The Faddeev and Schrödinger approaches to Efimov states—a numerical comparison

We compare effective hyperradial three-body potentials calculated using the S-wave part of the Faddeev equations to calculations using the full Schrödinger equation. As two-body model potential we test both a short-range potential and a Lennard-Jones potential with van der Waals tail. In the former case we find excellent agreement between the two methods for the lowest adiabatic state, indicating that the Faddeev method can be a useful tool also for numerical computations. For excited states the two methods show important differences, but agree for hyperradii larger than about five times the range of the potential (independent of the value of the scattering length and of the number of bound states). For the van der Waals potential, we focus on how well the Faddeev method reproduces the so-called van der Waals universality. We find that indeed the universality is manifest also using this method, but at a slightly different value of the universal parameter κ∗ . We further derive an efficient method to solve the integro-differential equation arising in the Faddeev method.


Introduction
Ever since their theoretical prediction in the early 1970s, Efimov states have fascinated [1].These three-body states with peculiar universal properties were sought for in many physical systems, until in 2006 when they finally were realised in a gas of ultracold Cesium [2].This discovery has stimulated a lot of theoretical and experimental interest (for a review of recent progress see e.g.[3][4][5]), and Efimov states have now been found in a number of alkali systems [6][7][8][9][10][11][12][13].
Whereas most of the universal properties of Efimov physics can be derived from a simple zero-range model (ZRM) Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.interaction between the atoms [14], quantitative predictions require a more realistic description.A commonly used theoretical approach to Efimov states, and three-body systems in general, is the adiabatic hyperspherical method [15].In this method, the three internal coordinates (after separating out the centre of mass, and three Euler angles defining the overall rotation of the system) are chosen as a hyperradius ρ giving the overall size of the system, and two hyperangles Ω describing its internal configuration.The hyperangular part of the problem is then solved for a fixed hyperradius, yielding hyperradial potential curves, in a manner similar to the Born-Oppenheimer approximation for diatomic molecules [16,17].
The adiabatic hyperspherical method comes in two varieties.Either the full Schrödinger equation is solved or the wavefunction is separated into three Faddeev components, giving in general three coupled equations [18].For three identical particles the coupled Faddeev equations reduce to just a single equation, which can be expanded in partial waves with the angular momentum l of the pairwise interactions.Whereas the couplings between different angular momenta in principle can be calculated, the problem is greatly simplified by restricting the solution to the part with l = 0.For Efimov states, this approximation is well motivated, as the relevant physics is dictated by low-energy scattering where it is well known that the s-wave dominates.
The simplest model that captures the Efimov physics is the Faddeev formalism combined with a zero-range potential with the scattering length a as its only parameter [19].It is useful to have a model which captures the essential physics, without the full numerical complexity of the Schrödinger equation.While Efimov physics can be derived from the ZRM, it cannot capture any physics at length scales shorter than the scattering length.
Generally the universal three-body theory includes a nonuniversal three-body parameter.In one formulation this parameter represents a cutoff at short length scales.However, it can be formulated in several different ways that in the ideal universal scenario are equivalent [14].Another form of the parameter is κ * = mE 3b /h 2 , where E 3b is the binding energy of the most tightly bound Efimov state for infinite scattering length.A third form, which is directly accessible in experiments, is the negative scattering length a − at which the Efimov state joins the continuum, thus giving rise to a peak in the threebody recombination rate, when the scattering length is tuned close to this value.The three-body parameter was thought to be essentially random, i.e. not following any law, and hence necessary to determine for each system individually.However, a few years after the realisation of Efimov states in ultracold atomic gases, it was unexpectedly found that experimental results for a − seemed to group around the value −9r vdW , where r vdW is the van der Waals length, characteristic of the range of the atom-atom interaction [20,21].Theoretical studies traced this universality to the acceleration from the van der Waals interaction, which causes a suppression of two-body correlations at short distances.This suppression then has the van der Waals length as its only parameter.In the effective three-body potentials this suppression is manifested as a barrier around 2r vdW , and consequently shields the three-body system from the non-universal physics at shorter distances [22][23][24].
Most experiments studying Efimov states in alkalis rely on magnetic tuning of the scattering length through the hyperfine coupling to a quasibound molecular state, i.e. a Feshbach resonance [25].As originally suggested by Petrov [26], and more recently supported by more and more experimental evidence [21,27,28], the universal properties of Efimov states with a tuned scattering length depend on the strength of the coupling between the closed molecular channel and the open scattering channel, or, in other words, the width of the resonance.This has inspired theoretical treatments that explicitly include the coupling between the two channels at the two-body level.This can be done either using the hyperspherical adiabatic method in a multi-channel generalisation of the treatment investigated in this paper, based on either the Faddeev equations with zero-range interactions [29][30][31][32], or the Schrödinger equation with zero-range [33] or van der Waals [12,13,34,35] interactions.A completely different approach is to use field theoretical techniques with creation/annihilation operators for both atoms and molecules, and an interaction term coupling a pair of atoms to a molecular state [36][37][38][39].This method has also been generalised to more than one molecular state, i.e. overlapping resonances [40].Yet another approach is to model the two-body collisions using a separable T-matrix [41].Of particular interest is models employing a separable potential constructed using a multi-channel generalisation of the method by Ernst, Shakin and Thaler [42], which have been used to study the van der Waals universality at both broad and narrow Feshbach resonances [39,43,44].
With a multitude of theoretical models and methods, it is important to establish the limits of their validity.In this paper we compare side-by-side results using three of the models mentioned above, i.e. the full Schrödinger equation (SE), the S-wave Faddeev method (SFM), and the ZRM.Our philosophy is that it is useful to adopt as simple model as possible, both because of the smaller numerical effort and the simplicity of interpretations.However, care must of course be taken to avoid losing important physics by oversimplifying the theory.Below we seek to determine just how far this simplification can go.This has informed our choice of method for a multichannel generalisation to study the van der Waals universality at narrow Feshbach resonances [35].

