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Quantum Scattering in Quasi-1D Cylindrical Confinement

Finite size effects alter not only the energy levels of small systems, but can also lead to new effective interactions within these systems. Here the problem of low energy quantum scattering by a spherically symmetric short range potential in the presence of a general cylindrical confinement is investigated. A Green's function formalism is developed which accounts for the full 3D nature of the scattering potential by incorporating all phase-shifts and their couplings. This quasi-1D geometry gives rise to scattering resonances and weakly localized states, whose binding energies and wavefunctions can be systematically calculated. Possible applications include e.g. impurity scattering in ballistic quasi-1D quantum wires in mesoscopic systems and in atomic matter wave guides. In the particular case of parabolic confinement, the present formalism can also be applied to pair collision processes such as two-body interactions. Weakly bound pairs and quasi-molecules induced by the confinement and having zero or higher orbital angular momentum can be predicted, such as p- and d-wave pairings.


I. INTRODUCTION
Matter at very small dimensions can be strongly affected by boundary and surface effects. Bulk 3D properties may then change substantially or even disappear. Reaching this regime poses a long term challenge motivated among others by the continuous technological drive towards ever smaller information processing devices [1]. Below a certain limit of small length scales, novel quantum electronic properties may appear and be useful in order to develop alternative devices [2].
Recent developments in atomic physics have addressed similar questions by using atom-optical devices (or atomchips) made out of surface-patterned substrates [3,4,5,6,7,8] or by means of laser manipulation of atoms for trapping [9] and atomic lithography [10]. Analogous to electronic matter waves, these devices and techniques have the potential to trap and manipulate atomic matter waves down to sub-micron scales. A vast range of phenomena with atoms can then be studied, several of them bearing close relationships to specific solid state systems.
Of particular interest for both bulk and atomic systems is the quasi-1D regime of quantum coherent single mode transport. In this ultimate limit, the degrees of freedom in the transversal trapping directions are effectively frozen in the quantum-limited ground state of the confining potential. Such one-dimensional system can show distinct physical properties e.g in quasi-1D quantum wires in two-dimensional electron gas (2DEG) systems [2,11,12,13,14,15,16,17] and Tomonaga-Luttinger liquids [18,19,20]. In elongated gases of ultracold atoms, the one dimensionality is revealed e.g. in magnetic guides [3,4,5,6,7,8], in phase fluctuations of quasi-condensates [21,22,23], the Tonks-Girardeau gas of impenetrable bosons [24,25,26,27,28,29] or one dimensional optical lattices [30,31]. In this broad con-text, we address two processes that are affected by the quasi-1D geometry. On one hand, the matter waves can be scattered off e.g. impurities or defects fixed along the guide. This scattering by central fields is an important variable in the transport properties and can be strongly affected by the confinement. This has been revealed by several studies of mesoscopic systems in the context of the Landauer approach to the quantized conductance of quantum point-contacts in the presence of impurities [13,14,15,16]. On the other hand, two-body collision processes should be dealt with when the matter wave consists of interacting particles. A full understanding of these low dimensional systems should therefore deal with both types of scattering. It should also derive ad hoc phenomenological parameters such as effective low dimensional coupling strengths from the real 3D character of the interactions. Another context in which such dimensional reduction plays a role is the transmission and reflection of electromagnetic [32] and elastic [33] waves in transmission lines or resonators.
New light has recently been shed on this subject by studies of atomic collisions in 1D [34,35,36,37] and 2D [38,39,40]. Focusing on a parabolic cylindrical confinement, for which the center of mass can be eliminated, pioneering predictions for collision processes are possible, e.g. that of confinement induced scattering resonances (CIR) in 1D. In such a resonance [34,35,36], a total reflection may take place between the colliding atoms. The s-wave (or zero-range) approximation to the scattering potential [34,35] may be partially lifted, provided the coupling of orbital angular momenta due to the cylindrical geometry are not considered [36]. Another property induced by the confinement and closely related to CIR is the formation of weakly bound quasi-molecules with zero (s-wave) orbital angular momentum [37], which have been recently observed with singlet fermionic atoms The scattering potential V (r) is limited to within the spherical shaded area at the center. (b) Total potential energy along the transverse xy-plane passing through the origin r = 0 (thick curve). For an attractive V (r), one deep E0 and another shallow EB bound state without confinement are indicated, with their respective probability density profiles (dotted curves).
in tight laser traps [31]. Coupling to the center of mass under non-parabolic confinement in the s-wave approximation is treated in [41] and can lead to the appearance of additional CIRs. In the independent and distinct context of impurity scattering studies, rectangular geometries predominate which are most suitable for the growth of semiconductor heterostructures. Similar properties of resonant scattering and weakly localized states are then predicted for non-interacting 2DEG systems. This is demonstrated in the s-wave approximation to the scattering potential in [13]. Finite range scatterers under general rectangular confinement are also considered, but in an already reduced two-dimensional space [14,15,16].
In this work, a detailed formalism is presented that extends the results for impurity scattering and collisions. It can treat general cylindrical confinement, parabolic or not, and incorporate all scattering phase-shifts beyond s-waves, as well as the full couplings of orbital angular momenta due to the broken spherical symmetry. A comprehensive assessment of the scattering process reveals the most important mechanisms and parameters at play. From this unified description, CIRs are seen to be a general low energy effect in quasi-1D geometries. It might be useful e.g. as an alternative gating mechanism in low power transistor-like devices that could incorporate quasi-1D structures [2,11,12,13,14,15,16,17,42]. The existence of weakly localized bound states and quasimolecules can, in turn, be extended to include higher orbital angular momentum pairing, whose binding energies and wavefunctions can be systematically computed.
This paper is organized as follows. In section II the present problem of scattering in confined geometries is formulated for both scattering by central fields and collision processes. In section III, the s-wave approximation to the scattering potential is briefly reviewed. In section IV, after an overview of the free space scattering, our formalism in terms of Green's functions and phase-shifts is developed. In section V, this formalism is applied to analyze two types of confinement induced effects, namely, CIR and weakly localized states. A discussion of the main approximations used is given in section VI, followed by the conclusions.

II. SCATTERING IN LOW DIMENSIONALITY
In a low energy scattering process under confinement, one expects the scattering potential to affect primarily the degrees of freedom not experiencing external confinement. The other degrees of freedom are then forced to occupy only one or very few confined states. However, the very existence of these confined states do play a role due to virtual transitions. The effects of the latter can become noticeable when properly examining the unconfined degrees of freedom. In this regard one is lead to calculate effective scattering amplitudes as given by Eq.(5b).
These amplitudes provide a complete description of the quasi-1D problem. For real energies, they describe the scattering properties of running waves. Weakly localized states are then found in the complex-energy plane.

