\frac{1}{2}{k}^{2}}\sum\limits _{\nu =0}^{\infty }\enspace {g}_{\nu }(k,{\mathcal{E}}_{p}){\tau }_{\nu }({\mathcal{E}}_{p}){\varphi }_{\nu }(p,\mathcal{E})\end{equation} \tag{ 15 }$$

in terms of products of functions depending on only one momentum variable. Here, ${\mathcal{E}}_{p}\equiv \mathcal{E}-{\alpha }_{x}{\alpha }_{y}\enspace {p}^{2}/2$ and the functions g_ν and τ_ν ≡ −η_ν /(1 − η_ν ) are determined by the eigenvalue equation

$$\begin{equation}\int \frac{\mathrm{d}{k}^{\prime }}{2\pi }V(k,{k}^{\prime })\frac{1}{{\mathcal{E}}_{p}-\frac{1}{2}{k}^{\prime 2}}\enspace {g}_{\nu }({k}^{\prime },{\mathcal{E}}_{p})={\eta }_{\nu }({\mathcal{E}}_{p}){g}_{\nu }(k,{\mathcal{E}}_{p}),\end{equation} \tag{ 16 }$$

where

$$\begin{equation}V(k,{k}^{\prime })={v}_{0}\int \mathrm{d}\xi \enspace f(\xi ){\mathrm{e}}^{-\mathrm{i}(k-{k}^{\prime })\xi }\end{equation} \tag{ 17 }$$

is the momentum representation of the heavy–light potential v(ξ), equation (2).

For a given r > 0 and ν ⩽ r the functions g_ν can be related to the wave functions ${\psi }_{\nu }^{(2)}$ of the two-body bound states, however with variable energy argument ${\mathcal{E}}_{q}$. Indeed, for a given v₀, they are connected to each other only when ${\mathcal{E}}_{q}={\mathcal{E}}_{\nu }^{(2)}$ [19, 20]. The term with ν = r describes the one close to the threshold and the terms with ν < r correspond to the deeply-bound heavy–light states. We can reveal the influence of the latter for the prediction of the correct three-body universality by excluding single terms with ν < r in the expansion, equation (15), and comparing the subsequent results to the ones without excluding them. This is performed in section 4.

For the functions φ_ν we derive in appendix A.3 the system of coupled integral equations

$$\begin{align}\hfill {\varphi }_{\lambda }(p,\mathcal{E})& =\pm \sum\limits _{\nu =0}^{\infty }\int \frac{\mathrm{d}q}{2\pi }\enspace {\tau }_{\nu }({\mathcal{E}}_{q})\enspace {\varphi }_{\nu }(q,\mathcal{E})\hfill \\ \hfill & \quad \times \frac{{g}_{\lambda }\left(q+\frac{\alpha }{1+\alpha }p,{\mathcal{E}}_{p}\right)\enspace {g}_{\nu }\left(p+\frac{\alpha }{1+\alpha }q,{\mathcal{E}}_{q}\right)}{\mathcal{E}-\frac{1}{2}{q}^{2}-\frac{1}{2}{p}^{2}-\frac{\alpha }{1+\alpha }pq}\hfill \end{align} \tag{ 18 }$$

which yields the same energy solutions ${\mathcal{E}}_{r,n}$ as the Schrödinger equation (7).

As mentioned above, we consider in this article three-body bound states with the boundary conditions, equation (12), embedded in a continuum associated with deeply-bound heavy–light states together with an unbound heavy particle. They therefore correspond to real energy solutions ${\mathcal{E}}_{r,n}$. The three-body bound state energy $\mathcal{E}$ enters the kernels in equation (18) as a parameter, hence the solutions ${\mathcal{E}}_{r,n}$ are obtained by varying over a suitable region${\mathcal{E}}_{r,n}< {\mathcal{E}}_{r}^{(2)}$ and requiring eigenvalue unity (minus unity) for bosons (fermions). The functions φ_ν are obtained as the corresponding eigenvectors.

Equation (18) is a system of coupled homogeneous Fredholm integral equations of the second kind. We solve it numerically by truncating the infinite number of expansion terms in equations (15) and (18) to a finite value ν_max, and approximating the continuous interval for p and q by a discrete grid. This reduces equation (18) to a finite system of coupled matrix eigenvalue problems. In our analysis ν_max = 10 has yielded sufficient convergence.