Theory
Here we briefly summarise the Schrödinger and Faddeev approaches to the adiabatic hyperspherical method for threebody systems.More in-depth analysis can be found, e.g. in [16,17,45] for the Schrödinger approach and [5,14,18] for the Faddeev approach.In this work we shall only be concerned with states having total orbital angular momentum J = 0, as these dominate in the ultracold regime and are the states giving rise to the Efimov effect.We consider the Hamiltonian for three atoms of mass m, interacting only via pairwise interactions V(r) where r i = |r j − r k | is the distance between atom j and atom k.
Since the interactions only depend on the relative coordinates of the atoms, we conveniently separate out the centre-of-mass motion from the internal dynamics of the atoms by introducing mass scaled Jacobi coordinates, where for equal masses µ jk = χ/2 and µ i,jk = 2χ/3.(Unfortunately, different conventions exist.Here we follow [17] and use χ = √ 3, so that µ i,jk = µ −1 jk = √ 2/3 1/4 .Some works, e.g.[19], instead use χ = 1.)A general three-body system can now be described by combining the two Jacobi vectors into a single position vector parameterised by six hyperspherical coordinates.Three of these are internal coordinates that describe the size and internal motion of the three-atomic system.The remaining three are external in the sense that they are used for specifying the spatial orientation of the system as a whole, i.e. the triangle formed by the three particles, relative to a space-fixed frame.The external variables are commonly taken to be the Euler angles α, β, and γ, and they can also be separated out if we only care about the internal motion of the atoms and not rotations of the system.
Thus we are left with three internal degrees of freedom, which can be expressed in terms of the hyperradius and two hyperangles.We use the Jacobi coordinates to define the hyperradius as Note that while the Jacobi coordinates depend on the configuration i chosen, the hyperradius is invariant.
There are many ways of defining the hyperangles, and the different definitions are in general not invariant under particle permutations.The choice of hyperangles usually depends on the method adopted for solving the three-body problem.We will use the customary choice of Delves coordinates [46] in combination with the Faddeev method and a modified set of Smith-Whitten coordinates [47,48] when solving the full Schrödinger equation.
In terms of the hyperspherical coordinates the Hamiltonian operator takes the form where µ 3b = m/χ is the reduced mass of the three atoms and Λ is called the hyperangular momentum operator, whose form depends on the choice of hyperangles.After substituting the rescaled wavefunction ψ = ρ 5/2 Ψ into (4), the Schrödinger equation describing the three particles takes the form (5)