A. Hamiltonian under confinement
Consider first the scattering of a single particle of mass µ and coordinates r = (x, y, z) by a central field fixed at the origin. Let V (r) be the spherically symmetric scattering potential and U (ρ) the cylindrically symmetric confining potential, where ρ = (x, y) denotes the transverse coordinates. The Hamiltonian is then given by If R V and R U are the ranges of V (r) and U (ρ), respectively, then |V | ≪ V 0 for r ≫ R V and U ≪ U 0 for ρ ≪ R U , where V 0 and U 0 are characteristic energy scales, such as the respective ground state energies (see fig. 1). Two particular examples are the parabolic confinement U (ρ) = µ ω 2 ⊥ ρ 2 /2, for which R U is of the order of the oscillator length a ⊥ = ( /µ ω ⊥ ) 1/2 and the squarewell (or hard-wall) such that U (ρ) → +∞ for ρ ≥ R U and zero otherwise. The first condition on the scattering and on the confining potentials assumed in this work is R V ≪ R U . In this sense, both the potentials V (r) and U (ρ) can be relatively general for this type of scattering process. This condition implies that there is a distinct region R V ≪ r ≪ R U where spherical symmetry still prevails and one can define scattering phase-shifts. This is an (intermediate) asymptotic region, where the effect of the confinement U (ρ) is felt only as a boundary condition to the scattering by V (r) (see section IV C). Fig. 1 depicts the geometry of the problem. Another important condition for the validity of the present approach to scattering in confined geometries is the condition k ∼ 1/R U of low total scattering energy E ≡ 2 k 2 /2µ (or Fermi energy) for the Hamiltonian H. Only few excited transverse states ϕ n above the ground state (see Eq.(2)) can be effectively populated. The scattering process is thus assumed to occur at low temperatures or under sufficiently tight confinement.
Alternatively, consider the case of collisions between e.g. two cold atoms of masses m and coordinates r i = (ρ i , z i ), i = 1, 2 . R = (r 1 + r 2 )/2 and r = r 1 − r 2 denote the center of mass (CM) and relative coordinates, respectively. The total Hamiltonian is where U c is the cylindrical confinement and V the twobody interaction potential. For this scattering process U c should be parabolic, so that the CM can be separated exactly. The Hamiltonian for the relative coordinates is then given by Eq. (1), where µ = m/2 becomes the (collisional) effective mass and U (ρ) = 2 U c (ρ/2) . For nonparabolic confinement, additional scattering resonances that can occur even at weak coupling to the CM [41] is not included in Eq. (1). However, at ultracold temperatures for which only the lowest transversal states of the guide are occupied, the probability to find the atoms in the center of the guide is largest. Consequently, it might be a reasonable zeroth-order approximation to assume on average that the CM coordinates are also confined to the center of the guide. Therefore Eq.(1) is a starting point to the dynamics of the relative coordinates for a general interaction V (r), in which the CM fluctuations around the mean value R = 0 are at the present stage neglected.

B. 1D scattering
In a purely 1D problem let the 1D interaction potential V 1D (z) have the finite range R 1D around the origin z = 0, such that V 1D ≈ 0 for |z| ≫ R 1D . The scattering problem for an incoming plane wave e ik0z can be described by the amplitudes f ± in the asymptotic behaviour of the solution ψ(z) in the transmission and reflection regions

Current conservation then yields
Particular cases are resonant transmission (f + → 0) and total reflection (f + → −1) at finite k 0 and finite potentials.
C. Effective 1D scattering The above 1D picture corresponds to a fixed unperturbed transversal state, e.g., the ground state ϕ 0 (ρ). The total wave function would then have the form ψ(z)ϕ 0 (ρ) where scattering effects would occur only in ψ(z). However, transitions to other states ϕ n (ρ), n = 1, 2, . . . , even as virtual transitions, arise from the coupling between the ϕ n 's caused by the scattering potential V (r).
Let Ψ(r) be the full 3D axially symmetric scattering solution to Eq.(1), where E ≡ 2 k 2 /2µ > 0 is the total scattering energy, u(ρ) ≡ 2µU (ρ)/ 2 and v(r) ≡ 2µV (r)/ 2 . The above mentioned couplings and virtual transitions are best seen by expanding the solution in the cylindrical basis defined by the ϕ n 's, and solving for each ψ n (z). Substituting into Eq.(4a), multiplying by ϕ * n , integrating and using Eq.(2) yields For a given total energy E, one defines the longitudinal wave vectors k n by where n E is the largest integer such that q nE < k < q 1+nE and denotes the highest "open channel". In the single mode regime n E = 0. As usual, the general solution ψ n (z) can be expressed in terms of a homogeneous solution and a particular inhomogeneous solution, ψ f,n (z). This inhomogeneous solution can be written in terms of the 1D Green's function describing an outward scattering for n ≤ n E and an exponentially decaying virtual state for the "closed channels" n > n E . Then for all n FIG. 2: A pictorial view of a scattering "path" of a particle coming from the left. Outside the effective 1D range |z| ≫ R ′ 1D , the particle is stabilized by the confinement back into e.g. the tansversal ground state (for the single mode regime nE = 0). Here the net effect of the scattering is contained in the amplitudes f − 0 and f + 0 of backward and forward scattering, respectively. Within the dashed circle, however, the spherical symmetry prevails. The phase information of the wave function within this region is then captured by f ± 0 .
The expansion Eq.(4b) of Ψ(r) takes then the form where the homogeneous part b n e iknz of each ψ n (z), for some constant b n , gives rise to the total incoming state and thus is limited to the open channels n ≤ n E only. The asymptotic condition is obtained in the limit |z| → ∞ by neglecting the exponentially decaying terms ψ f,n (z) in Eq.(4c) for n > n E . Using then |z − z ′ | = |z| ∓ z ′ in G n , for z → ±∞, one has where the effective 1D scattering amplitudes f ± n are defined by (for n ≤ n E ) for forward z > 0 and backward z < 0 scattering, respectively. Note first that although |z| ≫ |z ′ |, the phase k n z ′ in f ± n is not necessarily negligible since k n |z ′ | ∼ 2π for sufficiently large momentum k n and depending on the range R V of the interaction v(r ′ ). This phase is negligible only when the wavelength 2π/k n of the incoming wave is not able to resolve the details of the potential v(r ′ ) at low momenta. In this case, a zero-range or s-wave approximation to v(r ′ ) is valid. However, this approximation should break down when the scattering occurs at short wavelengths or when higher angular momenta become necessary. Note that there is no clearly defined 1D range R 1D as in the purely 1D case, since the asymptotics given by Eq.(5a) depends on the convergence behaviour of the series in Eq.(4c). If the convergence is slow, one expects an effective 1D range, say R ′ 1D , substantially larger than R V or even R U . It is an effective range for the 1D collision and is of the order of the minimum separation from the scattering center, such that Eq.(5a) is the dominant contribution to the scattering wave function (see fig. 2). As regards collision processes, this effective range sets some limits on the average equilibrium distance between the colliding particles.
The present formalism to quantum scattering under confinement is approximate but non-perturbative, in the sense that there is no small parameter around which an expansion is performed. The determination of its range of validity is chosen here by imposing the conservation of probability on the final wavefunctions. The probability conservation condition in 3D and the identity Ω d 3 r divj(r) = Sc+S1+S2 d 2 S · j(r) for a volume Ω enclosed by a large cylinder of surface area S c and by two transversal discs of surface areas S 1 at z 1 < 0 and S 2 at z 2 > 0 yields whereρ andk are cylindrical unit vectors. Applying Ψ = n ψ n ϕ n , the first integral on the rhs vanishes for large radius ρ → ∞ since each ϕ n (ρ) → 0. The other two integrals yield where j n (z) are 1D currents calculated with ψ n (z). Setting z 1,2 ∼ ∓R ′ 1D one can use the asymptotics for ψ n in Eq.(5a) so that the total conservation condition reads This equation extends the pure 1D result in Eq.(3). It also determines how the initial flux is distributed among the open channels. In other words, even when the initial state occupies only a single channel, i.e., b n = δ 0,n , new channels are occupied due to scattering, if the total energy is large enough n E > 0. This conservation condition can serve then as a gauge for the range of validity of our approximations to the amplitudes f ± n .