We start by analyzing the universality in the energy spectrum of the three-body system. In figure 2 we present the relative deviation Δ_r,n , equation (8), between the energy ratio _r,n , equation (9), obtained in the case of a finite-range heavy–light interaction and the energy ratio ${{\epsilon}}_{n}^{\star }$, equation (10), for a zero-range heavy–light interaction, equation (5), as a function of the two-body binding energy ${\mathcal{E}}_{r}^{(2)}$. Throughout the subfigures we indicate by filled red and empty blue symbols the results for an interaction of shape f_L, equation (3), and f_G, equation (4), respectively. The top row shows Δ_1,n near the first excited heavy–light threshold, ${\mathcal{E}}_{1}^{(2)}\to {0}^{-}$, the central row Δ_2,n for ${\mathcal{E}}_{2}^{(2)}\to {0}^{-}$, and the bottom row Δ_3,n for the third excited state, ${\mathcal{E}}_{3}^{(2)}\to {0}^{-}$. The left and right columns display the case of two heavy bosons (n = 0, 2, 4) or fermions (n = 1, 3, 5), respectively.

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For all three thresholds (r = 1, 2, 3) and all three-body bound states (n = 0, ..., 5), the filled red and empty blue data sets are very close to each other. We highlight that this is true throughout the entire range of two-body energies displayed here. Moreover, the deviation between the red and blue data sets is decreasing as the threshold is approached. Thus, universality is indicated by the fact that near each threshold, the same energy ratios are achieved for the different finite-range interactions.

Next, we compare the universal behavior near the different heavy–light thresholds. For all considered cases r = 1, 2, 3, the relative deviations Δ_r,n are decreasing strictly monotonically towards zero as ${\mathcal{E}}_{r}^{(2)}\to {0}^{-}$. Hence, the energy ratios _r,n associated to all three excited-state two-body thresholds converge towards the same limit values ${{\epsilon}}_{n}^{\star }$, table 1, as for the ground-state threshold. This explicitly implies that the universal limits of the three-body energies are identical for symmetric (r = 0, 2, ...) and antisymmetric (r = 1, 3, ...) weakly-bound states in the heavy–light subsystems. Moreover, we observe that for each three-body state labeled by n, the results for the considered thresholds r = 1, 2, 3 behave almost identical throughout the entire range of two-body binding energies.

In particular, the deviations Δ_r,n appear to decrease as a power law, as indicated by the linear dependence in the double-logarithmic scale. The corresponding slopes, which in return denote the exponent of the power law, are listed in table 2 and have been obtained via linear regression and for the potential of Gaussian shape f_G, equation (4). The slopes for r > 0 are smaller compared to those for r = 0. This relates to a slower convergence towards the universal three-body energies near excited-state thresholds compared to the ground-state one. The difference between the thresholds r = 0 and r > 0 is characterized by the presence of deeply-bound states in the latter cases. Hence these deeply-bound states have a crucial influence on how fast the universal limit is approached. Moreover, for each respective state labeled by n, we observe that the slopes are almost identical for all r > 0. We note that Δ_r,n , equation (8), is defined as an absolute value, hence our analysis does not distinguish whether the limit values ${{\epsilon}}_{n}^{\star }$ are approached from above or below.

We can further analyze the universal behavior for different excited three-body states characterized by the index n. In the case of fermions (odd n) the relative deviation of _r,n from ${{\epsilon}}_{n}^{\star }$ is much smaller and decreases much faster than in the case of bosons (even n) when approaching the threshold, as indicated by the different range of the vertical axes. Indeed, this is also reflected in the values of the corresponding slopes, table 2, which are almost a factor of ten larger for fermions compared to bosons. This striking difference in the universal behavior is a pure three-body effect due to the fact that the property of the two heavy particles being bosons or fermions does not change the heavy–light two-body problem. Finally, we observe a smaller deviation Δ_r,n for states with larger index n.

Having found that the universal limit of the three-body binding energies is the same when even-numbered and odd-numbered thresholds are approached, we now turn to the analysis in terms of the corresponding three-body wave functions. Since the wave functions carry the full information about a quantum state, we aim to obtain a deeper insight into three-body universality from their study.