Adiabatic hyperspherical representation.
The first step in solving for ( 5) is to expand the wavefunction ψ n (ρ, Ω) in terms of a complete orthonormal set of channel functions Φ ν (ρ; Ω) that depend parametrically on ρ, with the radial wavefunctions F νn (ρ) as expansion coefficients.
We treat the hyperradius as an adiabatic variable, and thus solve the hyperangular part of the problem for a fixed ρ.The channel functions Φ ν are then eigenfunctions of the adiabatic eigenvalue equation in which the adiabatic Hamiltonian is given by and the eigenvalues U ν (ρ) are adiabatic potential energy curves obtained by solving equation ( 6) at a number of different hyperradii.These hyperradial potentials contain most of the three-body physics and can be viewed as the three-body equivalent to the Born-Oppenheimer potential in the two-body problem.
The total wavefunction which is represented as an expansion in terms of the adiabatic states, can now be inserted into equation (5).After projecting out Φ µ and integrating over the hyperangular coordinates, the problem is reduced to solving a set of coupled ordinary differential equations given by Here P µν and Q µν are the nonadiabatic coupling terms defined as where the double brackets denote integration over the two hyperangles.It can easily be shown that P µν = −P νµ and hence P µµ = 0. On the other hand Q µµ ̸ = 0, so it is convenient to define the effective potential including this diagonal correction, We choose a modified set of Smith-Whitten's democratic coordinates, which allows for permutation symmetries for three identical particles to be imposed exactly.Following the mapping procedure described by Johnson [47], we define the hyperangles in terms of the Jacobi coordinate system i as {θ, ϕ i }, where (13) so that the two body-fixed Jacobi vectors lie in the xy plane.The range of θ is [0, π/2] and the span of ϕ i is restricted to the range [0, 2π] by requiring the wavefunction to be single valued.
The hyperangular coordinates can be understood by considering the geometry of the triangle formed by the three particles, with θ determining its shape, which is equilateral at θ = 0 and linear at θ = π/2, and ϕ i determining the particle arrangement at the vertices, where the particles permutate in a cyclical manner as ϕ i changes from 0 to 2π.In the case of three identical particles, translation and reflection symmetries further restrict the range of ϕ i to [0, π/3].For the sake of simplicity, we hereafter suppress the indices labelling each coordinate set.
To express the interaction for three particles V(ρ, θ, ϕ) as the pairwise sum in equation ( 5), we write the distances between the atoms in terms of the hyperspherical coordinates as We solve (6) by expanding θ and ϕ in B-splines as detailed in appendix 'Method for solving the Schrödinger equation'.

SFM 2.3.1. Hyperangles.
For the Faddeev method we instead use hyperangles Ω = {α i , η i } as defined by Delves [46].In terms of the Jacobi coordinate system i these are given by tan The squared hyperangular momentum operator takes the form where lxi and lyi are the angular momentum operators along x i and y i respectively.We are interested in states with total angular momentum J = 0, where Ĵ = lxi + lyi .Thus we have l xi = l yi = l and the expression above simplifies to This operator has eigenfunctions [49] Ψ and eigenvalues Here P l (x) is the Legendre function and 2 F 1 (a, b, c, x) is the hypergeometric function.
The kinetic part of the Hamiltonian is separable in the variables α i and η i , and can be expressed in any of the Jacobi systems i.The potential term, on the other hand, is not separable in the hyperangles, and involves all three Jacobi channels.However, when three-body interactions are neglected, the interaction can be written as a sum over the different channels (1).The Faddeev approach takes advantage of this fact by expanding the total wavefunction in three terms corresponding to the three Jacobi configurations.For identical particles all three terms take the same form, Using ( 7) and ( 19) and projecting with P l (cos η i ), the equation for the individual Faddeev components becomes where Adding the three components the Schrödinger equation ( 7) is obtained.However, it is not guaranteed that a solution to (24) leads to a physical solution of (7).There are also spurious solutions that when added together result in a Φ with zero norm.As these solutions clearly do not represent any physical state, care must be taken to exclude them.The difficulty has now been moved from the potential term to the integral over η i , where the integrand depends on coordinates in two different systems.The transformation between Jacobi systems is given by from which we deduce the transformations between hyperangles Using (27) we change the integration variable in (24) from η i to α j (holding α i fixed), where from (27) the limits of integration are α This set of coupled equations could in principle be solved by evaluating the couplings between different l:s, using the relations ( 27) and (28) to transform the arguments of the Legendre functions.If enough l-values are included the results should be the same as given by the Schrödinger equation.However, for Efimov physics, and ultracold collisions in general, the l = 0 term can be expected to dominate.Note also that the coupling integral enters multiplied by the potential, thus the coupling terms will vanish outside the range of the two-body interaction, which makes the method particularly well suited to shortrange potentials.
After restricting to l = 0 this becomes an equation of just one variable α, On the other hand this is now an integro-differential equation, whose solution requires different numerical techniques.In appendix 'Method for solving the Faddeev equations' we derive an efficient method to handle this complication.