III. SCATTERING IN THE s-WAVE APPROXIMATION
In many circumstances, the low energy scattering in free space can be well described by approximating the interaction V (r) by Fermi-Huang's zero-range potential [43,44] This approximation singles out the l = 0 angular momentum component of the collision and thus depends only on the s-wave scattering length a. The regularization ∂(r · )/∂r assures that the singularity 1/r of the full scattering solution to this V δ (r) is properly dealt with. For collisions in a strong parabolic confinement and using Eq.(6a), the series in Eq.(4c) also has a singularity for short distances r → 0 very similar to 1/r. By properly dealing with this singularity, with [34] or without any regularization [35], the amplitudes f ± n in Eq.(5b) can be calculated. In the single mode regime (n E = 0) and for small longitudinal momenta k 0 one obtains where C = 1.4603 . . . and a ⊥ is the harmonic oscillator length of the parabolic confinement [34,35]. Here, a total reflection f + 0 = −1 is predicted for large scattering lengths, on the order of the confinement length, such that a ⊥ = Ca , which is called the confinement induced resonance (CIR) [34]. This total reflection means that the colliding pair experiences a diverging effective 1D interaction along the longitudinal cylindrical axis (see section V D 1).
It will be seen later that such a singularitiy should actually be the same singularity 1/r. It arises from the 3D free space Green's function and becomes dominant as r → 0 away from the confining boundaries (see sections IV B and VI). This latter identification is one of the elements that will allow us to incorporate all partial waves l ≥ 1 besides the s-wave. In addition, the above conclusions concerning CIR under parabolic confinement will be seen to hold not only for general interactions V (r), as first examined in [36], but also for general non-parabolic confinement U (ρ) . In this sense the present formalism is of a more general character.

IV. SCATTERING PHASE-SHIFTS
In the following, a general formalism is developed in order to calculate the scattering amplitudes for any given finite short-ranged and spherically symmetric scattering potential V (r). This is done by expressing the amplitudes f ± n in terms of the scattering phase-shifts δ l associated to this V (r). Our formalism is also able to deal with a broad range of confinements U (ρ) as well as scattering energies above the transversal ground state. Particularly relevant is the straightforward accounting of angular momenta couplings due to the confinement. We begin by providing the connection to scattering in free space.

A. Phase shifts in free space scattering
The standard formalism for free space scattering by a central potential is recast in a form suitable for later comparison. The main idea is to express all quantities relevant for scattering in terms of the phase-shifts. The rationale is that these phase-shifts can be considered as intrinsic to a given scattering potential V (r), without much regard to details of external boundary conditions.
In the absence of any confinement, the scattering solution can be written as where Φ i (r) is the incoming state and G(r, r ′ ) is the free space Green's function. In terms of outward and inward scattered waves, the latter reads for some constants γ ± obeying γ + + γ − = 1. A single constant γ can be introduced by setting In free space scattering, the inward component proportional to γ − is usually absent (i.e., γ = 1). However, one expects this inflow of particles if there is e.g. an exterior confinement that forces the scattered particles back towards the center. This interpretation will be validated in section IV B (see also Fig 2). Φ i (r) and G(r, r ′ ) are determined by the boundary conditions imposed on Φ(r), e.g. that of an axially symmetric solution.
The scattering variables of interest here are the amplitudes f out,in of outward and inward waves, scattered along a direction given by θ and φ at large distances r ≫ R V from the scattering center. Using the expansion where k ≡ kr/r depends on the direction (θ, φ) of r. It can be clearly seen that these amplitudes have a shape similar to their equivalent effective 1D amplitudes f ± n in Eq.(5b). To express the amplitudes in terms of the phaseshifts an angular momentum decomposition of f out,in is necessary. Following standard procedures [45,46], we obtain from Eq.(8b) with the definition where j l and P l are the spherical Bessel functions and Legendre polynomials, respectively. The next step is to relate the elements τ l to the phase-shifts. Decompose Φ i (r) and G(r, r ′ ) in spherical coordinates. Substituting then into Eq.(7a) for r > r ′ ∼ R V and carrying out the having used Eq.(7c) and Eq.(9b) and the constants β and β ′ are related to Φ i (r). On the other hand, the phaseshifts arise when directly decomposing this same solution Φ(r) in spherical coordinates and looking at the radial solutions outside the interaction region r > R V , Here the phase-shifts δ l are calculated from the behaviour of the radial parts R l (r) inside the scattering region r < R V only, irrespective of the details of the outer boundary conditions [46]. The boundary conditions are matched by properly choosing the constants c l , i.e. by superposing the spherical components. Comparing these last two forms of Φ(r) yields, Considering the phase-shifts δ l (or the scattering potential v(r)) as given, the second relation Eq.(10b) solves for the amplitudes f out,in in Eq.(9a) (and also the integral equation Eq.(7a) at distances r > R V ). The first relation Eq.(10a) in turn determines how to superpose the spherical components in Φ = l c l R l P l in order to match the boundary condition contained in Eq.(8a).

