K Jänkälä

M.-H Mikkelä

M Huttula

Valence photoionization of free, neutral, and size varied alkali metal clusters

Keywords: numbers: 36, 40, -c, 79, 60, Bm

Valence photoionization of free, neutral, and size-varied alkali metal clusters

K Jänkälä, M.-H Mikkelä, M Huttula

Introduction

Clusters are particles that consist of finite number of atoms or molecules thus forming an intermediate phase of matter between individual atoms and solid state. Large clusters reflect properties of solid state metals whereas small clusters resemble the characteristic properties of molecules and atoms. Photoelectron spectroscopy is an established tool to investigate electronic structure of clusters and solids [1,2]. Photoelectron spectra (PES) provide information about the structure and bonding mechanisms of clusters [3,4]. Previously laser sources have been used for such experiments on free clusters, mostly on charged anionic or cationic species [5,3]. The laser sources provide high radiation intensities, but are limited to energies well below 10 eV. Synchrotron radiation storage ring facilities offer tunable UV to Xray region photons which can be applied for valence and core level photoelectron spectroscopy measurements. The higher photon energy allows also valence studies which are free from, for example, complicated threshold effects and plasmon resonances [7,8]. Binding energy values of inner level states can be used to carry out cluster size estimations using models describing ionization potentials of finite size metal droplets [9].

Metal cluster production methods have been developed during the past decades [5]. Recently described Exchange Metal Cluster Source (EXMEC) [10] has made it feasible to measure PES from free, neutral, and size varied metal clusters. The source Confidential: not for distribution. Submitted to IOP Publishing for peer review 5 April 2011 offers cluster size variation with a possibility to adjust the mean size distribution of produced neutral metal clusters.

Since the early days of cluster physics numerous theoretical papers have been published (see, e.g. [5,6] and references therein). Most of the studies concerned the properties of ground states and simulation of the mass spectra [11,12,5,13]. Recently, the emphasis has moved more into describing the dynamical processes (e.g. [14]). The most common way of interpreting photoelectron spectra of clusters is to use density of states (DOS) approach applying directly the quantum chemistry codes [15]. This, however, requires calculation of stable cluster structures which becomes increasingly demanding for larger clusters. In contrast to high level calculations one may use semiempirical methods. For example, jellium model has been recently used to predict the photoelectron angular distribution of size-selected anionic clusters [16,17]. In those studies the model was used to describe the anisotropy of photoelectron emission of anionic clusters near to valence ionization energies.

Our goal in this study is to investigate how the simplest possible theoretical cluster model describes the valence photoelectron spectra of neutral small to medium size metal clusters recorded far above the ionization threshold. The emphasis is addressed to study on how the inclusion of the dipole matrix elements and simple continuum wave function changes the result in comparison to the standard DOS approach.

Experiment

Experiments were carried out at MAX-laboratory at the soft X-ray undulator beamline I411, in Lund, Sweden [18,19]. A Scienta R4000 hemispherical electron kinetic energy analyzer was used for electron detection. Photoelectron spectra were recorded at 90 • angle relative to the polarization direction of linearly polarized radiation. The photon energy of 40 eV was used in all experiments. Alkali metal clusters were produced using the EXMEC source. Detailed description of the setup is given in Ref. [10]. A resistively heated oven was used to evaporate solid samples. The vapor pressure inside the stainless steel crucible was estimated to be in the range of 10 -2 -10 -3 mbar [START_REF] Honig | Vapor Pressure Curves of the Elements RCA Laboratories[END_REF]. In the EXMEC method a distribution of clusters with different sizes is produced. Major part of the formed clusters are close to the mean cluster size N which can be used as an effective size of the clusters [START_REF] Lewerenz | [END_REF]. To obtain the most prominent effects, parameters of EXMEC set-up were varied in a way that the smallest and the largest possible clusters were produced. Cluster size variation was carried out by changing the temperature of the nozzle, pressure of inlet Ar gas, and temperature of the crucible. The inner valence K 3p and Rb 4p spectra recorded at the same time and settings [22] were used to carry out tentative size estimations of the produced clusters applying quantum corrected conduction sphere approximation [9].

Theory

The simulations were done utilizing simple model that accounts for the photoionization process in metal clusters. The model is described in detail in Ref. [23] (see also [17]) and only the main principles are reviewed here. The valence states are calculated using jellium model where the Hartree-Fock mean field created by the nuclei and electrons is replaced by a square well potential with radius R 0 and depth V 0 . For the bound states the potential outside the well is a constant, whereas for the continuum states the constant is replaced by r dependent Coulomb tail. This jellium potential in atomic units is given by

V (r) =    s -V 0 , if r ≤ R 0 s, if r > R 0 (bound) -1/r, if r > R 0 (continuum), (1)
where s = -1/R 0 is the value of the Coulomb tail at r = R 0 . The width and depth of the well are calculated from

R 0 = r s N 1/3 + δ s (2)
V 0 = 1 2 9πN 4R 3 0 2/3 + W N + s, (3)
where the Wigner-Seitz radius r s and the spillout δ s parametrize the size of the cluster, and N is the number of atoms in the cluster. The average ionization potential W N is given by W N = W wf + 3/(8R 0 ), where W wf is the bulk work function. For K the size parameters are r s = 4.86 a 0 and δ s = 1.1 a 0 , and for Rb r s = 5.20 a 0 and δ s = 1.15 a 0 . The most convenient form of the solution of the Schrödinger equation with potential (1) is obtained in the spherical coordinates. In the inner region r < R 0 the radial wave functions are spherical Bessel functions of the first kind. For the bound states the solutions in the outer region r > R 0 are spherical Hankel functions of the first kind, and for the continuum states the solutions are linear combination of regular and irregular Coulomb functions [23].

The photoionization cross sections were calculated in the dipole approximation. The angular parts of the photoionization matrix elements can be worked out analytically. Following Ref. [23] the cross section is then

σ nl→ l = 4π 2 αω 3 (2l+1) l =l±1 (2l +1) l 1 l 0 0 0 2 × P l (r)|r|P nl (r) 2 , (4)
where P nl (r) and P l (r) are bound and continuum single electron radial wave functions, respectively. The standard definition of the 3j symbol is used, denotes the (positive) energy of the continuum electron, ω is the frequency on the incoming photon and α is the fine-structure constant. The calculations were done far above the ionization threshold. Therefore the step function Θ l and treatment of the plasmon resonances discussed in [23] were not needed. To maintain simplicity thermal excitations and ellipsoidal deformations were also neglected.

Discussion

Valence PES of K and Rb clusters for two different limit-sizes are presented in Figs.

1 and 2 together with the theoretical simulations. The PES include also atomic K 4s and Rb 5s photolines at 4.3 eV and at 4.2 eV which provide the binding energy calibration [25,24]. The observed valence PES of alkali metal clusters are seen to locate above the solid work function of K at 2.30 eV and Rb at 2.16 eV [26]. The mean cluster size estimations obtained from conduction sphere approximation [9] using the binding energies of 3p and 4p orbitals of K and Rb, respectively, were 30 and 650 for K, and 50 and 200 for Rb. The cluster sizes in the jellium model calculations were chosen to correspond electronic magic numbers close the estimations obtained from the conduction sphere approximation. The selected sizes were 34/40 and 676 for K, and 58 and 198 for Rb. The ionization potentials as well as the shape of the spectra display a clear cluster size dispersion especially in smaller clusters. Therefore in the case of smaller K clusters in Fig. 1(a) a reasonable agreement was obtained by using a sum of two cluster sizes 34 and 40 with equal weight distribution. Such a favoring of magic sizes in the size distribution may well be a property of the EXMEC method [10]. To account the several broadening factors in the experiment, such as experimental broadening, size distribution, thermal broadening, and nonspherical deformations, the calculated spectral lines were convolved with a 150 meV Gaussian function. The dotted lines depict the results obtained using the jellium model and the photoionization cross section (4) described above. To highlight the importance of including the continuum wave functions, dashed lines show the calculation without inclusion of the radial dipole matrix elements in Eq. ( 4). The approximation corresponds the DOS approach, where the intensities in photoelectron spectra are approximated using the number of electrons in each level.

Comparison of the experimental and calculated results in Figs. 1 and2 shows that the jellium model applied gives surprisingly good agreement with the experiment. In the cases of smaller sizes of K and Rb clusters (upper panels) the peak profile is quite symmetric due to less dense array of occupied states. On the other hand, in the case of larger clusters (lower panels) the valence states approach rapidly the solid Fermi level shown as a vertical line. The increasing density of occupied high l orbitals closer to the Fermi level explains the sharp rise and slowly decreasing intensity profile. In the cases of larger clusters the binding energies are slightly underestimated. However, one may see that the peak profiles are well described.

Previous experiments have shown that the jellium model is applicable in the description of photoionization of anionic clusters [5,16,17]. This may be affected by the excess charge in the anionic cluster and the fact that the cluster is in a neutral state after the ionization. The case of initially neutral clusters is essentially different. The jellium-like potential is less evident and the continuum electron is moving in a positive charge of the ionic cluster. Our result is Figs. 1 and2 show that the spherical well jellium model can indeed describe valence photoionization of initially neutral metallic clusters.

Comparison between the two calculations done with and without dipole matrix elements shows striking differences. Inclusion of the continuum waves and the radial dipole matrix elements improved the agreement remarkably in comparison to DOS approach in every case studied. The improvement is less evident in figures 1(a) and 2(b) which is due to slightly shifted binding energies in the calculations. The shift is caused by limitations of the potential. However, if the calculated spectra are shifted about +150 meV in binding energies, the agreement between the experiment and theory in for example 2(b) is remarkably good. It can be seen that the DOS spectrum tend to overestimate the relative intensities. This disagreement is systematically corrected by the radial matrix elements. As a specific example, in the case of smaller Rb clusters shown in Fig. 2(a), the ionization probability of an electron from the outermost 1g orbital around 2.6 eV is drastically overestimated by the DOS approach. Inclusion of the matrix elements modify the calculation in a way that the highest ionization probability is from the 1f orbital around 3.0 eV. Jellium orbitals 1g and 1f are nodeless and show very similar radial dependence, thus the result cannot be predicted without actual calculation of the radial matrix elements. Comparing the experiment and theory at higher binding energy region around 4 eV one can see that the theory underestimates the intensities. This is due to lack of energy levels in the region. Most likely electron correlation plays a more significant role for the deeper levels and shift some levels into this region. Analysis of such effects is however left for the future.

It is a known fact that the theoretical description of ionization matrix elements and poly-centric continuum wave functions of many-atom systems is a very demanding task, which is the reason why DOS approach has been widely used in clusters. Our result has shown that the dipole matrix elements may change the result drastically. This indicates that much of the disagreement found in the DOS approach using high level Hartree-Fock or density functional theory calculations could be due to the omission of the dipole matrix elements and continuum wave functions. Therefore we may conclude that the way to improve high level calculations describing photoionization in clusters could be in including even a simple one-center continuum wave function and the dipole matrix elements instead of forcing the calculation for better bound state wave functions.