ZRM
Below we shall also compare our results to the universal theory derived from the zero-range potential with strength proportional to the scattering length, For this contact interaction, the solution of ( 30) can be reduced to a transcendental equation [19] This form can be used to derive the universal properties of Efimov physics [14], since these exist at length scales larger than the range of the two-body potential.However, it can of course not capture any physics at shorter length scales, such as the van der Waals universality.Below we shall compare also to λ(ρ/a) calculated in this approximation.Clearly, only S-wave scattering contributes to this model.Thus, SE, SFM and ZRM represents a hierarchy of approximations, with SE the most accurate.It should be emphasised that all these models are subject to a number of additional approximations: the adiabatic approximation for the three-body system, the inclusion of only pairwise interactions, and the model potentials for the two-body system.

Long-range solutions
The adiabatic states are most easily classified according to their long-range behaviour, i.e. as ρ → ∞.Here there are two possibilities.The first possibility is that two atoms bind together, with the third atom far away.In this case the potential term will be reduced to the interaction between the bound atoms only, and the corresponding adiabatic potential will approach the energies of the two-body states.For large positive scattering lengths there is a very loosely bound two-body state with the universal binding energy [25] and the corresponding adiabatic state is labelled by ν = 0. Adiabatic states asymptotically converging to deeply bound states are labelled −1, −2, . . . .The other possibility is that the adiabatic state corresponds to a three-body continuum state.In this case we can set V = 0, and the asymptotic adiabatic potentials are given by the eigenvalues of Λ2 (22) where for identical bosons the allowed values are n = 0, 3, 6, . . .and l = 0, 2, 4, . . . .These states are numbered in ascending order starting with 0. From ( 7), ( 25) and ( 22) we find that λ ν (ρ) → λ nl + 4 as ρ → ∞.Thus the asymptotic limits of the continuum states can be found from the eigenvalues of Λ2 (22).
We can also analyse the long-range behaviour of (31).In this limit the sin-function on the right-hand-side must vanish.This happens for λ = 4n 2 , where n = 0 or a positive integer.However, for n = 0 and 2 also the left-hand-side vanishes, and thus λ = 0, 16 is a solution to (31) for any ρ and thus does not represent a physical state (these are examples of the spurious solutions mentioned above).For λ < 0, (31) also admits the long-range solution λ = −2ρ 2 /a 2 , leading to a potential approach the universal binding energy (32).
Note that the asymptotic forms above apply when ρ is larger than all other length scales in the system, including not only the range of the two-body potential, but also its scattering length.At ρ ≪ |a|, but still much larger than the range of the two-body potential, the right-hand-side of ( 31) can be neglected, and the universal solutions become λ = −1.0125,19.9389, 46.4900 . . . .The first value in this sequence gives rise to an attractive effective potential which can support an, in principle, infinite sequence of bound Efimov states.

Results
We compare hyperspherical potentials calculated using the Faddeev and Schrödinger methods using two different model potentials for the two-body interaction.