B. Green's functions under confinement
Just as the free space 3D scattering amplitudes in Eqs. (8) can be written in terms of the phase-shifts via Eqs. (9) and Eq.(10b), a very similar procedure can be applied to calculate the amplitudes f ± n characterizing the 1D effective dynamics. In this section a key element of the formalism is worked out, namely, the Green's function G c under confinement. It is shown that G c can be expressed in terms of the free space Green's function G as in Eq. (12). The final purpose, to be achieved in section IV C, is to express it as a series in spherical coordinates, valid at least in the center of the guide where the spherical symmetry prevails over the cylindrical one due to the main assumption of a short range scattering potential v(r) and a confining potential u(ρ) that is sufficiently flat at the center, R V ≪ R U . Such a series allows then the straightforward introduction of the phase-shifts.
Consider the solution Ψ to Eq.(4a) given by the expansion in Eq.(4c). One can rewrite Eq.(4c) in the integral form (keeping again the φ ′ -integration as in Eq.(9b) for convenience) where the incident state is and G c , the axially symmetric Green's function of the confining potential, has the form It is convenient to introduce a non-axially symmetric Green's function G u under the confining potential u(ρ). This G u then satisfies If the φ-independent solution Ψ(r) is written as an integral equation using this φ-dependent G u , it follows from comparison with the integral equation Eq.(11a) that G c and G u are related by 2π G c (r, r ′ ) = dφ ′ G u (r, r ′ ). Compare now the differential equations of G u and that of the free space Green's function G of Eq.(7b) in the region of the scattering potential such that r, r ′ ≪ R U . Since u(ρ) ≈ 0 by assumption, they should differ at most by a homogeneous term, say ∆ u , so that G u (r, r ′ ) ≈ G(r, r ′ )+∆ u (r, r ′ ), for r, r ′ ≪ R U , where ∆ u satisfies the homogeneous equation (∇ 2 + k 2 )∆ u (r, r ′ ) = 0 and we obtain where ∆ c ≡ dφ ′ ∆ u /2π. In order to obtain the function ∆ c and γ in Eq.(7c) in terms of the scattering parameters, we first write the closed channel (n > n E ) wavefunctions of G c in Eq.(11c) and G in Eq.(7b) using cylindrical Bessel functions. The latter represents a good approximation to these transversal eigenstates for a given potential U (ρ) close to the guide center ρ ≪ R U where U (ρ) ≈ 0. By subsequently comparing G c and G we can determine ∆ c . First we express the transversal eigenstates ϕ n (ρ) in terms of the normalized Bessel function according to where N n = |J 1 (r n+1 )| −1 and r n+1 is the (n + 1)-th root of J 0 , related to the transverse momenta q n by q n R U = r n+1 . A good expression for the roots r n+1 is given by the cosine approximation J 0 (x) ≈ (2/πx) 1/2 cos (x − π/4). This leads to The corresponding energy dispersion relation ǫ n ∼ n 2 is quadratic with respect to n compared to the linear one ǫ n ∼ 2n for, e.g., 2D harmonic oscillators (see e.g. [34]). This should not cause any confusion since this excited spectrum n > n E is summed over in Eq.(11c) and leads to the confinement independent free space Green's function G in Eq. (12). It is the lower part of the spectrum n ≤ n E , in turn, that relates to the term ∆ c .
Indeed, we substitute Eq.(13a) in the series for the closed channels in Eq.(11c) yielding for r, where δq n = π/R U is the increment of q n and we used |J 1 (r n+1 )| 2 = 2/πq n R U . For r, r ′ ≪ R U a continuum approximation to the closed channel series can now be applied. Separating the real and imaginary parts of G c (supposing ϕ n (ρ) real for n ≤ n E without much loss of generality) gives where q 1+nE > k designates the transversal momentum of the first "virtual" or closed channel n = 1 + n E . As for G, one uses now the identity [45] and the correct branch 0 ≤ Arg k 2 − q 2 < π in order to obtain for the integrated free space Green's function To compare with the Green's function G c in Eq. (14), we specialize Eq.(15) to r, r ′ ≪ R U and impose the condition for low total scattering energy. The value of γ follows from the comparison of the imaginary parts of Eqs. (14) and (15). Changing the integration variable from q to k 2 − q 2 in the first integration on the rhs of Eq.(15) and using the low energy condition in the imaginary parts of Eq. (14) and Eq. (15), γ can be identified in the limit r, r ′ → 0 In other words, since r, r ′ ≪ R U ∼ 1/k , both imaginary parts are approximated by a single and the same constant γk/4π, independent of r and r ′ . Although exact values of |ϕ n (0)| 2 can be employed for a given potential U (ρ), it is more instructive to use general approximations such as |ϕ n (0)| 2 ∼ 1/πR 2 U obtained through normalization. An estimate that takes into account the spatial variation and the few nodes of ϕ n is the square well approximation, namely, |ϕ n (0)| 2 ≈ N 2 n /πR 2 U . Therefore, without loss of generality, one can thus write It should be mentioned that one can improve on how these low energy poles q n of G c are treated. However, it suffices to use for now the above result in order to present the formalism. By comparing the real parts of Eq. (14) and Eq. (15), it is seen that ∆ c in Eq.(12) is given by where k n = k 2 − q 2 n andq ≡ k 2 −p 2 . The first term on the rhs stems from the offset between the lower limits of integration q 1+nE and k in Eq. (14) and Eq.(15), respectively. It accounts for the discreteness of the low lying transversal states due to the confinement. The second term in the square brackets is of order (k 2 /8π − nE n=0 N 2 n /2πR 2 U ) |z − z ′ | and thus is relatively smaller by a factor of |z − z ′ |/R U compared to the first term, whose order of magnitude (q 2 1+nE − k 2 ) 1/2 /4π ∼ 1/R U can be estimated by calculating the integral for r, r ′ ∼ 0 . Neglecting this second term yields where q ≡ (k 2 + p 2 ) 1/2 and p c ≡ (q 2 1+nE − k 2 ) 1/2 . The relation Eq. (12) is then proved and will be the basis of an angular momenta decomposition of the effective amplitudes f ± n in section IV C. In this regard, the weak dependence of ∆ c on the coordinates r and r ′ is kept, since this dependence introduces couplings between orbital angular momenta. This will be seen as ∆ c is decomposed in spherical coordinates in the following.