Conclusion

We have presented the valence photoelectron spectra of free, neutral, and size varied K and Rb clusters using synchrotron radiation. The model based on approximating the Hartree-Fock potential by a square well provides a good correspondence with the valence spectra far above the ionization threshold of neutral K and Rb clusters. It was shown that the inclusion of the radial matrix elements is important in obtaining reasonable agreement with the experiment. It demonstrates that the widely used DOS approach in more laborious theoretical frameworks could be significantly improved by including radial dipole matrix elements together with some approximate continuum wave function.

Figure 1 .

 1 Figure 1. Valence PES of size varied K clusters a) smaller and b) larger. Solid black lines present the experimental data. Dotted and dashed lines correspond the jellium model calculations. The intensities in dotted lines are calculated including the dipole matrix elements, whereas for dashed lies the intensity is obtained by the statistical weight of each jellium level. Position of the work function of solid K at 2.30 eV is shown as a vertical line. The smaller cluster PES in a) are simulated by sizes 34 (light gray line) and 40 (dark gray line). The larger cluster PES is in b) calculated using size 676.

Figure 2 .

 2 Figure 2. Valence PES of size varied Rb clusters a) smaller and b) larger. Solid black lines present the experimental data. Dotted and dashed lines correspond the jellium model calculations. The intensities in dotted lines are calculated including the dipole matrix elements, whereas for dashed lies the intensity is obtained by the statistical weight of each jellium level. Position of the work function of solid Rb at 2.16 eV is shown as a vertical line. The smaller cluster PES in a) is simulated using size 58 and the larger in b) using size 198.

Acknowledgments

This work has been financially supported by the Research Council for Natural Sciences of the Academy of Finland, the Swedish research Council (VR), the European Community-Research Infrastructure Action under the FP6 "Structuring the European Research Area" Programme (through the Integrated Infrastructure Initiative "Integrating Activity on Synchrotron and Free Electron Laser Science"), the Knut and Alice Wallenberg Foundation (KAW), the Göran Gustafsson Foundation, and the Carl Trygger Foundation. Author M.-H.M. would like to thank Vilho, Yrjö, and Kalle Väisälä Foundation for financial support. The authors would like to thank Prof. S. Aksela for providing the groundwork for evaporation methods. We thank Mr. P. Kovala for assistance during mechanical design. We acknowledge the staff of MAX-lab for their assistance during the experiments.


Ludovic Pricoupenko

Isotropic contact forces in arbitrary representation: Heterogeneous few-body problems and low dimensions

Keywords: numbers: 03.65.Nk, 03.65.Ge, 21.45.-v, 34.50.-s

The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for threebody systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov Ter-Martirosian equation and the Thomas collapse is avoided. Fourbody bound state eigenequations are derived and the 2D '3+1' bosonic ground state is computed as a function of the mass ratio.

I. INTRODUCTION

The non-perturbative zero-range limit of a pairwise interaction was introduced in the context of nuclear physics for the description of the deuteron [1][2][START_REF] Blatt | Theoretical Nuclear Physics[END_REF]. In this modeling, the finite range pairwise interacting potential is replaced by a zero-range potential and a peculiar asymptotic behavior is imposed on the wave function as the two interacting particles approach each other. More than 60 years after these pioneering works, the Bethe-Peierls approach appears to be especially relevant for modeling pairwise interaction in few-and many-body systems of ultra-cold atoms where details of the interaction are useless for understanding a large class of low-energy processes [START_REF] Olshanii | [END_REF][5][6][7][8][9][10][11][12][13][14][15][16][17]. This remarkable simplicity follows essentially from two features: first, the temperature in these systems is sufficiently low that s-wave scattering is the dominant process in two-body collisions [18]; second, the three-dimensional (3D) s wave scattering length (denoted a 3 ) in the two-body scattering can be tuned by use of a magnetic Feshbach resonance (FR) toward an arbitrary large value with respect to the range of the interatomic forces (denoted by b 3 ) and which is of the order of the van der Waals radius [19,20]:

b 3 ∼ µC 6 2 1/4 , (1)
where C 6 is the van der Waals coefficient of the interatomic potential and µ is the reduced mass of the two interacting particles. In the vicinity of a broad FR the two-body scattering cross section 4πa 2 3 is very large with respect to the non-resonant cross section (∼ 4πb 2 3 ) and this justifies the use of the 3D Bethe-Peierls approach only parameterized by a 3 .

Ultra-cold atoms in reduced geometries are the subject of intensive experimental and theoretical studies [START_REF]Proceedings of the School "Quantum Gases in Low Dimensions[END_REF].

Decisive progress have been obtained both in quasi-1D systems (for example with the observation of the Tonks Girardeau gas [22,23]) and in quasi-2D systems. A general review of low-dimensional systems can be found for example in Refs. [24,[START_REF] Hadzibabic | Course CLXXIII: 'Nano optics and atomics: transport of light and matter waves[END_REF]. Systems in reduced dimensions are achieved by using very anisotropic trapping potentials leading to a D-dimensional behavior where D = 1 (quasi-1D systems) or D = 2 (quasi-2D systems). In an ideal D-dimensional atomic waveguide, noninteracting atoms move freely along D direction(s) while they remain frozen in the lowest state of the transverse zeropoint motion characterized by a length denoted as a trans . The associated energy E trans = 2 /(2µa 2 trans ) defines the limit of energy beyond which higher transverse states are populated and a 3D-like behavior is progressively recovered for increasing energies. In the actual experimental state-of-the-art, atomic waveguides have a transverse atomic length which is large as compared to the 3D potential radius (b 3 ). Therefore for low energy processes i.e., for collisional energies E such that |E| ≪ 2 /(µb 2 3 ), the short-range pairwise interacting potential can be described through the 3D Bethe-Peierls approach. For collisional energies much smaller than E trans the transverse excited states of the trap are populated only via virtual processes and for large interatomic separations r (r ≫ a trans ) only the ground transverse state is occupied. In this regime of collisional energies and in the free Ddimensional subspace, scattering process can be deduced from a D-dimensional effective pairwise interaction characterized by a finite range denoted b D , which is of the order of the transverse length a trans . Hence for low energy D-dimensional processes where |E| ≪ 2 /(µb 2 D ) the effective low-dimensional interaction can be analogously to the 3D case, replaced by a zero-range force. Therefore in each dimension D less than or equal to 3, a Ddimensional Bethe-Peierls approach can be relevant and useful as a tool for exploring the properties of few-and many-body shallow states. The dimensional reduction of the effective interaction from 3D to 1D was first achieved in Ref. [START_REF] Olshanii | [END_REF] and from 3D to 2D in Ref. [5].

One of the purpose of this paper is to show that the zero-range potential approach can be handled in a very simple and unified way in any dimension D ≤ 3. The key tool used along these lines is introduced in Sec. II of the paper. It consists of the general expression of the Bethe-Peierls asymptotic condition which can be used in any representation for all dimensions D. Substitution of the δ-source terms of the zero-range approach by a family of well-behaved functions of vanishing but finite support allows one to avoid technical problems concerning double-limit calculations (i.e., evaluation of infinite series in the zero-range limit). While used usually in the configuration space, the zero-range approach can thus be implemented directly in the momentum representation for translation invariant systems. The link between this formulation of the Bethe-Peierls approach and the Λ potential introduced in Ref. [START_REF] Olshanii | [END_REF] is analyzed. The method is illustrated in Sec. III with the example of the dimensional reduction issue from 3D toward 1D and 2D. The Bethe-Peierls contact condition is directly expressed in the functional basis which diagonalizes the free Hamiltonian and known results are obtained in a simple manner. For N -body systems, integral equations which encapsulate the Bethe-Peierls contact condition are obtained in Sec. IV. It is also shown how to express this zero-range force approach in a second quantized representation. As an example, energy theorems [27][28][29][30][31] are derived by using this last formulation. Section-V of the paper is devoted to the three-and four-body shallow bound state issue. For three identical bosons in 3D, the original Skorniakov Ter-Martirosian (STM) equation [32] appears as a straightforward application of the results of the preceding section. A nodal condition is imposed on the three-body wave function in order to avoid the Thomas collapse [33] while keeping the simplicity of the zero-range approach [34]. Known results of universal theory [35] are recovered from the regularized STM equation. For two-mass component fermionic systems in 3D, critical mass ratios for the Efimov threshold are given in each partial wave by using the STM equation. Finally, the formalism is applied to derive four-body equations in the zero-range limit. In 2D, the eigenenergy equation for two-mass component tetramers is studied numerically. The energy of the ground s wave of '3+1' bosonic tetramers in the zerorange limit is computed as a function of the mass ratio.

In this paper the norm of any vector v is denoted v. For a uni-dimensional system (D = 1), v is algebraic and v = |v| is the absolute value of v.

II. ISOTROPIC CONTACT FORCES

A. Two-body transition matrix and scattering amplitude

This section reviews basic notions of two-body scattering in D-dimensional spaces which are useful in the formulation of the Bethe-Peierls approach.

Two colliding particles are described in their centerof-mass frame by the relative particle of reduced mass µ and relative momentum k. They are supposed to interact through a short range potential V , and the Hamiltonian for the relative particle is H = H 0 + V , where H 0 is the free Hamiltonian. In what follows, as a consequence of the small collisional energy, scattering is supposed to only occur in the s wave channel of the relative particle. For an incoming atomic wave of momentum k 0 and of collisional energy E = 2 k 2 0 /2µ (measured from the continuum threshold), the scattering state |Ψ k0 of the relative particle verifies the Lippmann-Schwinger equation:

|Ψ k0 = |k 0 + 1 E + i0 + -H 0 V |Ψ k0 . (2)
We now turn to the k representation with the convention r|k = exp(ik • r):

r|Ψ = d D k (2π) D k|Ψ exp(ik • r). (3)
In the k representation, the scattering states in Eq. ( 2) can be written as a function of the half on-shell transition matrix (or t-matrix) defined by:

k|T (E + i0 + )|k 0 = k|V |Ψ k0 , (4)
so that:

k|Ψ k0 = (2π) D δ(k -k 0 ) + k|T (E + i0 + )|k 0 E + i0 + - 2 k 2 2µ
.

The pairwise interaction is supposed to be short range with a typical radius b D . For a small energy |E| ≪ 2 /(µb 2 D ) and momentum |k|b D ≪ 1, the half onshell t-matrix only depends on the collisional energy E and coincides with the low energy on-shell t-matrix denoted as T D :

k|T (E + i0 + )|k 0 ≃ T D (E + i0 + ). ( 6
)
At the lowest order in energy (implying that |k 0 |b D ≪ 1), the expression of T D can be parameterized by the Ddimensional scattering length a D . It can be written as [36,37]:

T D (E + i0 + ) = Ω D 2 2µ ×                  a 3 (1 + ia 3 k 0 ) (D = 3) -1 ln(-ia 2 k 0 e γ /2) (D = 2) -ik 0 (1 + ia 1 k 0 ) (D = 1), (7)
where Ω D is the full D-dimensional space angle:

Ω 3 = 4π, Ω 2 = 2π, and Ω 1 = 2. ( 8
)
In Eq. ( 7) for the two-dimensional case, γ is the Euler's constant, and the 2D scattering length a 2 is always positive. The transition matrices in Eq. ( 7) are the basic objects for describing low energy scattering processes in D-dimensional few-and many-body systems. Scattering properties can be equivalently described with the D-dimensional scattering amplitudes defined by:

f 3 (k 0 ) = -a 3 /(1 + ik 0 a 3 ), (9) f 2 (k 0 ) = 1/ ln(-ik 0 a 2 e γ /2), ( 10
) f 1 (k 0 ) = -1/(1 + ik 0 a 1 ). ( 11
)
The notion of scattering amplitude is often used in the configuration space where the scattering states for large relative coordinates (r ≫ b D ) are deduced from Eq. ( 7):

r|Ψ k0 = e ik0•r + f D (k 0 ) ×              e ik0r r (D = 3) iπ 2 H (1) 0 (k 0 r) (D = 2)
e ik0r (D = 1). ( 12) In Eq. ( 12), for D = 2 the outgoing scattering wave function H

(1) 0 is the Hankel's function of order zero [38]; for D = 1, f 1 is also called the even scattering amplitude [START_REF] Olshanii | [END_REF].