Short-range potential
Our first model potential is the widely used form This potential has the advantage that the scattering length is given by an analytical expression [16], and that the absence of a repulsive core, as well as the exponential decrease at large r, make it relatively easy to treat numerically.The scattering length as well as the number of bound states, can be varied by changing the potential depth d.We calculate the hyperangular eigenvalues λ ν (ρ) (25) and the effective potentials W ν (12), expressed in energy units E 0 = h2 /mr 2 0 , as d is changed, comparing results calculated using SFM and SE.
For shallow potentials, not supporting any bound state, we find almost perfect agreement between SFM and SE for the lowest eigenvalue.An example is shown in figure 1, where the potential was tuned so that a/r 0 = −100.The difference in λ between the two methods is less than 0.013 (0.6%) at all ρ.This demonstrates that the potential exhibiting the Efimov effect is completely dominated by S-wave scattering between the atoms.Our calculation extends to ρ/r 0 = 1000, corresponding to ρ/|a| = 10.At this value λ 0 has not yet reached its asymptotic value λ 0 (∞) = 4, but it is well described by the ZRM, which hence can be used in the asymptotic region ρ ≳ 50r 0 = 0.5|a|.
Turning to the first excited state (figure 1, bottom row) we still find good agreement between SE and SFM for ρ/r 0 ≳ 5.
At these hyperradii λ 1 ∼ 30 and thus the hyperradial potential U 1 (ρ) = (λ 1 (ρ) − 0.25)/ρ 2 is repulsive, and its detailed form has very little impact on low-energy scattering.In the limit ρ → 0 both solutions approach the known eigenvalue given by ( 22), but this is of no practical consequence both because the corresponding three-body energy is very high, and because the two-body model potential will not approximate the true atom-atom interaction at short range.The ZRM is correct only for ρ/r 0 ≳ 10 2 , i.e. in the far asymptotic region.Estimating the range of validity of SFM as kR ≲ 1 for k = √ W 1 ≃ λ 1 /(2ρ 2 ) and some length scale R, we find that this agrees with R ∼ r 0 as would be expected from the partial wave expansion (29).Similarly, we find that ZRM is valid for k ≲ 0.05/r 0 , or in other words that on length scales ≳ 20r 0 the two-body interaction will appear as effectively a zero-range interaction.
In figure 2 we also show the nonadiabatic coupling matrix elements between the two states.For ρ/r 0 ≳ 1, i.e. roughly the same region as we observe differences in the excited-state potential, we note some differences in the coupling matrix elements, in particular in Q 12 .The magnitude of differences are always less than 0.01, and thus far smaller than the differences in the excited-state potential constructed from λ 1 .We conclude that the differences in the non-adiabatic couplings will have a minor influence on the eigenvalues of (9).
Tuning the scattering length to other values gives similar results.In figure 3 we show an example where the potential just supports a loosely bound dimer state (a/r 0 = 100).As expected, the lowest potential curve converges to the two-body binding energy (32).
As a final example, in figure 4 we also show results for a potential supporting a deeply bound two-body state, again with a/r 0 = 100.In this case the good agreement is transferred to the hyperradial potential converging to the dimer + atom threshold.This is expected since the bound two-body state has angular momentum l x = 0, which for total J = 0 leaves only the possibility l y = 0 for the third atom.However, since the dimer-bound-state energy sets the threshold for any pair of atoms, the ν = 0 potential supporting the Efimov effect differs considerably between the two methods, except at large ρ.
Overall we find that, for the lowest adiabatic state, SFM agrees very well with the full SE results.However, for higher states there are differences for ρ ≲ 10.In all cases ZRM connects to the full Schrödinger results at even larger ρ ∼ 10 2 r 0 , and at intermediate ρ it deviates much more than SFM.