C. Angular momenta decomposition
We are now in the position to express the effective 1D scattering amplitudes f ± n in terms of the scattering phase-shifts δ l of a given scattering potential. The first step is to decompose these amplitudes in Eq.(5b) in a similar way as was done in the free space case in Eqs. (9) by introducing "matrix" elements similar to τ l . Subsequently these quantities are related to the phase-shifts δ l in a manner analogous to Eqs. (10).
Applying the condition R V ≪ R U , the transversal states ϕ n in the integrand of the amplitudes Eq.(5b) should be well approximated by the Bessel functions Eq.(13a). This is because r ′ ∼ R V whereas |k n | ∼ q n ∼ 1/R U . It is enough then to find expansions of e ±iknz J 0 (q n ρ) in spherical coordinates, with k 2 = q 2 n +k 2 n . For this purpose one may invert the following identity [45] P l (cosθ) j l (kr) = 1 2i l π 0 dw sin w e ikr cosθ cos w × J 0 (kr sinθ sin w)P l (cos w) by using the orthonormality property of the Legendre polynomials. As a result, one obtains for r ≪ R U e iknz ϕ n (ρ) = ∞ l=0 i l (2l + 1) α ln j l (kr) P l (cos θ) , (17) where α ln ≈ N n P l (k n /k)/π 1/2 R U . For more quantitative results, these coefficients α ln can be evaluated numerically from the exact eigenfunctions ϕ n (ρ). One can then express the amplitudes f ± n in Eq.(5b) as the following series in angular momenta for n ≤ n E f ± n = f ng ± f nu (18a) ≡ l even 4π (2l + 1) α ln 2ik n T l ± l odd 4π (2l + 1) α ln 2ik n T l having used the parity property P l (−x) = (−) l P l (x). The "matrix" element T l is defined by The non-zero momenta l ≥ 1 contributions to f ± n stem from the (small) dependence on z ′ and ρ ′ of the integrand in the definition of f ± n . Their neglect amounts to assuming a point like zero range interaction v(r), for which only the s-wave remains. In the free space scattering case, the corresponding quantity equivalent to this T l is τ l given in Eq.(9b). The key difference now is that each T l will depend on all other T l 's, meaning that the confined scattering solution Ψ(r) in the definition of T l couples the angular momenta, while the free space solution Φ(r) in the definition of τ l does not.
The next step is to relate the T l to the phase-shifts δ l . From the results in Eqs. (16) for the approximation Eq.(12), the Green's function G c can be decomposed into spherical coordinates in the region r ′ , r ≪ R U by separately decomposing G and ∆ c . The free space part G can be decomposed following the procedures of section IV A. Then, for r ′ < r, j l (kr ′ ) [γ j l (kr) + i n l (kr)] × 2l + 1 4π P l (cos θ)P l (cos θ ′ ), (19) having used the expansion of e ±ik|r−r ′ | /4π|r − r ′ | in spherical harmonics and γ ± = (1 ± γ)/2. To evaluate ∆ c we use the analytic continuationθ = π/2 − iθ ′ of the first identity of this section to obtain where p = k sinh θ ′ and q = k cosh θ ′ such that k 2 = q 2 − p 2 . The expansion of ∆ c in Eq.(16b) for r, r ′ ≪ R U can then be written as × i l (2l + 1)j l (kr)P l (cos θ) , (20) where σ zz ′ ≡ sign(z − z ′ where α l ≡ nE n=0 b n α ln comes from the expansion of e iknz ϕ n (ρ) in the initial state Ψ i (r) in Eq.(11b) and γ l (z) is defined by The z-dependence of this γ l (z) arises from the sign of σ zz ′ in ∆ c . Because of this z-dependence, the above decomposition of Ψ(r) is not fully spherically symmetric. It reflects the fact that an approximate radial solution on the left side of the guide at some z 1 = −|z 0 | needs not coincide necessarily with one on the right side at z 2 = |z 0 | since the problem has no perfect spherical symmetry. A better understanding of the role of this γ l (z) arises when one tries to write it in terms of the T l 's that determine the amplitudes f ± n in Eq.(18a) and the solution in Eq. (21). Expanding e σ zz ′ pz ′ J 0 (qρ ′ ) (see above) in the integrand in the definition of γ l (z) one obtains γ l (z) = ∞ s=0 (2s + 1) pc 0 dp P l (ip/k)P s (ip/k) assuming the summation and integration can be interchanged (see section V C). If it happened that (σ zz ′ ) l+s = 1, then γ l (z) would be a linear sum of the elements T s and the scattering solution in Eq.(21) would have spherical symmetry. Such a constant γ l (z) could be possible if one could neglect in the series for γ l (z) terms for which the parity of P s is different from that of P l , such that for each l, only terms satisfying l + s = even would be retained. This condition can be related to the fact that the confining potential u(ρ) only couples angular momenta with the same parity, in the sense that l|û|s ≡ dθ ′ P l (cos θ ′ )u(r sin θ ′ )P s (cos θ ′ ) = 0 if l + s = odd. Therefore, a consistent approximation to the scattering solution Ψ(r) is found if one neglects odd parity combinations in the series defining γ l (z) and sets γ l (z) ≈ γ l , such that (2s + 1) P ls T s , l = 0, 1, 2, . . . , (22) where s[l] denotes the sum over even (odd) s for even (odd) l. Using Eq. (22), the scattering phase-shifts δ l can now be introduced, since the expansion in Eq.(21) becomes spherically symmetric. Indeed, this expansion can be conveniently rewritten by the introduction of constants c ′ l and δ l defined, for the moment, by (l = 0, 1, 2, . . . ) such that our solution has the form c ′ l [cos δ l j l (kr) − sin δ l n l (kr)] P l (cos θ). (24) In order to identify the δ l , one notes, on one hand, that this wavefunction Ψ(r) can be directly obtained as a spherically symmetric series l c l R l (r)P l (cos θ) in the region r ≪ R U , where u(ρ) is negligible, if one solves the Schrödinger equation moving outward from the origin r = 0 (see Fig 2). The radial part R l can then be separately determined and for R V ≪ r ≪ R U it should be given by Eq. (24), provided one chooses c l = c ′ l . On the other hand, the solution for the scattering potential v(r) without confinement is given by a series with the same radial parts R l but different constants c l = c ′ l (see Section IV A). The asymptotics of R l in this free space case is then also given by Eq. (24) so that δ l are indeed the phase-shifts of the unconfined scattering problem. The difference between the confined and free space solutions, in the region R V ≪ r ≪ R U , is thus accounted for by the different constants c ′ l and c l , respectively, related to distinct boundary conditions. Besides, δ l depends solely on the solution of R l in the interior region r < R V [46]. The above defining relations for c ′ l and δ l in Eqs. (23) are then equations that determine c ′ l and T l in terms of δ l , namely, Finally, using Eq.(22) one obtains the matrix equation relating T l to the full ensemble of 3D free space scattering phase-shifts δ l (l = 0, 1, 2, . . . ) (2s + 1)P ls T s .
The coupling of angular momenta brought about by the confinement is a result of ∆ c , which accounts for the confining geometry and the discreteness of the low lying transversal states that should be resolved at low energies. The main equations that allow us the analysis of the effective 1D scattering in confined geometries are Eqs.(5), Eq.(18a) and Eq. (25). The probability conservation condition Eq.(5c) serves to gauge the range of validity of the results. Although the above result is enough to present the formalism, improvements to Eq.(25) can be systematically made if necessary, e.g. by better dealing with the poles q n in Eq.(16a) and the constants α l .

V. CONFINEMENT INDUCED EFFECTS
We now use the above formalism to analyze confinement induced phenomena that occur both in the scattering by a central field and in collision processes. Although related to each other, two important phenomena can be distinguished: resonant scattering (sections V B and V D) and weakly localized states induced by the confinement (section V E). In the present paper, our formalism is focused on the single mode regime. It is shown also that previous results of atom-atom collisions under parabolic confinement are recovered, regarding not only the scattering resonances (CIR) [34,35,36] but also the confinement induced weakly bound states [37]. CIR in atomic collisions involving excited transversal states at energies k 2 > q 2 1 are treated in [35] for interactions in the s-wave approximation and in [36] for general V (r) . See also independent related works for 2DEG mesoscopic systems [13,14,15,16].

A. Single mode regime
In this regime, the total scattering energy E = 2 k 2 /2µ allows for only the transversal ground state ϕ 0 (ρ) to be effectively occupied k 2 = q 2 0 + k 2 0 < q 2 1 , i.e., to be an open channel. All excited states can only be virtually populated due to the scattering of the incoming state. Then n E = 0 and b n = δ n,0 . In terms of the even f 0g and odd f 0u angular momenta decomposition of f ± 0 given in Eq.(18a), namely f ± 0 = f 0g ± f 0u , the current conservation can be written as A suitable parametrization of f 0g,u is obtained by introducing 1D scattering parameters δ g and δ u , such that In this form, Eq.(26) is verified if δ g,u are arbitrary real numbers, since then Re{f 0g,u }+|f 0g,u | 2 = 0 vanishes separately. There is by now no a priori reason for the parameters δ g,u to be both real. However, in the case of general potentials v(r) and u(ρ) considered in this work, it will be seen in the next sections that Eqs. (27) with real δ g,u provide a valuable means to establish the boson-fermion and fermion-boson mappings as well as the conservation condition Eq.(26).