B. Bethe-Peierls asymptotic approach in arbitrary representation

Surprisingly the zero-range approximation has been essentially used in the configuration space while other representations can yield substantial simplifications. For example, as a consequence of translation invariance, the momentum representation is very well suited for solving the few-body problem in homogeneous space. In this section, it is shown how the zero-range approximation can be formulated in arbitrary representation.

While for finite range forces Eq. ( 6) is only valid for a small relative momentum k (i.e., for kb D ≪ 1), in the zero-range potential approximation this equality is extended for arbitrary large values of k. That way, the short range pairwise potential V is replaced by a formal zero-range potential such that the scattering states in this approximation coincide with Eq. ( 5) in the small relative momentum limit (kb D ≪ 1) or with Eq. ( 12) for large interparticle distances (r ≫ b D ). The simplest way to implement the zero-range approximation is to follow the Bethe-Peierls method [1], where for any state |Ψ described by the Hamiltonian H 0 + V , the pairwise short range potential V is replaced by a delta-source term |δ D with an amplitude S Ψ :

V |Ψ -→ S Ψ |δ D , (13)
and S Ψ is deduced from an asymptotic condition on the state |Ψ . In Eq. ( 13), |δ D is such that the usual δdistribution is obtained in the configuration space:

r|δ D = δ(r), (14)
and |δ D is also denoted by the formal ket of zero relative coordinates |0 r . From Eq. ( 5) the scattering state |Ψ k0 is written in the zero-range potential approach as

|Ψ k0 = |k 0 + T D (E + i0 + ) E + i0 + -H 0 |δ D . ( 15
)
Injecting Eq. ( 13) in the Lippmann-Schwinger equation (2) gives

|Ψ k0 = |k 0 + S Ψ k 0 E + i0 + -H 0 |δ D . ( 16
)
Identifying Eq. ( 15) and (16) shows that the source amplitude associated with a scattering state |Ψ k0 is nothing but the on-shell transition matrix:

S Ψ k 0 = T D (E + i0 + ).
In configuration space, scattering states in Eq. ( 15) or in Eq. ( 12) are singular for vanishingly small interparticle distances. As shown in what follows, this singular behavior defines the Bethe-Peierls asymptotic condition. From Eq. ( 15) one finds that all the scattering states for a given dimension D have the same singular behavior which reduces to

r|Ψ = r→0 A ×                  1 a 3 - 1 r + O(r) (D = 3) ln r a 2 + O(r) (D = 2) (r -a 1 ) + O(r) (D = 1). (17)
In Eq. ( 17) A is a function of the energy E but does not depend on r. It is important to note that Eq. ( 17) is obtained in the center-of-mass frame, meaning that the limit r → 0 is taken at fixed value of the center-of-mass of the two colliding particles. Conversely, for a given dimension D, Eq. ( 17) is the sufficient asymptotic condition which permits one to obtain the source amplitude S Ψ k 0 of a scattering state |Ψ k0 at energy E in Eq. ( 16). The asymptotic condition in Eq. ( 17) referred also as the "contact condition" or the Bethe-Peierls condition, remains unchanged for any linear combination of scattering states and can be thus used to find any eigenstate in the zero-range approximation for a system where an external potential is included within the free Hamiltonian H 0 . For a positive energy E, the Lippmann Schwinger equation is

|Ψ = |Ψ (0) + S Ψ E + i0 + -H 0 |δ D , (18)
where |Ψ (0) is the complementary solution i.e., it is an eigenstate of the free Hamiltonian H 0 at energy E (for a negative energy |Ψ (0) = 0 and the +i0 + prescription in Eq. ( 18) is useless). The relation between the number A in Eq. ( 17) and the source amplitude S Ψ in Eq. ( 18) is given by

A = 2µS Ψ Ω D 2 . ( 19
)
The crucial point of this section is to show that the asymptotic condition in Eq. ( 17) can be expressed in arbitrary representation. In what follows for convenience, the delta distribution is represented by the ǫ → 0 limit of a Gaussian weight: δ D (r) = lim ǫ→0 r|δ D ǫ and

r|δ D ǫ = 1 (2πǫ 2 ) D/2 exp - r 2 ǫ 2 . ( 20
)
With this particular choice, in the momentum space the representation of the delta term has the same expression [denoted χ ǫ (k)] for all dimensions:

k|δ D ǫ = χ ǫ (k) = exp - k 2 ǫ 2 4 . (21)
Matrix elements and states without index ǫ are considered in their zero-range limit which corresponds in this formalism to the limit ǫ → 0 (for example lim ǫ→0 |Ψ ǫ = |Ψ . For the formulation of the zero-range approximation it is also useful to introduce the 'reference state' denoted |φ Λ ǫ which results from the action of the two-body Green's function in free space on |δ D ǫ at the negative energy E Λ :

|φ Λ ǫ = 1 E Λ -H 0 |δ D ǫ where E Λ = - 2 Λ 2 2µ < 0. ( 22
)
In Eq. ( 22) the parameter Λ is chosen positive (Λ ∈ R + ) according to the usual prescription in scattering theory.

In the momentum representation:

k|φ Λ ǫ = - 2µ 2 × χ ǫ (k) k 2 + Λ 2 . ( 23
)
In the zero-range limit where ǫ → 0, in the configuration space the reference state has the same type of singularity as in Eq. ( 17):

r|φ Λ = 2µ Ω D 2                      - 1 r + Λ + O(r) (D = 3) ln Λre γ 2 + O(r) (D = 2) r - 1 Λ + O(r) (D = 1). ( 24
)
The contact condition in Eq. ( 17) can then be written in terms of the reference state as

lim r→0 lim ǫ→0 r|Ψ ǫ -S Ψ φ Λ ǫ = S Ψ T D (E Λ ) . (25)
In the configuration space, the state |Ψ ǫ -S Ψ φ Λ ǫ is a smooth function for (r, ǫ) close to (0, 0). It is thus possible to commute the r → 0 and ǫ → 0 limits in Eq. ( 25):

lim r→0 lim ǫ→0 r|Ψ ǫ -S Ψ φ Λ ǫ = lim ǫ→0 lim r→0 r|Ψ ǫ -S Ψ φ Λ ǫ . ( 26
)
For particles moving in the presence of an external potential, the source amplitude in configuration space depends on the center-of-mass coordinates of the interacting pair. Hence, in general situations the source amplitude is replaced by a state associated with the center-of-mass of the interacting pair:

S Ψ -→ |S Ψ . (27)
Finally, the Bethe-Peierls condition can be written without specifying any representation as:

lim ǫ→0 0 r |Ψ ǫ -S Ψ φ Λ ǫ = |S Ψ T D (E Λ ) . (28)
Equation ( 28) is by construction, invariant in a change of Λ ∈ R + : as is shown in the next section, one recovers the so-called Λ freedom of the Λ-potential [START_REF] Olshanii | [END_REF]. This freedom permits to simplify exact calculations without introducing any approximation, or also to improve approximate schemes [START_REF] Olshanii | [END_REF]40,41]. Moreover, the ǫ → 0 limit can be taken equivalently as follows,

lim ǫ→0 δ D ǫ |Ψ ǫ -S Ψ φ Λ ǫ = |S Ψ T D (E Λ ) , (29)
or also by substituting the ket in the left hand side of Eq. ( 29) by its ǫ → 0 limit:

lim ǫ→0 δ D ǫ |Ψ -S Ψ φ Λ = |S Ψ T D (E Λ ) . ( 30
)
The contact conditions in Eqs. ( 28)-( 30) can be expressed in any desired representation by inserting a closure relation in the scalar product concerning the relative particle. For example in the momentum representation, one can insert the closure relation

d D k (2π) D |k k| = I (31)
at the right of the bra δ D ǫ | in Eq. ( 30) and this gives:

lim ǫ→0 d D k (2π) D χ ǫ (k) k|Ψ -S Ψ φ Λ = |S Ψ T D (E Λ ) . (32)
In the next sections, it is shown that Eqs. ( 28)-( 30) allow one to obtain in a simple way the standard integral equations of few-body problems in the zero-range limit and also to compute the induced scattering resonances and related scattering problems in the presence of a harmonic transverse confinement. As a conclusion of this part, it is interesting (and useful) to note that for D = 3 or D = 1, it is possible to eliminate the presence of the reference function in the contact condition by performing a specific limit on Λ. For D = 3, in the Λ → 0 limit the contact condition can be written as:

lim ǫ→0 ∂ ǫ ǫ δ D=3 ǫ |Ψ = µ|S Ψ 2π 2 a 3 . (33)
Equation ( 33) can be written in the following alternative form:

Reg ǫ→0 δ D=3 ǫ |Ψ = µ|S Ψ 2π 2 a 3 , (34)
where Reg ǫ→0 extracts the regular part of δ D=3 ǫ |Ψ in the limit where ǫ → 0. For D = 1, elimination of the reference function occurs in the limit where Λ tends to ∞.