Lennard-Jones potential
While the Efimov effect can be realised in many physical settings, the recent experimental surge has been in systems of ultracold atoms.The characteristic property of the atom-atom potentials is that they are not short ranged, but have the longrange van der Waals form V(r) → −C 6 /r 6 , which corresponds to a van der Waals length scale usually defined as Comparison of λν (ρ) (25) (left) and the effective potential Wν (12) (right) calculated using the SE (black solid line), SFM (red dashed line), and the ZRM (blue dotted line).The top graphs show results for the lowest adiabatic state (ν = 0).The attractive well supports bound Efimov states.The bottom graphs shows results for the first excited adiabatic state (ν = 1).This hyperradial potential supports no bound states, and hence represents a three-body continuum state.Here the two-body interaction was given by the short-range potential (33) with d tuned so that a = −100r 0 and no bound two-body states exist.(10) and Q 12 (11) between the two states displayed in figure 1, using the same colour coding.
The corresponding energy scale is E vdW = h2 /(mr 2 vdW ).In our calculations we have used the Lennard-Jones 6-10 potential, which in scaled units takes the form By changing the parameter C = C 10 /(C 6 r 4 vdW ) we can tune the potential to support different numbers of bound states and different scattering lengths.
We explore the van der Waals universality at infinite scattering length, and for different numbers of bound states.1, but with the potential tuned so that a/r 0 = 100, thus supporting one loosely bound state with binding energy 1/a 2 .The inset shows that the three-body effective potential indeed converges to the two-body binding energy at large ρ.Effective three-body potentials calculated using either SE or SFM are displayed separately in figure 5. Note that except for details of the numerical implementation the SE results should be identical to those of figure 1 in [22].Clearly, both methods show a high degree of universality as the number of bound states is changed.It should be noted that calculations using SE give adiabatic potential curves U ν with a large number of avoided crossings.At these crossings the diagonal correction Q νν becomes very large, causing spikes in the effective potential W ν (12).The reason is that while due to the non-crossing rule potentials of the same symmetry cannot cross, the properties of the physical state continue through the crossing more or less unchanged.Thus staying on the same curve gives a rapid change in the properties of the state.Clearly, this is an artefact of the adiabatic approximation.We have therefore followed [22] and diabatically continued the curves through these crossings.For calculations using SFM these crossings are much less prevalent, probably because the number of curves is smaller since only dimer states with rotational quantum number l x = 0 are represented by the model.This makes the SFM results easier to interpret.
A closer inspection, state by state, does however reveal that there are small consistent differences between the two methods, see figure 6.The SFM results gives a somewhat shallower potential and its barrier crosses zero at a slightly larger ρ.In order to assess whether these small differences have any bearing on the van der Waals universality, we have calculated the parameter κ * using these diabatised potentials.The results are presented in table 1.We find that our results using SE are in good agreement with [22], where results within 15% of κ * = 0.226(2) are reported.The results using SFM are also internally roughly consistent, but deviate significantly from the results calculated with SE.This indicates that the suppression of the two-body correlation at short interatomic separations, which is thought to be responsible for the van der Waals universality, is present also in the SFM.However, at separations where this effect arises there is a substantial amount of l i ⩾ 2 components in the total wavefunction which are neglected in the SFM.

Numerical efficiency
From a numerical point of view the main difference between SE and SFM is that the adiabatic functions in the the latter only depend on the hyperangle α while the former depend on two hyperangles (θ, ϕ).Thus, with the basis functions in appendix 'Method for solving the Schrödinger equation', the size of the basis is N and grows quadratically with the number of B-splines in each dimension, while for SFM the growth is linear.
Thus, one expects that the SFM method will in general be much faster.To some extent one would expect that the potential gain by the SFM method is reduced because equation (29) does not result in a symmetric eigenproblem, and thus, unlike SE, its computation cannot take advantage of efficient numerical algorithms that exist for the symmetric case.
The computational time does of course depend drastically on the number of basis functions N b = N   small compared to those of more deeply bound states, but also because of the repulsive core of the LJ potential.In table 2 we summarise the measured processor times required by the two methods for these extreme cases.These results were obtained using the time command, and citing the user time (note that if several cores are used in parallel the real time can be significantly shorter).The computations were carried out on a Mac Pro 2013 running an Intel Xeon E5 processor.Eigenvalues were found using LAPACK routines as implemented in the mkl library from Intel.We find that when the nonadiabatic coupling matrices P and Q [( 10) and (11)] are not calculated the SFM method is 14 to 25 times faster.We also find that some of this gain disappears when the computation of nonadiabatic couplings is included.The reason for this is that for the SFM method we used standard finite difference methods to calculate the derivatives of the hyperangular wavefunctions, which requires evaluations at ρ ± dρ for some small dρ.For the SE method we could instead take advantage of a much more efficient method based on the Hellmann-Feynman theorem [50].