For this purpose, we compare the three-body wave functions for the two finite-range potentials of shape f_L, equation (3), and f_G, equation (4), tuned near the excited-state thresholds r = 1, 2, 3, to the ones resulting from a contact heavy–light interaction with f_δ , equation (5), by means of the fidelity F_r,n defined by equation (11). As previously mentioned we use the results for the contact interaction as scale-invariant representatives for the case r = 0. Moreover, in order to compare the universal behavior for weakly-bound excited heavy–light states of the same symmetry, we consider the overlap ${\left\vert \langle {\psi }_{3,n}\vert {\psi }_{1,n}\rangle \right\vert }^{2}$ of the three-body wave functions for the two odd-numbered thresholds r = 1 and r = 3.

In figure 3 we present the fidelity F_r,n as a function of the two-body binding energy ${\mathcal{E}}_{r}^{(2)}$. Throughout the subfigures we indicate by filled red and empty blue symbols the fidelities for a heavy–light interaction of shape f_L, equation (3), and f_G, equation (4), respectively. Analogous to figure 2, the top three rows show the results for three different heavy–light thresholds ${\mathcal{E}}_{1}^{(2)}\to {0}^{-}$, ${\mathcal{E}}_{2}^{(2)}\to {0}^{-}$ and ${\mathcal{E}}_{3}^{(2)}\to {0}^{-}$, respectively. In the last row we separately display the overlap |⟨ψ_3,n |ψ_1,n ⟩|². The left and right columns distinguish the case of two heavy bosons (n = 0, 2, 4) or fermions (n = 1, 3, 5), respectively.

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A comparison of the fidelities between the two different finite-range heavy–light interactions shows a similar behavior as for the three-body energies. In all subfigures the corresponding red and blue data sets are very close to each other and thus suggest the universality of the wave functions of three-body bound states.

Next, we analyze the universal behavior of the fidelities for different weakly-bound states in the heavy–light subsystems. In contrast to the results for the energies, which show the same universal limit for all values of r, we observe that the fidelities near the three excited-state thresholds display a fundamentally different behavior. A fidelity of unity indicates that the two wave functions overlap perfectly and are therefore identical. On the other hand a vanishing fidelity indicates that the two wave functions ψ_r,n and ${\psi }_{n}^{\star }$ are orthogonal. For the even-numbered threshold (r = 2) the fidelities F_2,n are increasing and approaching unity as the threshold is approached, as demonstrated in figures 3(c) and (d). Contrarily, for the odd-numbered thresholds (r = 1 and r = 3) the fidelities F_1,n and F_3,n are far below unity, as evident from figures 3(a), (b), (e) and (f). However, the overlap |⟨ψ_3,n |ψ_1,n ⟩|² between the three-body wave functions near the odd-numbered thresholds approaches again unity as ${\mathcal{E}}_{r}^{(2)}\to {0}^{-}$, see figures 3(g) and (h).

To conclude, these results for the fidelities indicate that the three-body wave functions approach two distinct universal limits, one for symmetric, and another one for antisymmetric weakly-bound heavy–light states. The universal limits for the symmetric case are those for the zero-range contact interaction. On the other hand, the universal limits for the antisymmetric case might correspond to those obtained for an odd-wave pseudopotential [29]. We recall that such a dependence on the symmetry of the weakly-bound heavy–light state has not become apparent in the universal limit of the respective three-body energies.

Finally, we analyze the universal behavior for different excited three-body states characterized by the index n. Near the even-numbered threshold (r = 2), the fidelities F_2,n are larger for fermionic heavy particles (odd n) than for bosonic ones (even n) and the limit value for ${\mathcal{E}}_{2}^{(2)}\to {0}^{-}$ is reached faster. This is in accordance with the results for the energies shown in figure 2. In contrast, near the odd-numbered thresholds (r = 1 and r = 3) the overlap |⟨ψ_3,n |ψ_1,n ⟩|² shows no significant difference between bosons and fermions. The fidelities F_1,n and F_3,n are far below unity and therefore we refrain from analyzing them in more detail.

In order to reinforce our results, we display contour plots of the wave functions in figure 4. We restrict ourselves here to the potential shape f_G, equation (4), of the heavy–light interaction and to the most deeply-bound states for the case of bosons(n = 0) and fermions (n = 1). The wave functions are displayed in momentum space

$$\begin{equation}{P}_{1}\equiv \frac{{p}_{1}}{\sqrt{2\left\vert {\mathcal{E}}_{r}^{(2)}\right\vert }}\enspace ,\qquad {K}_{23}\equiv \frac{{k}_{23}}{\sqrt{2\left\vert {\mathcal{E}}_{r}^{(2)}\right\vert }}\end{equation} \tag{ 19 }$$

scaled by the two-body energy ${\mathcal{E}}_{r}^{(2)}$ of the corresponding weakly-bound heavy–light state.