B. Boson-fermion and fermion-boson mappings
Consider now a collisional process between identical particles. It is convenient to split Ψ(r) into symmetric ψ g (z)ϕ 0 (ρ) and antisymmetric ψ u (z)ϕ 0 (ρ) parts, with respect to r → −r, ψ g (z) and ψ u (z) can be deduced by using the form Eq.(18a) for the amplitude f ± 0 in terms of its even f 0g and odd f 0u sectors. For the symmetric part, we obtain, where Eq.(27a) has been employed. The antisymmetric part is given by where the plus (minus) sign in ψ u (z) refers to z > 0 (z < 0) and Eq.(27b) has been used.
If the colliding particles are e.g. spin polarized Ψ(r) must be (anti-) symmetrized. For bosons, the correct effective 1D scattered wavefunction in the asymptotic region |z| > R ′ 1D is ψ g (z). The resonance condition can be identified as f 0g = −1 or δ g = |π/2| and it follows from Eq.(28a) that the wavefunction becomes ψ g (z) = −i sin (k 0 |z|), which is with respect to its modulus the wavefunction of a pair of non-interacting fermions, as can be seen by setting f 0u ≡ 0 (or δ u ≡ 0) in Eq.(28b). This is the well-known fermionic mapping of strongly interacting, inpenetrable bosons in 1D [24,25,26,28]. The bosons would not be allowed to be located at the same position z = 0, supposing the asymptotic solution Eq.(28a) could be extended towards the origin |z| ≪ R ′ 1D . For fermions, a reciprocal mapping exists at resonance From Eq.(28b), the wavefunction becomes ψ u (z) = ∓ cos (k 0 z), which is the wavefunction of a pair of noninteracting bosons [36]. Clearly, from the exact, fully antisymmetric wavefunction of a pair of spin polarized fermions, the probability of finding both of them at the origin r = 0 is strictly zero. The bosonization implied by this confinement induced resonance means that the asymptotic behavior of the fermions far from the interaction region is that of free bosons. These mappings between identical colliding particles are valid also under a longitudinal confinement along the z-axis. In order to neglect couplings to the center of mass, this longitudinal confinement can be chosen to be parabolic. The characteristic oscillator length R should then be flat enough, such that R ≫ R ′ 1D . The asymptotics of the symmetric and antisymmetric eigenstates with (positive) energies E g,u = 2 k 2 g,u /2µ are given by Eq.(28a) and Eq.(28b), respectively. The corresponding eigenvalues k 2 g,u = q 2 0 + k 2 0g,u can be estimated by solving the equations ψ g,u (R ) ≈ 0 for k 0g,u , namely, cos (k 0g R + δ g ) = 0 and sin (k 0u R + δ u ) = 0. If the parameter δ g = δ g (k 0g ) is made equal to ±π/2 by varying, e.g., R V , R U , or R , one sees that ψ g (z) is once more mapped onto an antisymmetric eigenstate without interaction v(r) ≡ 0 (or δ u ≡ 0), as already seen above for free longitudinal motion. Analogously, ψ u (z) is mapped onto a non-interacting symmetric eigenstate as δ u = δ u (k 0u ) → ±π/2 . This latter type of bosonic mapping of fermionic eigenstates under longitudinal confinement was first verified numerically in [36] for the collision interaction V (r) = d/ cosh (r/b) 2 . Note that, if one is not restricted to the case of collisions, this result can be formally quite general regarding not only the interaction potential v(r) , but also some sufficiently flat transversal and longitudinal confinements.
The considerations in the preceding section assumed the general form Eqs. (27) for f 0g,u . In this section, f ± 0 is explicitly calculated by solving the matrix equation Eq.(25) for the elements T l . From k ∼ 1/R U , it follows that kR V ∼ R V /R U ≪ 1, which is the condition of low scattering energy in 3D free space scattering and the phase-shifts tan δ l = tan δ l (k) ∼ k 2l+1 ∼ 1/R 2l+1 U are generally small [46] for large R U (or small R V ). One expects then that orbital angular momenta higher than the leading contributions, e.g. the s-and p-waves, should not significantly change the main features arising from the latter. Indeed, the calculation of f 0g and f 0u can be done separately, since the even and odd angular momenta in Eq. (25) are uncoupled from each other. In the single mode regime, Eq.(16a) becomes γ = 2/d 2 U k k 0 and d U ≡ R U /N 0 , where k 0 = (k 2 −q 2 0 ) 1/2 and f ± 0 in Eq.(18a) depends only on the ratio t l ≡ T l /k 0 . The matrix equation Eq. (25) in this case can be rewritten in terms of these t l . Thus for all l , where the diagonal term s[l] = l is excluded from the series on the rhs. Assume now in this equation for t l that this series converges. As l increases, α l = α l0 = P l (k 0 /k)/π 1/2 d U (see Eq. (21) and Eq.(17)) decrases, while P ls (see Eq. (22)) should decrease mildly due to the oscillation of P l (ix) . Then the rhs of this equation for t l should decrease relatively smoothly with l . On the other hand, the phase δ l decreases exponentially δ l ∼ 1/R 2l+1 U with l at low energies, so that cot δ l increases exponentially. Then t l on the lhs should be exponentially small with increasing l . This is consistent with the assumption of convergence of the series on the rhs [see also discussion on γ l (z) after Eq. (21)]. Apart from this, one can use the smallness of the phase-shifts |δ 2 | ≪ |δ 0 | and |δ 3 | ≪ |δ 1 | in order to explicitly diagonalize the finite subspaces l = 0, 2 and l = 1, 3 separately. Then the couplings to l = 2 and l = 3 are seen to have a minor effect to the leading terms l = 0 and l = 1, respectively. Hence we put The even sector f 0g is then given by the leading term, e.g, the s-wave, for which k cot δ 0 ≈ −1/a , where a is the s-wave 3D free space scattering length [46]. From Eq.(18a) one obtains The odd sector f 0u is approximated by e.g. the p-wave.
In this case, the relevant quantity [47] is the scattering volume V p related to δ 1 by k cot The range of validity of these results can be obtained by imposing e.g. the conservation of probability. As discussed in connection to Eq.(26) and Eqs. (27), the 1D scattering phase-shift δ g is real for the result Eq.(30a). Then a sufficient condition for Eq.(26) to hold is since then 3[P 1 (k 0 /k)] 2 ≪ 1 so that f 0u ≈ 0 and the parameter δ u ≈ 0 can be considered also real. In this first region of low longitudinal momenta, the angular momenta couplings in the matrix equation for t l should become negligible due to the factor k 0 that multiplies the series d 2 U s[l] =l (2s + 1)P ls t s containing other angular momenta. The present uncoupled solution to t l should then be a good approximation. The second region is at relatively large momenta, when 3[P 1 (k 0 /k)] 2 ∼ 1, i.e., when since then δ u can also be taken as real. The decoupling between odd angular momenta rests then on the smallness of the phase-shifts δ l . These are, nevertheless, the most interesting regions for the momentum k 0 that are relevant to resonant scattering in symmetric and antisymmetric states, respectively. For more detailed applications these regions can be enlarged by improving on Eq.(25).