Conclusions
We have developed an efficient method for solving the l = 0 component of the Faddeev equation using B-splines.We compare the results obtained from this method to solutions of the full Schrödinger equation, and find very small differences for short-range potentials, as long as only the lowest hyperradial potential is considered.This may be enough for many purposes, but for two-body potentials supporting deeply bound dimers we find that the Faddeev and Schrödinger results deviate significantly for the hyperradial potential associated with Efimov physics.Nevertheless, in all cases the Faddeev method agrees with the Schrödinger results at sufficiently large hyperradii.This can potentially simplify calculations for large ρ, but still not large enough for the ZRM to be valid.We have also investigated whether the choice of method has any impact on the results for the van der Waals universality.Here we have compared the parameter κ * which is proportional the square root of the binding energy of the most tightly bound Efimov state at infinite scattering length.Our results for κ * are only approximate, since they are based on the diabatised potentials, and hence does not include all the couplings between different adiabatic states.Nevertheless, our results using SE, which are likely to be most affected by such couplings, agree well with those reported in [22].The results using SFM, on the other hand, are consistently about 25% smaller.We have not here evaluated the related parameter a − , but see no reason to doubt that similar considerations apply for it as well.
The SFM method offers many practical advantages, both in its numerical implementation, and in the interpretation of its results.However, our investigation suggests that care must be taken when it is used in quantitative comparisons.

Figure 1 .
Figure 1.Comparison of λν (ρ)(25) (left) and the effective potential Wν (12) (right) calculated using the SE (black solid line), SFM (red dashed line), and the ZRM (blue dotted line).The top graphs show results for the lowest adiabatic state (ν = 0).The attractive well supports bound Efimov states.The bottom graphs shows results for the first excited adiabatic state (ν = 1).This hyperradial potential supports no bound states, and hence represents a three-body continuum state.Here the two-body interaction was given by the short-range potential(33) with d tuned so that a = −100r 0 and no bound two-body states exist.

Figure 2 .
Figure 2. The nonadiabatic coupling matrix elements P 12(10) and Q 12(11) between the two states displayed in figure1, using the same colour coding.

Figure 3 .
Figure 3. Same as in figure1, but with the potential tuned so that a/r 0 = 100, thus supporting one loosely bound state with binding energy 1/a 2 .The inset shows that the three-body effective potential indeed converges to the two-body binding energy at large ρ.

Figure 4 .
Figure 4. Same as figure 3, but with the two-body potential supporting one deeply bound state (ν = −1) with binding energy E b = 4.04 in addition to the universal two-body bound state with E b ≃ 1/a 2 (ν = 0) (see also inset).Also displayed is the first continuum state (ν = 1).

Figure 5 .
Figure 5. Hyperradial potentials calculated using the Lennard-Jones potential (34), tuned to a = ∞ and supporting different numbers of bound states.Left panel: SE, Right panel: SFM.Note that the curves have been diabatised (see text).The potential is expressed in units of the van der Waals energy (see text).
a smooth potential, and no bound states, such as in figure1, it suffices with N b = 20 to get fully converged results.On the other hand the results in figure5require, when ρ is large, N b = 80 for convergence to within two decimal places.This is mainly because the computed energy is very

Figure 6 .
Figure 6.Hyperradial potentials calculated using the Lennard-Jones potential (34), tuned to a = ∞ and supporting different numbers of bound states.Black solid line, SE; red dashed line, SFM.Note that the curves have been diabatised (see text).

Table 1 .
The universal parameter κ * calculated using either SE of SFM for a two-body potential supporting n bound states. 1

Table 2 .
A comparison between computational times (in seconds per hyperradial point of the potential) required by the SE and SFM for different number N b of B-splines per dimension.For further details, see text.