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The four rows distinguish the two-body thresholds r = 0 (top), r = 1 (center-top), r = 2 (center-bottom) and r = 3 (bottom). The diagrams are split into two halves, the left one (a) corresponds to bosons (n = 0) whereas the right one (b) corresponds to fermions (n = 1). Each half is split again into two columns, the left column in each half displays the wave functions for the two-body binding energy $\left\vert {\mathcal{E}}_{r}^{(2)}\right\vert =1{0}^{-3}$, and the right column for $\left\vert {\mathcal{E}}_{r}^{(2)}\right\vert =1{0}^{-7}$. The contours of zero value for the wave functions are highlighted by black dashed lines.

First we discuss the even-numbered thresholds. The three-body wave functions near the ground state threshold r = 0 (top) and the excited-state threshold r = 2 (center-bottom) look already very similar for the respective state n. This similarity is further enhanced when the two-body threshold is approached. This behavior is in line with the fidelities increasing towards unity as presented in figures 3(c) and (d). Hence, from this we deduce that the three-body wave functions for the even-numbered heavy–light thresholds approach the same universal limit.

Next, we consider the odd-numbered thresholds. As presented in figure 4, the three-body wave functions forr = 1 (center-top) and r = 3 (bottom) are very similar, but look fundamentally different from those obtained near the even-numbered thresholds r = 0 (top) and r = 2 (center-bottom). This is in agreement with the high overlap shown in figures 3(g) and (h) and the low values of the corresponding fidelities (figures 3(a), (b), (e) and (f)). As a result, this expresses that the same universal limit of the three-body wave functions is approached near all odd-numbered heavy–light thresholds. However, this limit is distinct from the limit near even-numbered thresholds.

Subsequently, we analyze the three-body wave functions depending on the index n in order to highlight differences and similarities between the cases of two heavy bosons (n = 0) and fermions (n = 1). This property of the two heavy particles can be identified for all values of r from the symmetry with respect to the line K₂₃ = 0, as evident from figure 4 and summarized by equation (14). As pointed out already in section 2.4, this statement translates in coordinate space to the same symmetry with respect to the exchange y₂₃ → −y₂₃. As an example, the three-body wave functions in coordinate representation are displayed in reference [17] for the contact interaction (r = 0). However, already from figure 4 it is apparent that the bosonic wave functions have parity +1, whereas the fermionic ones have parity −1, which is summarized by equation (14).

For these two different kinds of heavy particles, we now discuss the speed of convergence towards a scale-invariant three-body bound state when a particular two-body threshold is approached. With scale-invariance we mean here that the wave functions expressed in the scaled momenta, equation (19), behave like a constant as a function of ${\mathcal{E}}_{r}^{(2)}$. Near the ground-state threshold (r = 0) both bosonic (figure 4(a)) and fermionic (figure 4(b)) wave functions look identical for the presented two-body binding energies $\vert {\mathcal{E}}_{r}^{(2)}\vert =1{0}^{-3}$ and $\vert {\mathcal{E}}_{r}^{(2)}\vert =1{0}^{-7}$. This is because they have already reached the scale-invariant limit [17]. However, for the excited-state thresholds (r > 0), the structure of the bosonic wave functions (figure 4(a)) still changes quite significantly when decreasing the two-body binding energy. On the other hand, the fermionic wave functions (figure 4(b)) are already much closer to the scale-invariant regime. This behavior of the wave functions also enables a better understanding of the results presented in figure 3. Indeed, the fidelities for r = 2 are lower for bosons, figure 3(c), than for fermions, figure 3(d). Consequently, we observe a major difference between bosons and fermions in their speed of convergence towards the scale-invariant regime. This is in line with the results for the energy spectrum, discussed in subsection 3.1. In conclusion, we note that the difference in the wave functions having an extremum (bosons, figure 4(a)) or zero (fermions, figure 4(b)) at the line K₂₃ = 0 might explain this faster convergence of the energies and wave functions for the fermionic case in comparison to the bosonic one.