D. Confinement induced resonances -CIR
As presented in sections V A and V B in general terms, the CIR is one remarkable effect arising due to the presence of a strong confinement. In the following, it is explicitly expressed in terms of the parameters determining a given scattering potential and a given confinement.

Even angular momenta
The symmetric wavefunction in the single mode regime is ψ g (z)ϕ 0 (ρ) and only the symmetric sector f 0g contributes. It can be used to describe e.g. spin-polarized bosons or fermions in an antisymmetric spin state. In the expression Eq.(30a) for this f 0g , P 00 is given in Eq. (22), thus having used the definition of p c in Eq.(16b) and the single mode condition n E = 0 . At small scattering lengths |a| ≪ R U , one can neglect this factor P 00 in Eq.(30a), since then |a|P 00 ≪ 1 , and This is also true at "high" energies |k 0 | when k → q 1 such that P 00 ≈ 0. One must, however, be careful about the validity of the present approximations in this region of large momenta |k 0 |, as mentioned above in the context of Eq.(31b). The only possibility for resonance f 0g = −1 is then at zero energy k 0 = 0 . At large scattering lengths R V ≪ |a| ∼ R U , this factor P 00 plays an important role in the scattering process at low momenta |k 0 | ≪ 1/R U for which P 00 ≈ q 2 1 − q 2 0 is maximum. Indeed, one obtains then, for |k 0 | ≪ 1/R U , where the constant C ′ depends only on the first closed and the last open channels, namely Recalling that f ± 0 ≈ f 0g , since f 0u ≈ 0 for such small momenta |k 0 | (see Eqs. (30)), this result implies that the scattering can be described by means of an effective 1D interaction potential V 1D (z) of zero range In other words, Eq.(33a) is the solution to f ± 0 for the hypothetical pure 1D scattering problem under this potential V 1D (z) and is valid also for distinguishable particles.
As discussed in section III for atom-atom collisions, this same 1D potential V 1D (z) at low momenta |k 0 | was first shown in [34] to be a direct result of parabolic confinement and of the 3D zero range interaction potential V δ (r) of Eq.(6a). The occurence of this collisional resonance at a ⊥ = Ca for general interactions V (r) was then shown in [36].
For other types of scattering processes, e.g. in mesoscopic 2DEG systems [13,14,15,16], Eq.(33a) shows therefore that at low longitudinal momenta k 0 the existence of a resonance of the type f + 0 ≈ f 0g = −1 is characterized by an infinite effective 1D coupling strength and is indeed not restricted to atom-atom collisions under parabolic confinement or to two-dimensional quantum point-contacts with a single impurity [15]. It is a general effect of single mode scattering at low velocities under quasi-1D confinement.

Odd angular momenta
For odd angular momenta, the procedure is similar to the previous analysis. The antisymmetric part ψ u (u) shows a scattering resonance when the associated amplitude f 0u = −1 . At this resonance, the system can be mapped to a non-interacting symmetric wavefunction as discussed in connection to Eq. (29). The term P 11 is given in Eq. (22), from which the actual condition of CIR in the amplitude f 0u in Eq.(30b) can be worked out for given scattering and confining potentials. Note that due to the factor 3[P 1 (k 0 /k)] 2 in the numerator of Eq.(30b), the odd component f 0u is non-negligible only at relatively large longitudinal momenta |k 0 | ∼ 1/R U , in contrast to the even amplitude f 0g discussed above.
For atom-atom collisions under parabolic confinement, the existence of CIR beyond the s-wave was first predicted by neglecting couplings to orbital angular momenta and isolating a single partial wave l > 0 and its phase-shift δ l as the sole dominant contribution [36]. A direct consequence of contributions from non-zero partial waves l > 0 is, among others, the possibility to define effective p-wave zero-range scattering potentials analogous to V δ (r) and V 1D (z), but which act only on antisymmetric wave functions of a colliding pair [48,49].

E. Weakly bound states
Besides CIR, a second effect of strong cylindrical confinement is the prediction of weakly bound and localized states. For atom-atom collisions this implies the formation of quasi-molecules. As first shown in [37] for zerorange atom-atom interactions under parabolic confinement, these bound states exist even when no bound state occurs in free space. They have recently been observed in optical traps loaded with 40 K atoms [31]. Similar bound states localized around impurities are found in independent studies of mesoscopic systems [13,14,15,16]. In this section, a localized state for general cylindrical confinement is calculated from the even, more precisely zero, angular momentum sector.