One obtains:

lim ǫ→0 δ D=1 ǫ |Ψ = - µa 1 2 |S Ψ . (35)
C. Link with the s wave Λ potential

A zero-range s wave pseudo potential was introduced by Fermi [42] in order to perform calculations in the firstorder Born approximation. Thus, the Fermi pseudo potential cannot be used for a non-perturbative approach (for large values of a 3 , for example). In the modern formulation of the zero-range pseudo potential, the Bethe-Peierls condition is included in the Schrödinger equation via the zero-range pseudo potential itself [2,[START_REF] Blatt | Theoretical Nuclear Physics[END_REF]. It has been shown in the configuration space that there exists in each dimension D a family of zero-range potentials: the so-called Λ potentials [START_REF] Olshanii | [END_REF]. This section, links the Bethe-Peierls approach and the Λ potential in arbitrary representation. For this purpose, a family of operators R Λ ǫ is introduced. They act on a state |Ψ ǫ as:

R Λ ǫ |Ψ ǫ = |δ D ǫ lim ǫ→0 δ D ǫ |Ψ ǫ -S Ψ φ Λ ǫ . (36)
For a state verifying the contact condition in Eq. ( 29) one has:

R (1/ãD) ǫ |Ψ ǫ = 0 ( 37
)
where ãD is related to the scattering length a D as

ãD = a D for D = 3 or D = 1 a 2 e γ /2 for D = 2 . ( 38
)
These regularizing operator satisfy two other properties: (i) for a regular state (S

Ψ = 0) thus R Λ ǫ |Ψ = |δ D ǫ 0 r |Ψ and (ii) R Λ ǫ |φ Λ ǫ = 0. Explicit expressions for R Λ
ǫ can be found without specifying any representation as follows:

R Λ ǫ |Ψ ǫ = |δ D ǫ lim ǫ→0 r Λ ǫ δ D ǫ |Ψ ǫ , (39)
where r Λ ǫ is an operator defined in each dimension as:

r Λ ǫ [ • ] =                      ∂ ǫ + π 2 Λ ǫ • (D = 3) 1 + ǫ 2 ln(e γ Λ 2 ǫ 2 /2)∂ ǫ • (D = 2) 1 + π 2 1 Λ ∂ ǫ • (D = 1).
(40) Expressions of r Λ ǫ in Eq. ( 40) depend on the choice made for the short range functions r|δ D ǫ , given here by Eq. ( 20). The source term in the stationary Schrödinger equation can be expressed in terms of R Λ ǫ by using Eqs. (28 and 36):

(H 0 -E) |Ψ ǫ + T D (E Λ )R Λ ǫ |Ψ ǫ = 0. ( 41
)
In the limit ǫ → 0, the pseudo potential in this equation coincides exactly with the Λ-potential introduced in Ref. [START_REF] Olshanii | [END_REF], where the coupling constant g Λ is nothing but the transition matrix at energy E Λ :

g Λ = T D (E Λ ). (42)

D. Context of resonant scattering

The zero-range approximation is especially interesting in the regime of parameters where the pairwise potential leads to a resonant scattering at low collisional energy. In this regime, the cross section proportional to |f D | 2 has a large value at small relative momentum (k 0 b D ≪ 1) and expressions in Eqs. ( 9)-( 12) are good approximations. For D = 3, this corresponds to the regime where the scattering length a 3 is large in absolute value as compared to the potential radius b 3 and the maximum is at k 0 = 0. This regime can be achieved by using the FR mechanism (see Ref. [20] for a general review of FR in ultracold atoms). The FR involves the coupling between atoms (in the "open channel") and a molecular state (in the "closed channel") characterized by a size of the order of the potential range b 3 . This two-channel description provides the expression of the scattering length (a 3 ) as a function of the external magnetic field B in the vicinity of a given resonance located at B = B 0 which can be parameterized by the formula [43]

a 3 = a bg 1 - ∆B B -B 0 . (43)
In Eq. ( 43), ∆B is the width of the magnetic FR and a bg is the "background" scattering length i.e., the scattering length away from the FR. The magnetic width (∆B) can be also characterized by the "width radius" R ⋆ defined by

R ⋆ = 2 ma bg δµ∆B , ( 44
)
where δµ is the difference of magnetic moment for an atomic pair in the open-and the closed-channel [20,44].

In the vicinity of a broad resonance, R ⋆ is of the order of (or smaller than) b 3 the 3D scattering amplitude can be approximated by Eq. ( 9) in a large interval of momentum where |k 0 |b 3 ≪ 1.

As shown in the next section, the resonant behavior in the low dimensional atomic waveguide can be reached by tuning the length a trans of the transverse confinement and/or the 3D scattering length. For D = 1 the resonant regime occurs in the limit where the 1D-scattering length a 1 is small with respect to the transverse length (i.e., |a 1 | ≪ b 1 ) and also for a vanishing relative momentum k 0 → 0. In 2D, the maximum in |f 2 | 2 occurs at the momentum k 0 = 2/(e γ a 2 ) (where |f 2 | = 2/π) and the resonant regime is thus reached at low energy if and only if the 2D scattering length is large as compared to the transverse length (i.e., a 2 ≫ b 2 ).

In the resonant regimes defined above, the probability of scattering for two colliding particles is large so that the pairwise short range potential affects the form of the wave function at interparticle distances which are large with respect to the potential radius b D . More precisely, the scattering states in Eq. ( 5) can be decomposed in an incoming part |φ inc = |k 0 and a scattered part |φ scatt . At large distance and in the resonant regime:

r|φ scatt r|φ inc = r≫bD    |a 3 |/r (D = 3) √ e γ a 2 / √ πr (D = 2) 1 (D = 1). (45)
For D = 3 or D = 2 this ratio is greater than or of the order of unity for b D ≪ r < |a D |, where |a D | is arbitrarily large and is equal to unity for arbitrary large distance in one-dimensional systems (D = 1). The zero-range potential approach is a formalism which permits one to evaluate accurately the wave function in configurations where particles are outside the potential radius while configuration where two or more particles are inside the potential radius are not reliably described. Consequently, the Bethe-Peierls asymptotic approach is very well suited for studying systems in the resonant regime where the wavefunction is modified by the interaction at interparticle distances which are large as compared to the potential radius.

In the 3D space and for a positive scattering length, the on-shell transition matrix T 3 in Eq. ( 7) has a real pole at negative energy

E 2 = -2 /(2µa 2
3 ). In the resonant regime, this pole is associated with the existence of a shallow dimer which is thus very well described in the zero-range approach (the probability that the relative pair has a radius greater than the potential radius b 3 is exp(-2b 3 /a 3 ) ∼ 1). For D = 2, the pole is at E 2 = -2 2 /(µa 2 2 exp(2γ)), and in the resonant regime it is also the signature of the existence of a shallow dimer. For D = 1, in the resonant regime (a 1 → 0) the approximation of the transition matrix in Eq. ( 7) doesn't possess a negative low-energy pole and next-order terms in the energy expansion are needed to find the pole corresponding to the lowest bound state [45].

III. DIMENSIONAL REDUCTION

In this section, the dimensional reduction issue is solved by using the zero-range approximation with the Bethe-Peierls condition expressed through Eq. ( 34). This section gives alternative derivations of the results given in Refs. [START_REF] Olshanii | [END_REF][5][6][7]. The problematic of the 3D→1D (respectively 3D→2D) dimensional reduction is as follows: two particles move freely in 1D (respectively 2D) while they are confined in the transverse direction by an atomic waveguide built from a 2D (respectively 1D) trapping potential. The collisional energy is such that at large interatomic distances, the particles are confined in the ground state of the trapping potential. Solving the issue amounts to finding the low energy transition matrix of this low dimensional quasi-2D (resp quasi-1D) scattering process as a function of the waveguide parameters and of the 3D scattering length a 3 . In what follows only the case of harmonic trapping where the center-of-mass and relative motions decouple is considered. In the center-ofmass frame the source amplitude S Ψ is constant and the scattering state can be written as:

|Ψ = |Ψ (0) + |Ψ int , (46)
where |Ψ (0) is the incoming wave and the "interacting part" is

|Ψ int = S Ψ E + i0 + -H 0 |0 r . (47)
In Eq. ( 47), the source amplitude S Ψ is found from the Bethe-Peierls condition by solving the equation:

0 r |Ψ (0) + Reg ǫ→0 δ D=3 ǫ |Ψ int = µS Ψ 2π 2 a 3 . ( 48
)
These quasi-1D and quasi-2D scattering problems have been initially solved by using the Bethe-Peierls method in the configuration space in Ref. [START_REF] Olshanii | [END_REF] and Refs. [5,6]. In this section, an alternative derivation is given in order to illustrate the formalism introduced in the previous section.

A. Linear atomic waveguide

The 3D→1D reduction problem, is considered here in the case where the two colliding atoms are confined in an isotropic two-dimensional harmonic trap while they move freely along the third direction (z). In this case, the problem can be solved in the center-of-mass frame where the noninteracting Hamiltonian is

H 0 = - 2 2µ ∂ 2 z + H 2D , (49)
and H 2D is the Hamiltonian for the transverse motion:

H 2D = - 2 2µ (∂ 2 x + ∂ 2 y ) + 1 2 µω 2 ⊥ (x 2 + y 2 ) -ω ⊥ . (50)
The linear atomic waveguide is characterized by the transverse length:

a ⊥ = µω ⊥ , (51)
and the zero-range approximation of the D = 3 pairwise potential is justified in the limit where a ⊥ ≫ b 3 . In Eq. ( 50), the zero-point energy has been subtracted. That way, the energy E of the scattering states is measured with respect to the continuum threshold. In this quasi-1D scattering problem at energy E = 2 k 2 0 /2µ, the incoming wave for the relative particle (state |Ψ (0) ) in Eq. ( 46) has a momentum k 0 along z and is in the groundstate of the transverse 2D oscillator:

|Ψ (0) = |n x = 0 |n y = 0 |k z = k 0 , (52)
where n x and |n x (respectively n y and |n y ) are the quantum number and eigenstate of the 1D harmonic oscillator of atomic frequency ω ⊥ and mass µ along x (respectively y). The system is quasi-1D for collisional energies such that:

E < E 1D trans , (53)
where

E 1D trans = 2 ω ⊥ .
Equation ( 53) defines the monomode regime of the atomic waveguide such that outgoing particles are trapped at large relative distances in the ground state of the transverse trap. The contribution of the incoming wave in Eq. ( 48) is:

0 r |Ψ (0) = φ 0 (0) 2 = 1 √ πa ⊥ , (54)
where φ n (x) = x|n is given by:

φ n (x) = exp -x 2 2a 2 ⊥ π 1/4 √ a ⊥ H n x a ⊥ . (55)
In the calculation of δ D=3 ǫ |Ψ int , the quantum numbers of the noninteracting Hamiltonian are introduced directly by inserting the closure relation:

+∞ -∞ dk z 2π |k z k z | ∞ nx=0 |n x n x | ∞ ny=0 |n y n y | = I. ( 56
)
Only even values of n x and n y give a non vanishing contribution, and one obtains

δ D=3 ǫ |Ψ int = S Ψ ω ⊥ +∞ -∞ dk z 2π ∞ p,q=0 exp - k 2 z ǫ 2 4 × δ D=1 ǫ |2p δ D=1 ǫ |2q φ * 2p (0)φ * 2q (0) τ - k 2 z a 2 ⊥ 4 -p -q , (57)
where the dimensionless energy variable τ is defined by

τ = E 2 ω ⊥ + i0 + . ( 58
)
The behavior of δ D=3 ǫ |Ψ int in the limit where ǫ vanishes is more easily obtained in the domain of negative collisional energies (τ < 0) than in the domain of positive collisional energies E > 0 and are related the each to the other by analyticity. For τ < 0 one can use the identity

1 k 2 z a 2 ⊥ 4 + n + p -τ = ∞ 0 du e -u k 2 z a 2 ⊥ 4 +n+p-τ , (59)
which allows one to decouple the discrete summations from the integration over k z in Eq. ( 57). From Eq. ( 55), one can deduce the following limit:

lim ǫ→0 ∞ p=0 δ D=1 ǫ |2p φ * 2p (0)e -pu = |φ 0 (0)| 2 √ 1 -e -u . (60)
Using Eq. ( 60) and integrating over k z in Eq. ( 57) gives:

δ D=3 ǫ |Ψ int = ǫ→0 -µS Ψ 2π 3/2 2 a ⊥ ∞ 0 du u + ǫ 2 a 2 ⊥ e uτ 1 -e -u . (61)
In the limit where ǫ → 0, this expression diverges as 1/ǫ and its regular part coincides with a Hadamard's finite part [START_REF] Gelfand | Generalized Functions[END_REF][START_REF] Schwartz | Théorie des distributions[END_REF]:

Reg ǫ→0 δ D=3 ǫ |Ψ int = µS Ψ 2π 3/2 2 a ⊥ P. f. ∞ 0 du √ u e uτ 1 -e -u .
(62) One recognizes in Eq. ( 62) an expression similar to the integral representation of the Hurwitz zeta function ζ(s, z) defined for ℜ(s) > 1 and 0 < z < 1 by [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]:

ζ(s, z) = 1 Γ(s) ∞ 0 dt t s-1 e -tz 1 -e -t . (63)
However, in Eq. ( 63) the integral diverges for s = 1/2. The Hadamard's finite part in Eq. ( 62) permits one to achieve a meromorphic continuation in the variable s of Eq. ( 63) and to identify the function ζ(1/2, z) [START_REF])/(1 -s), where the function g is supposed to decrease sufficiently rapidly to zero at t = ∞ for ensuring the convergence of the integrals[END_REF]. Hence,

S Ψ = 2π 1/2 2 µ ζ 1 2 , -τ + a ⊥ a3 . ( 64
)
For positive energies and in the monomode regime (0 < E < 2 ω ⊥ ), the scattering state written in the configuration representation in Eq. ( 46) has a non evanescent contribution at large relative length z ≫ a ⊥ , in the subspace of the transverse ground-state only. This property allows one to identify a quasi-1D scattering process with:

z, n x = 0, n y = 0|Ψ = |z|≫a ⊥ exp(ik 0 z)- iµS Ψ √ π 2 a ⊥ k 0 exp(ik 0 |z|), (65)
and a ⊥ plays the role of a 1D-potential radius (i.e., b 1 ∼ a ⊥ ). The low-energy scattering amplitude f 3D→1D of this quasi-1D system is thus:

f 3D→1D (k 0 ) = - iµS Ψ √ π 2 a ⊥ k 0 . (66)
In the |k z -representation, the scattering states read

k z , n x = 0, n y = 0|Ψ = (2π)δ(k z -k 0 )+ S Ψ √ πa ⊥ E + i0 + - 2 k 2 z 2µ
, (67) so that the quasi-1D scattering transition matrix T 3D→1D can be expressed as

T 3D→1D = S Ψ √ πa ⊥ . (68)
Finally, from Eqs. ( 64) and ( 68) one obtains:

T 3D→1D = 2 2 µa ⊥ × 1 ζ( 1 2 , -τ ) + a ⊥ a3 . (69)
In the low energy limit (k 0 a ⊥ ≪ 1),

ζ 1 2 , -τ ≃ τ →0 1 √ -τ + ζ 1 2 (70)
and for positive energies √ -τ = -ik 0 a ⊥ /2. From Eq. ( 7), one can then identify a 1D-scattering length which is a function of the transverse length a ⊥ and of the 3D scattering length a 3 [45]:

a 1 = - a ⊥ 2 a ⊥ a 3 + ζ 1 2 . (71)

B. Planar atomic waveguide

In the 3D→2D reduction problem, the two colliding particles are confined in a planar harmonic waveguide of frequency ω z along the z direction while they move freely in the two other directions. The noninteracting Hamiltonian in the center-of-mass frame reads:

H 0 = - 2 2µ (∂ 2 x + ∂ 2 y ) + H z , (72)
with, the 1D-harmonic trap along z,

H z = - 2 2µ ∂ 2 z + µ 2 ω 2 z z 2 - ω z 2 . ( 73
)
In Eq. ( 73) the zero point energy of the transverse trap has been subtracted, so that similarly to the previous 3D→1D reduction problem, the energy of an eigenstate is measured with respect to the continuum threshold and coincides with the collisional energy. In what follows, |n z is the eigenstate of quantum number n z for the 1D harmonic oscillator of frequency ω z and mass µ. The subsequent derivation supposes that the 3D zero-range approximation is justified, i.e., that the atomic waveguide is such that:

a z ≫ b 3 , (74)
where a z is the characteristic length of the atomic waveguide:

a z = µω z . ( 75
)
In the quasi-2D scattering problem the incoming state is:

|Ψ (0) = |k 2D = k 0 |n z = 0 , (76)
where k 0 is the 2D-wave relative wavevector of the incoming wave and k 2D = k x êx + k y êy is the 2D collisional relative momentum. The collisional energy is E = 2 k 2 0 /2µ and the monomode regime condition is given by:

E < E 2D trans ( 77
)
where E 2D trans = ω z . Equation ( 77) ensures that the system is quasi-2D; i.e., the colliding particles are asymptotically trapped in the ground state of the transverse trap. The behavior of δ D=3 ǫ |Ψ int in the limit where ǫ → 0 is obtained by using the same techniques as for the 3D→1D reduction problem. The quantum numbers of the free Hamiltonian are introduced by insertion of the closure relation:

d 2 k 2D (2π) 2 |k 2D k 2D | ∞ nz=0 |n z n z | = I. (78)
In the 3D Bethe-Peierls asymptotic condition, only even values of n z contributes and one obtains:

δ D=3 ǫ |Ψ int = µS Ψ a 2 z 4π 2 ∞ 0 k 2D dk 2D × ∞ p=0 exp - k 2 2D ǫ 2 4 δ D=1 ǫ |2p φ * 2p (0) τ - k 2 2D a 2 z 4 -p , (79)
where the dimensionless energy variable τ is defined by

τ = E 2 ω z + i0 + . ( 80
)
The discrete summation in Eq. ( 79) is performed in the domain τ < 0 by using the transformation of Eq. ( 59) together with the identity of Eq. ( 60). After integration over k 2D , the regular part of δ 3 ǫ |Ψ int can be expressed in the limit where ǫ → 0 as

Reg ǫ→0 δ 3 ǫ |Ψ int = - µS Ψ 2π 2 |φ 0 (0)| 2 J(τ ), (81)
where the function J(τ ) is defined in the domain τ < 0 by:

J(τ ) = P.f. ∞ 0 du u exp(τ u) 1 -exp(-u) . (82)
From Eq. ( 48), one finally obtains:

S Ψ = 2π 2 φ 0 (0) µ 1 a3 + |φ 0 (0)| 2 J(τ ) . ( 83
)
Projection of the wavefunction on the ground-state of the 1D transverse harmonic oscillator k 2D , n z = 0|Ψ gives the quasi-2D transition matrix:

T 3D→2D = - π 2 µ √ πaz 2a3 + J(τ ) 2 . ( 84
)
In the domain of positive energy Eq. ( 80) is continued analytically from the identity

J(τ ) = ln - B 2πτ + ∞ n=1 ln n n -τ (2n -1)!! (2n)!! , (85)
where B ≃ 0.9049 is defined by [39]:

ln Be γ 2π = ∞ 0 du u -1 √ 1 -e -u - 1 u 3/2 - 1 1 + u , (86)
and γ = 0.5772 . . . is the Euler's constant. The 2D low energy condition is given by:

|E| ≪ ω z ( 87
)
and a z plays the role of a 2D-potential radius, i.e., b 2 ∼ a z . Only the first logarithmic term in Eq. ( 85) contributes at the lowest order in the limit of Eq. (87), and one can identify the resulting expression of the quasi-2D transition matrix T 3D→2D with the 2D low energy transition matrix T 2 in Eq. ( 7). The 2D scattering length a 2 of this quasi-2D scattering problem can thus be expressed in terms of a z and of the 3D scattering length a 3 [6,7,41] as:

a 2 = a z e -γ 2π B exp - a z √ π 2a 3 . ( 88
)
It is interesting to discuss a little bit further the condition for having a low energy resonant behavior in this quasi-2D system. The 2D cross-'section' (this is a length) is proportional to |T 3D→2D (k 0 )| 2 /k 0 and a maximum occurs in the low-energy regime Eq. (87) for:

a z a 3 = 1 √ π ln πE B ω z . ( 89
)
The right hand side of Eq. ( 89) is large and negative which shows that the resonance occurs for sufficiently large values of |a 3 | with respect to a z in the domain of negative 3D scattering lengths (a 3 < 0). At fixed value of the collisional energy E and of the 3D scattering length, the resonance can be reached by tuning the harmonic frequency of the atomic waveguide: this is the so-called quasi-2D confinement induced resonance first found in Ref. [5].

IV. FORMALISM FOR N -BODY SYSTEMS A. k representation

In this section the Bethe-Peierls method is applied to few-and many-body systems in the k representation. The system is composed of N particles of respective masses m 1 , m 2 , . . . m N and of momenta k 1 , k 2 , . . . k N . For convenience the set of momenta is denoted by the short-hand notation {k}. The relative and total momenta of a given pair of particles (ij), are denoted by

k ij = m j k i -m i k j m i + m j , K ij = k i + k j , (90)
and the reduced mass of the pair (ij) is denoted µ ij :

µ i,j = m i m j m i + m j (91)
It is also convenient to introduce the set of momenta denoted

{ξ ij } with {ξ ij } = ξ ij 1 , ξ ij 2 .
. . ξ ij N where:

ξ ij n =            K ij m i 2(m i + m j ) for n = i K ij m j 2(m i + m j ) for n = j k n otherwise . (92)
This last notation permits one to isolate in the kinetic energy the contribution of the relative particle (ij) from the N -1 other momenta:

N n=1 k 2 n m n = k 2 ij µ i,j + N n=1 ξ ij n 2 m n . (93)
A non vanishing source amplitude |S i⇌j Ψ is associated with any pair of particles interacting via the short range pairwise potential in the many-body state |Ψ ǫ . In the case where the system is composed of particles with spin, in what follows |Ψ ǫ denotes the projection over a given spin configuration of the many-body state including the symmetry imposed by the quantum statistics. The free Hamiltonian of the system (which may include an external potential) is denoted H 0 , and the N -body stationary state at energy E verifies:

(H 0 -E)|Ψ ǫ = - i<j |(ij) : δ D ǫ |S i⇌j Ψ . (94
k ij |(ij) : φ Λ ǫ = - 2µ i,j 2 × χ ǫ (k ij ) k 2 ij + Λ 2 (96)
In what follows Eqs. ( 94) and (95) are used in the case where there is no external potential and |Ψ (0) denotes the complementary solution of Eq. ( 94) which satisfies the boundary conditions of the problem considered (for example an incoming plane-wave for a scattering problem). The many-body wavefunction can then be written as

{k}|Ψ ǫ = {k}|Ψ (0) + G E ({k}) i<j χ ǫ (k ij ) {ξ ij }|S i⇌j Ψ (97)
where G E ({k}) is the N -body Green's function in the k representation:

G E ({k}) = 1 E + i0 + - N n=1 2 k 2 n 2mn (98)
For each interacting pair, the contact condition Eq. ( 95) gives in the k representation an integral equation in terms of the source amplitudes and of the complementary solution |Ψ (0) . Interestingly, one can extract the contribution of the interacting pair (ij) in the contact condition Eq. (95) without any integration by using the fact that this equation is invariant in a change of Λ ∈ R + (i.e., by using the Λ freedom). To this end, one expresses the Green's function in Eq. (97) as:

G E ({k}) = 1 E (ij) col + i0 + - 2 k 2 ij 2µi,j , (99)
where

E (ij)
col is the collisional energy of the pair (ij) defined by:

E (ij) col = E - N n=1 2 ξ ij n 2 2m n . ( 100
)
It is the energy of the pair (ij) in its own center-of-mass frame while the other particles do not interact for a given set of momenta {k} and a total energy E.