Finally, we address another aspect of the universal behavior of the three-body bound states. Indeed, in each column of figure 4, that is for fixed two-body binding energies ${\mathcal{E}}_{r}^{(2)}$, the wave functions for both bosons and fermions associated with the odd-numbered thresholds (r = 1 and r = 3) look very similar. We point out that this is true although the bosonic ones have not yet reached the scale-invariant regime. This also explains why the overlap |⟨ψ_3,n |ψ_1,n ⟩|² between the three-body wave functions near odd-numbered thresholds does not display a significant difference as shown in figure 3(g) for the bosonic, and in figure 3(h) for the fermionic case. We expect the same behavior when comparing the results for any two excited-state thresholds for which the associated weakly-bound states have the same symmetry, i.e. r = 2 and r = 4.

We start by considering the three-body Schrödinger equation (7) in representation-free form

$$\begin{equation}\left({H}_{0}+{V}_{31}+{V}_{12}\right)\vert \psi \rangle =\mathcal{E}\vert \psi \rangle .\end{equation} \tag{ A1 }$$

Here, H₀ is the kinetic energy operator without the center-of-mass motion, whereas V₃₁ and V₁₂ describe the pair-interactions between particles 3 and 1, and particles 1 and 2, respectively.

According to the Faddeev approach [20, 58], the solution of the Schrödinger equation (A1) is given by the superposition

$$\begin{equation}\vert \psi \rangle \equiv \vert {\phi }^{(2)}\rangle +\vert {\phi }^{(3)}\rangle \end{equation} \tag{ A2 }$$

of |ϕ⁽²⁾⟩ and |ϕ⁽³⁾⟩, related to the so-called Faddeev components

$$\begin{equation}\vert {{\Phi}}^{(2)}\rangle \equiv {G}_{0}^{-1}\vert {\phi }^{(2)}\rangle \end{equation} \tag{ A3a }$$

$$\begin{equation}\vert {{\Phi}}^{(3)}\rangle \equiv {G}_{0}^{-1}\vert {\phi }^{(3)}\rangle .\end{equation} \tag{ A3b }$$

Here, ${G}_{0}={(\mathcal{E}-{H}_{0})}^{-1}$ is the three-body Green function corresponding to H₀. The Faddeev components obey the Faddeev equations

$$\begin{equation}\vert {{\Phi}}^{(2)}\rangle ={T}_{31}{G}_{0}\enspace \vert {{\Phi}}^{(3)}\rangle \end{equation} \tag{ A4a }$$

$$\begin{equation}\vert {{\Phi}}^{(3)}\rangle ={T}_{12}{G}_{0}\enspace \vert {{\Phi}}^{(2)}\rangle ,\end{equation} \tag{ A4b }$$

that is a system of two coupled integral equations.

The matrices T₃₁, T₁₂ are the T-matrices of the three-body system if only the pair-interaction V₃₁ between particle 3 and 1, or V₁₂ between particle 1 and 2 respectively, is present. Hence, T₃₁ and T₁₂ fulfill the Lippmann–Schwinger equation [20]

$$\begin{equation}{T}_{31}={V}_{31}+{V}_{31}{G}_{0}{T}_{31}\end{equation} \tag{ A5a }$$

$$\begin{equation}{T}_{12}={V}_{12}+{V}_{12}{G}_{0}{T}_{12}.\end{equation} \tag{ A5b }$$

Three-body systems can be described in three different arrangements of relative motions, corresponding to three sets of so-called Jacobi momenta [20]. For {i, j, l} = {1, 2, 3} and the two cyclic permutations thereof we denote by

$$\begin{equation}{k}_{ij}\equiv \frac{{m}_{j}{k}_{i}-{m}_{i}{k}_{j}}{{m}_{i}+{m}_{j}}\end{equation} \tag{ A6 }$$

the relative momentum between the particles i and j, whereas

$$\begin{equation}{p}_{l}\equiv \frac{({m}_{i}+{m}_{j}){k}_{l}-{m}_{l}({k}_{i}+{k}_{j})}{{m}_{i}+{m}_{j}+{m}_{l}}\end{equation} \tag{ A7 }$$

is the momentum of particle l relative to the center-of-mass of the pair of particles i, j. The momentum and mass of each particle are denoted by k_i and m_i respectively, with i = 1, 2, 3.