s-wave binding energy and wavefunction
For the free space l = 0 angular momentum, the bound state of the (attractive) scattering potential V (r) has an energy E Bf given by a ≫ R V whose relationship to the scattering length a via κ Bf ≈ 1/a holds only when a ≫ R V > 0 [46]. At negative scattering lengths a < 0, one can speak of a "virtual" state close to be incorporated to the spectrum of V (r). Under lateral confinement and along the radial direction of the cylindrical trap, the tail of its wavefunction e −κ Bf r is changed to vanish at a finite distance from the origin, i.e. at the edge r = ρ = R U of the guide when |a| ∼ R U (see Fig. 1). This lateral squeeze lifts E Bf < 0 by an amount ǫ 0 . It can be sufficient for this state to pass the threshold without confinement E = 0 as R U decreases further. This confined localized state with energy E B satisfies Eq.(4a) with k 2 replaced by 2µE B / 2 . This replacement is equivalent to redefining k 0 to be the imaginary number k 0B ≡ ± i(q 2 0 − 2µE B / 2 ) 1/2 . Since the wave e ik0B z should be absent from the solution Ψ and e ik0B |z| should show a bound state like exponential decay, 1/f ± 0 (k 0B ) must vanish at a positive imaginary value of k 0B , such that the interaction part e ik0B |z| outweights e ik0B z . Let then k 0B ≡ ix B /a, so that x B > 0 if a > 0 and x B < 0 if a < 0 . In terms of this x B , E B can be rewritten as From Eqs. (30), a root of 1/f ± 0 arises either from the even sector 1/f 0g or from the odd sector 1/f 0u . In order to recover the l = 0 free space state E Bf , this root should come from the even sector Eq.(30a). Using then the substitution k 2 → 2µE B / 2 = q 2 0 − x 2 B /a 2 in the expression Eq.(32) for P 00 , the equation for x B becomes where |a| arises as 1/a 2 is taken out of the square root in P 00 . Another enlarged form is the quartic equation For each |s|, one should pick up the correct positive (negative) root x B if s > 0 (s < 0), as indicated in Eq.(35). The weak confinement limit is |s| → 0 . A finite root x B ≈ +1 exists only for a > 0 as can be seen from Eq.(36a). This means that E B ≈ E Bf , when R U → ∞ or when a → 0 + as expected. For a < 0 , a trial Taylor expansion x B ≈ x 1 s + x 2 s 2 in Eq.(36b) yields x 1 = 0 and x 2 = −2. A localized state with energy is thus expected also for negative scattering lengths. In this case, the confinement has been able to turn the"virtual" state E Bf of V (r) into a real bound state localized around the scattering center. The strong confinement limit is |s| ≫ 1 (assuming the basic requirement R V ≪ R U is not violated). Setting then x B ≈ x ∞ /s −1 in Eq.(36b) and reexpressing it in terms of s −1 gives for positive and negative a , respectively, x ∞ = ±{[(16 + C ′4 ) 1/2 − C ′2 ]/2} 1/2 . For both positive and negative roots x B , the same fraction of the transversal ground state energy ǫ 0 is reached as |s| → ∞ For parabolic guide d U = a ⊥ [see Eq.(16a) with the exact value |ϕ n (0)| 2 = 1/πa 2 ⊥ ]. Then C ′ = 2 and q 0 a ⊥ = √ 2, both obtained by directly using the exact energies ǫ 0 ≡ 2 q 2 0 /2µ = ω ⊥ and ǫ 1 ≡ 2 q 2 1 /2µ = 3 ω ⊥ in terms of ω ⊥ . Hence E B = (2 − √ 2)ǫ 0 = 0.586ǫ 0 in good quantitative agreement with [31,37]. For square well confinement, it follows from Eq.(13b) and Eq.(33b) that C ′ = (20/3) 1/2 = 2.582 and q 0 d U = (3/2) 1/2 = 1.225, thus E B = 0.631ǫ 0 . These energies are plotted in Fig. 3 as function of the confinement length scale d U .
The spatial shape Ψ B (r) of this confinement induced weakly bound state in the region far from the scattering center is setting aside overall constants and where x B is the proper root of Eq. (36). This asymptotic form is common to both positive and negative scattering lengths a, with a x B > 0.
Closer to the origin, the wavefunction can be calculated by analytic continuation of the intermediate asymptotics in Eq. (21) and Eq. (22). There the function T 0 = T 0 (k 0 ) diverges at k 0 = i x B /a, which is the condition for the pole of f ± 0 . After this analytic continuation, the step towards Eq. (24) can no longer be taken (it was taken in order to calculate T 0 = T 0 (k 0 ) for real k 0 ). In this series Eq (21), this divergence then singles out terms proportional to the constant T 0 (i x B /a) → ∞ . As a result, apart from overall constants, the wavefunction Ψ B (r) has the following form (for R V ≪ r ≪ R U ) where κ 2 B ≡ 2 µE B / 2 and E B is given in Eq. (35). For E B < 0, it can be seen that both imaginary values of κ B yield the same result. In particular, the free space bound state is recovered in the limit a/d U → 0 + . Using E B ≈ E Bf ≈ − 2 /2µa 2 in Eq.(37b), one has Ψ B (r) ∝ − e −r/a r , a/d U → 0 + .
On the opposite limit of strong confinement |a|/d U → ∞, E B tends to a positive fraction λ of ǫ 0 , and the wavefunction becomes a superposition of both terms j 0 ( √ λ q 0 r) and n 0 ( √ λ q 0 r).

Higher angular momenta
Localized states from higher angular momenta contributions and their couplings among each other can in principle be systematically obtained by solving Eq. (25) and analyzing the poles of the full amplitude f ± 0 = f 0g ± f 0u . For instance, a d-wave pole should come from retaining the l = 2 contribution. This and other poles from the odd sector f 0u are treated in detail elsewhere.

VI. DISCUSSION
Here the role of the continuum approximation in section IV B is discussed by comparing with previous models for scattering under confinement. In the interior of the guide r, r ′ ≪ R U where the confining potential is negligible, the Green's function G c (r, r ′ ) under confinement approaches the 3D free space Green's function G(r, r ′ ), as shown in Eq. (12). Therefore, the singularity of G c (r, r ′ ) as r, r ′ → 0 is not only coincident with (see section III), but it is essentially the well-known singularity 1/|r − r ′ | of the 3D scattering scenario without confinement. In addition, one expects that the details of the confinement, whether parabolic or not, should not be important for the physical understanding. In fact, the spectrum of excited transversal states sums up to yield G(r, r ′ ) (integrated over the axial angle φ ′ ), as the continuum limit shows. The only condition is a short-ranged scattering potential R V ≪ R U or a correspondingly "flat" confinement.
The remarkable effects induced by the confinement are in turn directly related to the low lying transversal states as also demonstrated in [15] for a two-dimensional problem. The discreteness of these states is then captured in the factor γ and in the correction term ∆ c (r, r ′ ), which introduce renormalizations and couplings of orbital angular momenta not accounted for by G(r, r ′ ) alone.
For a more quantitative analysis, consider the specific case of atom-atom collisions in the low energy s-wave approximation and under parabolic confinement [34,35,37]. As discussed in section III, the discrete summation over the transversal states can be dealt with in order to extract the singularity of the scattering solution and to determine the value of the constant C = 1.4603 . . . in the effective scattering amplitude f ± 0 . On the other hand, the corresponding value of C in the continuum limit for this case is C ≡ lim s→∞ ( [34]), in agreement with C ′ = 2 calculated in section V E. This value C ′ = 2 arises by using the exact harmonic oscillator energies ǫ 0 = ω ⊥ and ǫ 1 = 3 ω ⊥ . It can be improved on by e.g. numerically computing ∆ c . As already pointed out in section IV B, the exact excited spectrum ǫ n ∼ 2n is substantially different from the "flat" potential approximation given in Eq.(13b). However, the discrete sum over this exact parabolic spectrum can be shown [34,35] to have an equal singular behavior 1/|z| as in free space (see section III). Hence also within the discrete approach, this demonstrates the major role played by the lowest transversal states, whereas the sum over the excited states turns out to be qualitatively confinementindependent.

VII. CONCLUSIONS AND PERSPECTIVES
An analytical treatment of quasi-1D quantum scattering by spherically symmetric and short but finite range potentials in general cylindrical confinement is developed. The full scattering wavefunction is calculated nonperturbatively without partially resuming perturbative series. All phase-shifts of the scattering potential can be readily incorporated. This formalism provides a unified physical picture of the process of confined quasi-1D scattering at low energies and can be applied to impurity scattering in mesoscopic 2DEG systems and two-body collisions. Following the reasoning related to Eq.(12) one expects to be able to treat non-cylindrical geometries and quasi-2D scattering in an analogous way. By computing then a few coefficients and functions such as α ln and ∆ c (or ∆ u ) and only the lowest eigenstates of the confining potential, reasonable numerical results are also possible. Scattering resonances such as total reflexion, as well as weakly localized states can be determined. For the particular case of parabolic confinement, the formalism presented here can also be used to obtain e.g. confinement induced two-body weakly bound states. Of particular interest is the study of the poles of the scattering amplitude which correlate to orbital angular momenta beyond s-waves and how non-parabolic geometries can affect these unconventional pairings due to couplings to the center of mass coordinates.