For a negative energy (E < 0) in which case |Ψ (0) = 0, without loss of generality one can make the particular choice Λ 2 = -2µ i,j E (ij) col / 2 in Eq. (95). That way, the term involving the reference state φ Λ exactly cancels with the term associated with the source amplitude of the pair (ij) (|S i⇌j Ψ ). By analytical continuation, the same integral equation holds for E > 0 (in which case |Ψ (0) = 0). Finally, assuming that the ǫ → 0 limit is well defined, for each interacting pair one obtains the following integral equation:

n<p (n,p) =(i,j) d D k ij (2π) D G E ({k}) {ξ np }|S n⇌p Ψ = {ξ ij }|S i⇌j Ψ T D (E (ij) col + i0 + ) - d D k ij (2π) D {k}|Ψ (0) . ( 101
)
One has to notice that the integrals in the first line of Eq. ( 101) are performed with the constraint that {ξ ij } is held fixed (but {ξ np } is not !).

B. Second quantization

In this section only fermionic particles of the same mass m and two spin-components are considered. Generalization to other systems gives similar equations. The creation and annihilation operator of an atomic wave of momentum k for a spin σ are denoted a † k,σ and a k,σ with the standard anti-commutation rule:

{a k,σ , a † k,σ ′ } = (2π) 3 δ(k -k ′ )δ σσ ′ and {a k,σ , a k,σ ′ } = 0.
The general expression of the Λ-potential is a simple way to write the Hamiltonian in its second quantized form. Using the definition of the Λ potential for finite values of ǫ in Eq. ( 41), the Hamiltonian can be expressed as:

H ǫ = d D k (2π) D σ 2 k 2 2m a † k,σ a k,σ + T D (E Λ ) 2 d D K (2π) D × d D k ′ (2π) D χ ǫ (k ′ )a † 1 2 K-k ′ ,↑ a † 1 2 K+k ′ ,↓ × lim ǫ→0 r Λ ǫ d D k (2π) D χ ǫ (k)a 1 2 K-k,↓ a 1 2 K+k,↑ • , ( 102
)
where the dot (•) after the annihilation operator a 1 2 K+k,↑ reminds that the limit ǫ → 0 depends on the many-body state on which H ǫ applies.

In the hypothesis where the few-or many-body state |Ψ ǫ is well defined in the zero-range limit (where only the Bethe-Peierls contact condition is used), the mean energy of the system Ψ ǫ |H ǫ |Ψ ǫ is a regular function of ǫ and the energy theorem [27][28][29][30][31] follows from the identity:

lim ǫ→0 r (1/ãD ) ǫ [ Ψ ǫ |H ǫ |Ψ ǫ ] = H . ( 103
)
The state |Ψ ǫ verifies the contact condition for each interacting pair thus using the property in Eq. ( 37), action of the operator lim ǫ→0 r

(1/ãD ) ǫ

[•] on the interacting term in Ψ ǫ |H ǫ |Ψ ǫ gives exactly zero and:

H = lim ǫ→0 r (1/ãD ) ǫ d D k (2π) D σ 2 k 2 2m a † k,σ a k,σ ǫ .
(104) The energy of the system can be thus expressed in terms of the one-body density:

n k,σ = lim ǫ→0 a † k,σ a k,σ ǫ . ( 105
)
In the limit where ǫ → 0, one obtains from Eq. ( 97) the high-momentum behavior of n k,σ :

n k,σ = k→∞ S k 4 + O 1 k 6 , ( 106
)
where S is spin independent since the interaction only occurs between particles of different spin. Hence, the integrand in Eq. ( 104) behaves as 1 k 2 at high momentum and the action of the regularizing operator gives:

H = d D k (2π) D σ 2 k 2 2m n k,σ - 2 Sã 2 D 2m(1 + k 2 ã2 D )
, (107) where the use of the length ãD defined in Eq. ( 38) permits one to gather in a single form the energy theorems for the three different dimensions.

V. FEW-BODY BOUND STATES A. Efimov states

In this section, the generic case where the Efimov effect occurs is considered: three identical bosons of mass m interact in 3D with a pairwise interaction of vanishing range with respect to the 3D scattering length a 3 [START_REF] Efimov | [END_REF]51]. These states have been observed for the first time in ultra-cold atoms [52][53][54][55]. In what follows, some universal properties of Refs. [35,[START_REF] Efimov | [END_REF]51] are recovered from the STM equation by using the nodal condition introduced in Ref. [34]. However, it is worth pointing out that deviations from universal predictions which are observed in experiments can be taken into account through finite range models [56][57][58][59]. The energy of a trimer E = E 3 in its center-of-mass frame is negative (and less than the dimer's energy E 2 if it exists) so that the complementary solution in Eq. ( 101) is zero (|Ψ (0) = 0). The binding wavenumber q > 0 of a trimer is defined from:

E 3 = - 2 q 2 m . ( 108
)
The Bose statistics imposes that the source amplitudes for each pair of particle coincide with the same function:

{ξ 12 }|S 1⇌2 Ψ = {ξ 13 }|S 1⇌3 Ψ = {ξ 23 }|S 2⇌3 Ψ . (109)
As a consequence of translation invariance, the source amplitudes in the center-of-mass frame can be written as:

{ξ 12 }|S 1⇌2 Ψ = (2π) 3 δ(k 1 + k 2 + k 3 ) F(k 3 ). ( 110
)
The integral eigenequation Eq. ( 101) takes the form: F(k)

f 3 i q 2 + 3k 2 4 = 8π d 3 u (2π) 3 F(u) u 2 + k 2 + k.u + q 2 .
(111) Equation ( 111) is the so-called STM equation [32]. This equation is rotationally invariant and can thus be studied in each momentum sector. In Ref. [60], Danilov showed that as it stands Eq. ( 111) is ill defined: it supports a continuum of negative energy solutions in the s wave sector of F (k). Hence, the Bethe-Peierls asymptotic method which is at first sight adapted for modeling the threeboson resonant problem does not permit one to derive a well-defined eigenequation. The s wave component of the source amplitude F(k) is denoted as,

dΩ 4π F(k) = φ(k), (112)
and φ(k) verifies the integral equation, φ(k)

f 3 i q 2 + 3k 2 4 = 2 π ∞ 0 du φ(u)K q (k, u), (113)
where the s wave kernel of the STM equation is given by:

K q (k, u) = u k ln u 2 + k 2 + q 2 + ku u 2 + k 2 + q 2 -ku . ( 114
)
The fact that there exists a continuum of negative energy solutions means that the Bethe-Peierls model is not self-adjoint for the three-boson problem. Danilov found a way to restore the self-adjointness by introducing a supplementary condition on the high momentum asymptotic behavior of the function φ(k) [60]. The method of Danilov is based on the fact that for all values of q and a = 0, Eq. ( 113) supports solutions with the asymptotic behavior:

φ(k) ∼ k→∞ Ak is0-2 + Bk -is0-2 , (115)
where A and B are two constants and s 0 solves the equation sin(πs 0 /6) = √ 3s 0 cos(πs 0 /2)/8 (s 0 ≃ 1.00624 . . . ). The zero-range approach is made self-adjoint if one fixes the asymptotic phase shift between the two conjugate behavior k ±is0 for all values of a and q. However, Minlos and Fadeev showed that even with this supplementary phase-shift condition the spectrum is not bounded from below [61]: this is the so-called Thomas collapse which is characteristic of zero-range forces [33]. In 1970, Efimov solved the three-boson problem in the resonant regime by introducing the notion of a three-body parameter which is also in the zero-range limit, a way to fix the asymptotic phase-shift in Eq. ( 115). In the k representation it is referred in what follows as κ ⋆ and is defined through the asymptotic behavior of φ(k):

φ(k) ∝ k→∞ 1 k 2 sin s 0 ln k √ 3 κ ⋆ . (116)
By construction the three-body parameter is not unique [Eq. ( 116) is invariant in a change κ ⋆ → κ ⋆ exp(π/s 0 )]. At unitarity (i.e., |a| → ∞ and b 3 → 0), Efimov showed that the spectrum of trimers is characterized by an accumulation point at zero energy: binding wavenumbers of the trimers are related the each to the others by a scaling factor,

q n = q p e -(n-p)π/s0 , (117)
where (n, p) ∈ Z 2 . Interestingly, in Ref. [62] the analytical expression for the source amplitude of the trimers has been found at unitarity:

φ(k) = 1 k q 2 + 3k 2 4 sin s 0 arcsinh k √ 3 2q . (118)
In particular, this result shows that the choice made for the definition of κ ⋆ in Eq. ( 116) is such that the spectrum of the zero-range theory at unitarity is q n = κ ⋆ exp(nπ/s 0 ) with n ∈ Z. The shape of the function k 2 φ(k) of Eq. ( 118) is given in Fig. (1). For increasing values of k starting from k = 0, the first zero of φ(k) is located at the momentum k = 2q √ 3 sinh(π/s 0 ). Using, this property, it is possible to filter a set of solutions of Eq. (113) satisfying Eqs. ( 116) and (117) for sufficiently high quantum numbers (shallow states) and with a spectrum bounded from below. This is done by imposing a nodal condition on the eigenstates,

φ(k reg p ) = 0, (119)
where the node k reg p is chosen among the set:

k reg p = κ ⋆ √ 3 e pπ/s0 p ∈ Z. (120)
The position of the node in Eq. ( 120) fixes the minimum energy of the spectrum. For example if one chooses the nodal condition for p = 2, using the fact that exp(π/s 0 ) ≃ 2 sinh(π/s 0 ), the minimum energy E 0 is almost equal to -2 κ ⋆ 2 exp(2π/s 0 )/m [with a relative error 2 exp(-2π/s 0 ) ≃ 3.8 × 10 -3 ) and the spectrum at unitarity is asymptotically (i.e., for large values of n]

given by

E n = - 2 κ ⋆ 2 m exp -2(n -1)π s 0 , n ∈ N. (121)
For a finite scattering length a 3 and for k|a 3 | ≫ 1, the eigenfunctions φ(k) of Eq. ( 113) have the same behavior as the unitary solutions of Eq. ( 118) so that Eq. ( 119) can be also used as a filtering condition and permits one to recover the universal spectrum of the zero-range theory for energies much larger than E 0 . In principle, the zerorange theory also called 'universal theory' in the literature is recovered by imposing the filtering condition at an arbitrary large node k reg p . To summarize this discussion, the nodal condition in Eq. ( 119) has two roles: first, it imposes the Danilov's asymptotic phase shift for states of sufficiently high quantum number and second, it imposes a minimum energy to the spectrum. For a realistic finite range force, the possible values of the binding wavenumbers obtained from the zero-range theory in Eq. ( 117) are such that q n b 3 ≪ 1, the ground state energy in Eq. ( 121) has thus to be chosen higher than -2 /(mb 2

3 ). The nodal condition in Eq. (119) applied in Eq. ( 113) can be transformed into an integral condition:

0 = - 2 π ∞ 0 du φ(u)K q (k reg p , u). (122)
Subtracting Eq. (122) from Eq. (113) gives a regularized STM equation [34]:

φ(k) f 3 i q 2 + 3k 2 4 = 2 π ∞ 0 du K q (k, u) -K q (k reg p , u) φ(u), (123)
which encapsulates the nodal condition. The Danilov-Efimov contact condition (116) is implemented exactly in the STM equation in the limit where the integer p tends to infinity in Eq. (123). Numerical solutions of Eq. ( 123) are obtained by introducing an ultraviolet cutoff Q in the integral. One can verify that results are insensitive to the choice made on the cut-off for Q ≫ k reg p . For instance considering the unitary limit, taking the nodal condition at k reg 1 and an uv cut-off at 5 × 10 2 × κ ⋆ , one finds the relative error for the ground state equals to 3.8 × 10 -3 ≃ 2 exp(-2π/s 0 ) and for the first excited states (n ≤ 8) one finds a relative error less than 10 -4 . In practice, the value of the scaling factor exp(2π/s 0 ) ≃ 515 is relatively large so that for the nodal condition at k reg 2 , the spectrum is very close to the universal spectrum beginning from the second branch (n ≥ 1) and for an inverse scattering length 1/|a 3 | smaller or of the order of κ ⋆ . In Fig. (2), the second Efimov branch of the regularized STM Eq. ( 123) with the nodal condition at k reg 2 has been plotted as a function of (1/a 3 ). As in Ref. [35], the thresholds for the appearance of the trimers are denoted by a * at the atom-dimer continuum limit and by a ′ * at the three-atom continuum limit. As a consequence of the choice of the nodal condition at k reg p for a finite p, results slightly differ from the zero-range theory in Ref. [35]. The atom-dimer scattering problem (regime FIG. 2: Solid line: Second branch of the trimer's spectrum obtained with the nodal condition at k reg 2 . Vertical dashed line: Trimer's appearance thresholds. Oblique dashed line: Atom-dimer continuum limit. The value of the scattering length at the trimer's appearance threshold a * ≃ -1.51/κ ⋆ and a ′ * ≃ .0713/κ ⋆ are close to the results of the "universal theory" in Refs. [35,62], a * ≃ -1.507/κ ⋆ and a ′ * ≃ .0707/κ ⋆ . The difference is due to the fact that the zero-range theory is recovered only for a nodal condition at an arbitrarily large k reg p .

where a 3 > 0) can be also solved by using the same filtering technique. The atom-dimer scattering length is computed as a function of the atomic scattering length a 3 for a given value of the three-body parameter κ ⋆ . To this end, the atom-dimer collisional energy is set to zero (q = 1/a 3 ) and the ansatz for the source amplitude is [32]

φ(k) = 2π 2 δ(k) k 2 + 4π g(k) k 2 . ( 124
)
Using the regularized STM equation one obtains 3g(k)a 3

8 1 + 1 + 3 4 (ka 3 ) 2 = a 2 3 1 + (ka 3 ) 2 - a 2 3 1 + (k reg p a 3 ) 2 + ∞ 0 du πu 2 K 1/a3 (k, u) -K 1/a3 (k reg p , u) g(u). (125)
The atom-dimer scattering length denoted (a ad ) is the zero momentum limit of the function g(k) and is plotted in Fig. (3) for the nodal condition taken at k reg 2 . Figure much smaller than 1/k reg 2 i.e., in a regime where the nodal condition permits to recover the universal theory with high accuracy. The scattering length a ad diverges at the threshold of appearance of an Efimov's trimer [a 3 ≃ a * exp(nπ/s 0 ), n ∈ N] and thus exhibits the log-periodicity which is a characteristics of the Efimov physics.

In Ref. [63] another regularizing technique of the STM equation was derived in the framework of the effective field theory. In this last formulation, the role of the integral counter-term explicitly depends on the uv integral cut-off of the integral (Q) in such a way that the parameter has a fixed value for all Q. In Refs. [64,65] a subtraction technique has been also introduced in order to regularize the atom-dimer scattering problem. In these last references, the subtraction is made at zero momentum and therefore imposes the exact value of the atom-dimer scattering length. Thus, this regularization scheme does not correspond to the nodal condition of Eq. (119). It is interesting to write down the equation for the atom-dimer scattering amplitude g(k)

obtained within this scheme: 3g(k)a 3

8 1 + 1 + 3 4 (ka 3 ) 2 - 3a ad a 3 16 = a 2 3 1 + (ka 3 ) 2 -a 2 3 + ∞ 0 du πu 2 K 1/a3 (k, u) - 2u 2 a 2 3 1 + u 2 a 2 3 g(u), (126)
and to compare it with Eq. (125).

B. Heteronuclear trimers

As a consequence of interesting predictions [8,16,17,[66][67][68][69][70][71][72][73], heteronuclear systems play an important role in ultra-cold physics. In this section, the eigenequation for heteronuclear trimers is considered without external potential. However, it is worth noticing that an external potential can deeply change the physical properties of the system as it is the case for example in systems of mixed dimensions [69,70] The present study is also restricted to '2+1' few-body systems where the two-body interaction is non-vanishing for heterogeneous pairs of particles only. As an example in the case of (bF ↑ ) interacting pairs, other pairs like (bF ↓ ) or (bB) are not interacting. For fermions neglecting (F ↑ F ↑ ) or (f ↑ f ↑ ) interactions is an exact assumption which follows from the Pauli principle. However, neglecting (BB) or (bb) interactions is not an exact hypothesis and only means that these interactions are negligible with respect to a heteronuclear interaction. Thus the true inter-atomic forces of the heteronuclear interacting pairs are in the vicinity of a s wave resonance while interaction for pairs of identical bosons is neglected. For convenience, the following notations are introduced for the different combinations of particles masses m i :

M (ij) = m i + m j (127) µ i,(jk) = m i M (jk) m i + M (jk) (128)
µ (ij),(kl) = M (ij) M (kl) M (ij) + M (kl) . (129)
In this section, two identical atoms of mass M (particles 1 and 2) interact with another one of mass m (particle 3). for trimers is obtained for a negative energy E = E 3 ,

E 3 = - 2 q 2 2µ 2,3 < 0, (130)
and if a dimer of binding energy E 2 exists then E 3 < E 2 .

In the center-of-mass frame, the source amplitude associated with the pair ( 23) is:

{ξ 23 }|S 2⇌3 Ψ = (2π) D δ(k 1 + k 2 + k 3 ) F(k 1 ). ( 131
)
The other source amplitude with the momentum coordinates represented schematically in Fig. [START_REF] Olshanii | [END_REF], is deduced from Eq. ( 131) by using the exchange symmetry between particle 1 and particle 2:

{ξ 13 }|S 1⇌3 Ψ = s 13 (2π) D δ(k 1 + k 2 + k 3 ) F(k 2 ), (132)
where s 13 = 1 when particles 1 and 2 are bosons and s 13 = -1 if they are fermions. For simplifying the notations, it is useful to introduce the mass ratio y defined by

y = µ 2,3 m 3 = M M + m . (133)
Assuming that the ǫ → 0 limit is well defined, the eigenequation is obtained from the contact condition for the pair (2 : M ; 3 : m) in Eq. ( 101) where |Ψ (0) = 0:

-s 13 2 F(k) 2µ 2,3 T D (E col k ) = d D u (2π) D F(u) u 2 + k 2 + 2yk.u + q 2 .
(134) In Eq. (134) the identity d D k 23 = d D k 2 has been used (the contact condition is performed at a fixed value of K 23 ) and E col k is the collisional energy:

E col k = E 3 - 2 k 2 2µ 1,(23) < 0. (135)
A detailed study of the three-body bound states in 2D has been performed in Ref. [67], and in this subsection the discussion is centered on the 3D case. By using the rotational symmetry of the kernel in Eq. ( 134), one can fix an arbitrary direction: êq and expand the source amplitude F(k) in terms of partial waves as:

F(k) = ∞ l=0 P l (ê k • êq )F l (k). ( 136
)
Each component F l (k) verifies the integral equation

2 s 13 (-1) l+1 F l (k) 2µ 1,3 T 3 (E col k ) = ∞ 0 du uF l (u) yπk Q l u 2 + k 2 + q 2 2yku , (137
) where Q l is a Legendre function of the second kind. One can notice that the standard STM Eq. ( 111) is obtained by setting formally s 13 = 2 and y = 1/2 in Eq. (137). For a sufficiently large value of the mass ratio, the Efimov effect appears in non zero partial waves [74,75]. As in the three-boson case, in this regime and in the limit of large momentum k Eq. (137) supports a pair of solutions F l (k) ∼ k -2±is (s ∈ R). The critical values of the mass ratio of the Efimov theshold in the l wave [denoted M m crit l ] are plotted in Fig. (5). They have been deduced from Eq. (137) in the Appendix A and the results found with this method coincide with the ones computed in Refs. [9,17,75].

C. Heteronuclear tetramers

Recent theoretical [73,[76][77][78][79][80] and experimental [81,82] progress has been achieved in the four-body problem with ultra-cold atoms. In this section, a zero-range eigenequation for four-particle bound states or "tetramers" is considered. The binding energy is denoted E 4 and

E 4 = - 2 q 2 2µ 1,2 , (138)
where E 4 < E 3 and/or E 4 < 2E 2 if a trimer and/or dimer exists. A particle (i) is characterized by a momentum k i and a mass m i equals to m or M . In the center-of-mass frame, the source amplitude associated with the pair ( 12) can be written as:

{ξ 12 }|S 1⇌2 Ψ = (2π) D δ( 4 n=1 k n )F(K 12 , k 34 ). ( 139
)
The integral equation satisfied by the function F is ob-FIG. 6: Schematic representation of the momentum coordinates used for the source amplitude associated with the contact condition between particles 1 and 2 for a system of fermions with two-mass components.

tained from Eq. (101) (here (ij) = (1, 2)). A specific conis represented schematically in Fig. (6). The summation on the left hand side of the integral equation is composed of source terms which are deduced from Eq. (139) by using the permutation symmetry and the statistics of the particles. In order to have a general equation for the different possible configurations, a statistical factor denoted s np is introduced for each pair of particles. Depending on the system, for two interacting particles n and p, the statistical factor is s np = ±1 or s np = 0 for noninteracting particles. Using this notation, the different source amplitude are written in Tab. (I). Different possible configurations of the two-mass component system and the corresponding statistical factors are listed in Table II. Equation (101) can be simplified to

d D k 12 (2π) D n<p p>2 s np F (K np , k kl ) κ 2 + k 2 12 = -2 F(u, v) 2µ 1,2 T D (E (12) col )
,

where u = K 12 , v = k 34 . In Eq. (140) the momentum κ is related to the collisional energy by E (12) col = 2 κ 2 /(2µ 1,2 ), thus: II. Using the notations defined in Table III, the integral equation which encapsulates the Bethe-Peierls contact condition can finally be written as:

κ 2 = q 2 + µ 1,2 µ (12),(34) u 2 + µ 1,2 µ 3,4 v 2 . ( 141
) Pair (ij) Permutations {ξ ij }|S i⇌j Ψ (12) (2π) D δ( 4 n=1 kn) F(K12, k34) (13
Configuration (m1, m2, m3, m4) (s13, s14, s23, s24, s34) BBBB (m, m, m, m) (1, 1, 1, 1, 1) (f or b)BBB (m, M, M, M ) (1, 1, 0, 0, 0) BbBb (M, m, M, m) (0, 1, 1, 0, 1) F ↑ f ↓ F ↑ f ↓ (M, m, M, m) (0, -1, -1, 0, 1) Bf ↑ Bf ↑ (M, m, M, m) (0, -1, 1, 0, -1) (f or b)F ↑ F ↑ F ↑ (m, M, M, M ) (- 1
d D k (
κ 2 + k 2 = -2 F(u, v) 2µ 1,2 T D (E (12) col )
,

In the particular case where the system is composed of two heterogeneous dimers made of fermions (F ↑ f ↓ F ↑ f ↓ ), one recovers the same integral equation as in Eq. ( 16) of Ref. [12]. In 3D depending on the statistics of the particle and of the mass ratio, Eq. ( 142) is not in general well defined as a consequence of the Efimov-Thomas effect. For systems composed of four particles in 2D, bound states of particles of same mass has been considered in [77]. In the present work, the binding energies

In order to check the numerical computation, the particular case of four identical bosons which has been already obtained by several authors [76,77] has been considered. In this configuration, two bound states have been found with binding energies which are close to already published results: E 4 /E 2 = 197 and 24, to be compared with E 4 /E 2 = 197.3 and 25.5 in Ref. [76] or with E 4 /E 2 = 194 and 24 in Ref. [77]. Results for three identical bosons interacting resonantly with another particle is shown in Fig. (7). In 3D for the same bBBB configuration, similarly to the three-boson original STM equation, Eq. (142) does not constitute a well-defined problem. The nature and properties of eigenstates of this system in the zerorange limit are yet unsolved. The bF ↑ F ↑ F ↑ or f ↓ F ↑ F ↑ F ↑ configurations have been studied recently in Ref. [73] and a pure four-body Efimov effect (i.e., without a three-body Efimov state) involving a four-body parameter has been found for a mass ratio of 13.384 < M/m < 13.607.

VI. CONCLUSIONS

In this paper, it has been shown how the zero-range approach can be formulated for different problems in a unified framework without technical intricacy for all dimensions D ≤ 3. In the few-body problem, general integral eigenequations are obtained straightforwardly in the momentum representation. Efimov effect is deduced from the Skorniakov Ter-Martirosian equation by using a nodal condition and a subtracting technique which is close to regularizing schemes used in the Effective Field Theory.

Various open issues remain to be solved for few-body systems in the limit of zero-range forces. For example, the question whether or not the 3-body parameter is enough for describing 4-bosons properties in 3D is still under debate [79,80]. For heteronuclear few-body systems, taking into account the external trapping (which is a natural concern in regards to experiments) makes the problem even richer. The external potential differs for different atomic species and this degree of freedom allows one to explore realistic intermediate situations between the fully three-dimensional homogeneous configuration and the limit of mixed dimensions where interesting predictions have been already made in the threebody case [69,70]. The present formalism can be also used to model systems where interaction occurs between three spin components like in 6 Li experiments [83][84][85][86]. Another open issue concerns the possible occurrence of Efimov states for more than four particles in 3D. The four-body Efimov phenomenon has been found to occur for a specific interval of mass ratio in the 3+1 fermionic problem in Ref. [73]. Therefore, one can wonder whether or not a pure Efimov effect in a N + 1 fermionic system can also occur and at which critical mass ratio for N ≥ 3. Despite the difficulty of this problem, the integral equation of the system in the limit of zero-range forces can be easily deduced from the present formalism and is given below as a concluding remark. Each of the N polarized fermions of mass M labeled by i (1 ≤ i ≤ N ) interact with only one impurity (particle i = N + 1) of mass m. In the center-of-mass frame, the source amplitude for the pair N, (N + 1) can be written as:

(i = 1 • • • N -1)
are deduced from this ansatz by using the fermionic statistics. The general integral equation for this problem which encapsulates the Bethe-Peierls asymptotic condition is then obtained from Eq. (101):

d D k N (2π) D F (k N , k 2 , k 3 , • • • k N -1 ) + F (k 1 , k N , k 3 , • • • k N -1 ) + • • • + F (k 1 , k 2 , • • • k N -2 , k N ) -E -i0 + + 2 2M N i=1 k 2 i + 2 2m N i=1 k i 2 = F (k 1 , k 2 , • • • k N -1 ) T D (E col + i0 + ) - d D k N (2π) D Ψ (0) (k 1 , • • • k N ), ( 145
)
where

E col = E - 2 2M N -1 i=1 k 2 i - 2 2(M+m) N -1 i=1 k i 2 , Ψ (0
) is a complementary solution associated with the free Hamiltonian at energy E in the center-of-mass frame (it is equal to zero for E < 0) and F is antisymmetric under the exchange of two coordinates (k i , k j ).

)

  In Eq. (94), |(ij) : δ D ǫ is the state |δ D ǫ for the relative particle (ij) of momentum k ij , and the source amplitude |S i⇌j Ψ is a state for the N -1 other particles of momenta {ξ ij }. If the pair (ij) does not interact then |S i⇌j Ψ = 0, otherwise it satisfies the Bethe-Peierls contact condition: lim ǫ→0 (ij) : δ D ǫ | |Ψ -|(ij) : φ Λ |S i⇌j Ψ where the reference state |(ij) : φ Λ ǫ is defined in the k representation as:

FIG. 1 :

 1 FIG.1: Shape of the expession of k 2 φ(k) in Eq. (118) plotted in a semi-log plot as a function of k/q.

3 FIG. 3 :

 33 FIG. 3: Atom-dimer scattering length a ad computed with the nodal condition at k reg 2 and plotted in a semi-log scale as a function of κ ⋆ /a3 for a3k reg 2 ≪ 1.

  . Particles (labeled by i) can be either bosons or fermions and have a mass m i which can take two possible values: m i = m or m i = M . Bosons (respectively fermions) of mass M are denoted by B (respectively, by F ) while bosons (respectively, fermions) of mass m are denoted by b (respectively, by f ). Fermionic particles are supposed to have two possible internal states denoted ↑ and ↓. The configurations studied here are thus constructed from the set of particles {b, B, f ↑ , f ↓ , F ↑ , F ↓ }.

FIG. 4 :

 4 FIG.4: Schematic representation of the momentum coordinates used for the source amplitude associated with the contact condition between particles 1 and 3.

FIG. 5 :

 5 FIG. 5: Critical mass ratio M m

3 , 2 ↔

 32 4 s34(2π) D δ( 4 n=1 kn) F(K34, k12) TABLE I: Source terms for each pair (ij) deduced by using the permutation symmetry of the four-body wave function from the source term of the interacting pair (12) in Eq. (139). The statistical factors sij are given in Table

  , -1, 0, 0, 0) TABLE II: Different possible 4-body configurations. The notation is as follows: B (respectively F ) means that the atom is a boson (respectively fermion) of mass M , b (respectively f ) means that the atom is a boson (respectively a fermion) of mass m. The fermions have two possible internal states ↑ and ↓. For each configuration, interaction is non vanishing only between one type of heterogeneous pair. The statistical factors sij appears in the integral equation (142).

FIG. 7 :

 7 FIG. 7: Ground state branch for the three bosons of mass M interacting with one impurity of mass m in 2D as a function of the mass ratio.

  {ξ N,(N +1) }|S N ⇌N +1 Ψ = (2π) D δ N +1 i=1 k i × F (k 1 , k 2 . . . k N -1 ), (144) and other source amplitudes |S i⇌N +1 Ψ

  2π) D s 13

		F(k, v 13 ) κ 2 + σ 2 13	+ s 14	F(k, v 14 ) κ 2 + σ 2 14	+ s 23	F(k, v 23 ) κ 2 + σ 2 23
	+s 24	F(k, v 24 ) 24 κ 2 + σ 2	+ s 34	F(-u, k)

TABLE III :

 III Coordinates appearing in Eq. (142).

	Pair (ij)			vij					σij
	(13)	m4u M3,4	+ v -	m4k M2,4	µ1,2 m1	-	µ3,4 m3	u -v + k
	(14) -	m3u M3,4	+ v +	m3k M2,3	µ1,2 m1	-	µ3,4 m4	u + v + k
	(23)	m4u M3,4	+ v +	m4k M1,4	µ1,2 m1	-	µ3,4 m4	u + v -k
	(24)	m3u M3,4	-v +	m3k M1,3	µ1,2 m1	-	µ3,4 m3	u -v -k

ACKNOWLEDGMENTS

It is a pleasure to thank Y. Castin, M. Jona-Lasinio, C. Mora and F. Werner for discussions. LPTMC is UMR 7600 of the CNRS and its Cold Atoms group is associated with IFRAF.

Appendix A: THRESHOLDS FOR THE '1+2'-BODY EFIMOV EFFECT

In this appendix, the values of the mass ratio for which an Efimov effect occurs are deduced from Eq. (137). To this end, zero energy solutions at unitarity |a 3 | = ∞ are considered (solutions for finite energy have the same high momentum behavior). In this regime, Eq. ( 137) is scale invariant and this allows one to search for power law solutions:

, where ν is a function of the mass ratio. For convenience, the following dimensionless parameter t is introduced:

Thus, the scattering amplitude is f 3 (κ k ) = 1/(k cos t) and Eq. (137) gives for each partial wave l an eigenvalue equation: λ(ν, l, t) = 0 where,

(A2) The integrand in Eq. (A2) is positive, hence for a Bose (respectively Fermi) statistics only even (respectively odd) values of l can support a solution. For x > 1,

where W l (x) is a polynomial of order l. The polynomial W l does not contribute in the integral of Eq. (A2) and λ(ν, l, t) can be thus expressed in terms of a sum of functions of the form:

with 0 ≤ t ≤ π 2 and |ℜ(γ)| < 1. In Eq. (A3), I(γ, t) can be calculated explicitly as:

In the l wave, Efimov states are characterized by a power law such that ν = -l + is, where s is real. For example, for two identical bosons interacting with another particle (s 13 = 1), in the s wave sector,

(A5) Equation (A5) admits solution of the form ν = is for all values of the mass ratio and s → 0 for t → 0. For higher l waves, Efimov states appear above a critical value of the mass ratio. The threshold is obtained by searching the value of the parameter t = t crit l such that:

In the p-wave sector which concerns the case of two identical fermions interacting with another particle (s 13 = -1) one gets

and the threshold is obtained from the equation

, which gives the critical mass ratio M m crit l=1 ≃ 13.6 found in Refs. [8,75]. For higher partial waves, the critical values of the mass ratio M m crit l found by this method coincide with the ones computed in Ref. [17] where the hyperspherical method was used. Results are reported in Fig. (5).

In the regime where an Efimov effect occurs, the regularizing technique can be achieved by fixing the asymptotic phase shift which is a function of the three-body parameter (denoted κ ⋆ l ) in the partial wave l:

A set of solutions satisfying Eq. (A8) can be filtered from Eq. ( 137) by imposing the nodal condition:

Exact zero-range theory is obtained in the limit where p → ∞. Universal results are recovered if the node k reg p,l is chosen at a large value as compared to all the low energy scales (k reg p,l ≫ 1/|a|, q). Using the subtracting scheme of the three-boson problem, Eq. (A9) can be also incorporated into the integral equation Eq. (137):