The relations between the Jacobi momenta read

$$\begin{equation}{k}_{12}({k}_{23},{p}_{1})=-\frac{{\alpha }_{y}}{2}{k}_{23}+{\alpha }_{x}{p}_{1}\end{equation} \tag{ A8a }$$

$$\begin{equation}{p}_{3}({k}_{23},{p}_{1})=-{k}_{23}-\frac{1}{2}{p}_{1}\end{equation} \tag{ A8b }$$

$$\begin{equation}{k}_{31}({k}_{23},{p}_{1})=-\frac{{\alpha }_{y}}{2}{k}_{23}-{\alpha }_{x}{p}_{1}\end{equation} \tag{ A9a }$$

$$\begin{equation}{p}_{2}({k}_{23},{p}_{1})={k}_{23}-\frac{1}{2}{p}_{1}\end{equation} \tag{ A9b }$$

$$\begin{equation}{k}_{12}({k}_{31},{p}_{2})=-\frac{\alpha }{1+\alpha }{k}_{31}-{\alpha }_{x}{\alpha }_{y}{p}_{2}\end{equation} \tag{ A10a }$$

$$\begin{equation}{p}_{3}({k}_{31},{p}_{2})={k}_{31}-\frac{\alpha }{1+\alpha }{p}_{2},\end{equation} \tag{ A10b }$$

with the mass ratio α = M/m and the coefficients

$$\begin{equation}{\alpha }_{x}\equiv \frac{1+2\alpha }{2(1+\alpha )}\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {\alpha }_{y}\equiv \frac{2}{1+\alpha }.\end{equation} \tag{ A11 }$$

The system of Faddeev equations (A4a) and (A4b) can be reduced to a single equation using the exchange symmetry between the two identical heavy particles in the three-body system. The exchange of particles 2 and 3 is most easily described in the set of Jacobi momenta k₂₃ and p₁ by the transformation k₂₃ → −k₂₃. From equations (A8a)–(A9b) we deduce

$$\begin{equation}{k}_{31}(-{k}_{23},{p}_{1})=-{k}_{12}({k}_{23},{p}_{1})\end{equation} \tag{ A12a }$$

$$\begin{equation}{p}_{2}(-{k}_{23},{p}_{1})={p}_{3}({k}_{23},{p}_{1}).\end{equation} \tag{ A12b }$$

We distinguish two cases: the identical particles 2 and 3 are either bosons or fermions. Depending on this property, the total three-body wave function ψ(−k₂₃, p₁) = ±ψ(k₂₃, p₁) has to preserve (bosons) or flip (fermions) its sign under exchange of the two particles. This fact can be expressed in terms of the Faddeev components via equation (A2) which yields

$$\begin{align}\hfill {{\Phi}}^{(2)}(-{k}_{12},{p}_{3})+{{\Phi}}^{(3)}(-{k}_{31},{p}_{2})& =\pm \left[{{\Phi}}^{(2)}({k}_{31},{p}_{2})\right.\hfill \\ \hfill & \left.+{{\Phi}}^{(3)}({k}_{12},{p}_{3})\right]\hfill \end{align} \tag{ A13 }$$

after the application of ${G}_{0}^{-1}$.

As a result, we can relate the two Faddeev components to each other

$$\begin{equation}{{\Phi}}^{(3)}({k}_{12},{p}_{3})=\pm {{\Phi}}^{(2)}(-{k}_{12},{p}_{3})\end{equation} \tag{ A14a }$$

$$\begin{equation}{{\Phi}}^{(3)}({k}_{31},{p}_{2})=\pm {{\Phi}}^{(2)}(-{k}_{31},{p}_{2}).\end{equation} \tag{ A14b }$$

With help of equations (A8a)–(A9b) this allows us to cast the total wave function in the form

$$\begin{align}\hfill \psi ({k}_{23},{p}_{1})& ={\phi }^{(2)}\left(-\frac{{\alpha }_{y}}{2}{k}_{23}-{\alpha }_{x}{p}_{1},{k}_{23}-\frac{1}{2}{p}_{1}\right)\hfill \\ \hfill & \quad \pm {\phi }^{(2)}\left(\frac{{\alpha }_{y}}{2}{k}_{23}-{\alpha }_{x}{p}_{1},-{k}_{23}-\frac{1}{2}{p}_{1}\right).\hfill \end{align} \tag{ A15 }$$

Since Φ⁽³⁾ can be obtained from Φ⁽²⁾ via equation (A14), it is sufficient to only obtain the Faddeev component Φ⁽²⁾ from its Faddeev equation (A4a). In momentum representation this equation reads

$$\begin{align}\hfill {{\Phi}}^{(2)}({k}_{31},{p}_{2})& =\pm \iiiint\frac{\mathrm{d}{k}_{31}^{\prime }\mathrm{d}{p}_{2}^{\prime }\mathrm{d}{k}_{31}^{\prime \prime }\mathrm{d}{p}_{2}^{\prime \prime }}{{(2\pi )}^{4}}\hfill \\ \hfill & \quad \times \langle {k}_{31},{p}_{2}\vert {T}_{31}(\mathcal{E})\vert {k}_{31}^{\prime },{p}_{2}^{\prime }\rangle \enspace \langle {k}_{31}^{\prime },{p}_{2}^{\prime }\vert {G}_{0}(\mathcal{E})\vert {k}_{31}^{\prime \prime },{p}_{2}^{\prime \prime }\rangle \hfill \\ \hfill & \quad \times {{\Phi}}^{(2)}\left[-{k}_{12}({k}_{31}^{\prime \prime },{p}_{2}^{\prime \prime }),{p}_{3}({k}_{31}^{\prime \prime },{p}_{2}^{\prime \prime })\right].\hfill \end{align} \tag{ A16 }$$

Using the momentum representation of the free-particle three-body Green function [20, 60]

$$\begin{equation}\langle {k}^{\prime },{p}^{\prime }\vert {G}_{0}(\mathcal{E})\vert {k}^{{\prime\prime}},{p}^{{\prime\prime}}\rangle ={(2\pi )}^{2}\enspace \frac{\delta ({k}^{\prime }-{k}^{{\prime\prime}})\enspace \delta ({p}^{\prime }-{p}^{{\prime\prime}})}{\mathcal{E}-\frac{1}{2}{k}^{\prime 2}-\frac{1}{2}{\alpha }_{x}{\alpha }_{y}{p}^{\prime 2}}\end{equation} \tag{ A17 }$$

as well as the relation of the three-body T-matrix to the off-shell two-body t-matrix [20, 60]

$$\begin{equation}\langle k,p\vert {T}_{31}(\mathcal{E})\vert {k}^{\prime },{p}^{\prime }\rangle =2\pi \enspace \delta (p-{p}^{\prime })\enspace t\left(k,{k}^{\prime },\mathcal{E}-\frac{1}{2}{\alpha }_{x}{\alpha }_{y}{p}^{2}\right),\end{equation} \tag{ A18 }$$

allows us to perform the integrations over ${p}_{2}^{\prime }$, ${k}_{31}^{\prime \prime }$ and ${p}_{2}^{\prime \prime }$ in equation (A16). Expressing further via equations (A10a) and (A10b) the momenta −k₁₂ and p₃ in terms of k₃₁ and p₂ for the arguments of Φ⁽²⁾(−k₁₂, p₃), we arrive at the integral equation

$$\begin{align}\hfill {{\Phi}}^{(2)}(k,p)& =\pm \int \frac{\mathrm{d}{k}^{\prime }}{2\pi }\frac{t\left(k,{k}^{\prime },\mathcal{E}-\frac{1}{2}{\alpha }_{x}{\alpha }_{y}{p}^{2}\right)}{\mathcal{E}-\frac{1}{2}{k}^{\prime 2}-\frac{1}{2}{\alpha }_{x}{\alpha }_{y}{p}^{2}}\hfill \\ \hfill & \quad \times {{\Phi}}^{(2)}\left(\frac{\alpha }{1+\alpha }{k}^{\prime }+{\alpha }_{x}{\alpha }_{y}p,{k}^{\prime }-\frac{\alpha }{1+\alpha }p\right).\hfill \end{align} \tag{ A19 }$$

Here we have omitted the indices of the momenta k₃₁ and p₂.

By introducing the change of variables, ${k}^{\prime }\equiv q+\frac{\alpha }{1+\alpha }p$, we cast equation (A19) into the form

$$\begin{align}\hfill {{\Phi}}^{(2)}(k,p)& =\pm \int \frac{\mathrm{d}q}{2\pi }\enspace \frac{t\left(k,q+\frac{\alpha }{1+\alpha }p,\mathcal{E}-\frac{1}{2}{\alpha }_{x}{\alpha }_{y}{p}^{2}\right)}{\mathcal{E}-\frac{1}{2}{q}^{2}-\frac{1}{2}{p}^{2}-\frac{\alpha }{1+\alpha }pq}\hfill \\ \hfill & \quad \times {{\Phi}}^{(2)}\left(p+\frac{\alpha }{1+\alpha }q,q\right)\hfill \end{align} \tag{ A20 }$$

eliminating the dependence on p in the second argument of Φ⁽²⁾ inside the integral.

We now transform equation (A20) into the form of a Fredholm integral equation of the second kind. For most potentials, the off-shell t-matrix $t(k,{k}^{\prime },\mathcal{E})$ appearing in equation (A20) cannot be written as a product of functions of k or k' only. However, it is possible to expand the t-matrix into a sum of separable products [19–21]

$$\begin{equation}t(k,{k}^{\prime },\mathcal{E})\equiv \sum\limits _{\nu =0}^{\infty }\enspace {\tau }_{\nu }(\mathcal{E}){g}_{\nu }(k,\mathcal{E}){g}_{\nu }({k}^{\prime },\mathcal{E}).\end{equation} \tag{ A21 }$$

Here, the functions g_ν and τ_ν ≡ −η_ν /(1 − η_ν ) are determined by the integral equation

$$\begin{equation}\int \frac{\mathrm{d}{k}^{\prime }}{2\pi }V(k,{k}^{\prime })\frac{1}{\mathcal{E}-\frac{1}{2}{k}^{\prime 2}}\enspace {g}_{\nu }({k}^{\prime },\mathcal{E})={\eta }_{\nu }(\mathcal{E}){g}_{\nu }(k,\mathcal{E})\end{equation} \tag{ A22 }$$

with the momentum representation

$$\begin{equation}V(k,{k}^{\prime })={v}_{0}\int \mathrm{d}\xi f(\xi ){\mathrm{e}}^{-\mathrm{i}(k-{k}^{\prime })\xi }\end{equation} \tag{ A23 }$$

of the heavy–light potential v(ξ) = v₀ f(ξ). The functions g_ν are orthonormal with respect to the relation

$$\begin{equation}\int \frac{\mathrm{d}k}{2\pi }\frac{{g}_{\nu }(k,\mathcal{E})\enspace {g}_{{\nu }^{\prime }}(k,\mathcal{E})}{\mathcal{E}-\frac{1}{2}{k}^{2}}=-{\delta }_{\nu ,{\nu }^{\prime }},\end{equation} \tag{ A24 }$$

where δ_ν,ν' denotes the Kronecker symbol.

By inserting equation (A21) into (A20), we arrive at the expansion

$$\begin{equation}{{\Phi}}^{(2)}(k,p)\equiv \pm \sum\limits _{\nu =0}^{\infty }\enspace {g}_{\nu }(k,{\mathcal{E}}_{p}){\tau }_{\nu }({\mathcal{E}}_{p}){\varphi }_{\nu }(p,\mathcal{E})\end{equation} \tag{ A25 }$$

for Φ⁽²⁾ with ${\mathcal{E}}_{p}\equiv \mathcal{E}-{\alpha }_{x}{\alpha }_{y}{p}^{2}/2$ and

$$\begin{equation}{\varphi }_{\nu }(p,\mathcal{E})\equiv \int \mathrm{d}q\enspace \frac{{g}_{\nu }\left(q+\frac{\alpha }{1+\alpha }p,{\mathcal{E}}_{p}\right){{\Phi}}^{(2)}\left(p+\frac{\alpha }{1+\alpha }q,q\right)}{\mathcal{E}-\frac{1}{2}{q}^{2}-\frac{1}{2}{p}^{2}-\frac{\alpha }{1+\alpha }pq}.\end{equation} \tag{ A26 }$$

Finally, by substituting equation (A25) into (A26), we derive a system of coupled integral equations

$$\begin{align}\hfill {\varphi }_{\lambda }(p,\mathcal{E})& =\pm \sum\limits _{\nu =0}^{\infty }\int \frac{\mathrm{d}q}{2\pi }\enspace {\tau }_{\nu }({\mathcal{E}}_{q}){\varphi }_{\nu }(q,\mathcal{E})\hfill \\ \hfill & \quad \times \frac{{g}_{\lambda }\left(q+\frac{\alpha }{1+\alpha }p,{\mathcal{E}}_{p}\right)\enspace {g}_{\nu }\left(p+\frac{\alpha }{1+\alpha }q,{\mathcal{E}}_{q}\right)}{\mathcal{E}-\frac{1}{2}{q}^{2}-\frac{1}{2}{p}^{2}-\frac{\alpha }{1+\alpha }pq}\hfill \end{align} \tag{ A27 }$$

in only a single variable for the functions ${\varphi }_{\nu }(p,\mathcal{E})$.