# Two-body continuum states in non-integer geometry

E.R. Christensen2, E. Garrido1, A.S. Jensen2 1Instituto de Estructura de la
Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain  2Department of Physics
and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark

###### Abstract

Wave functions, phase shifts and corresponding elastic cross sections are
investigated for two short-range interacting particles in a deformed external
oscillator field. For this we use the equivalent $d$-method employing a non-
integer dimension $d$. Using a square-well potential, we derive analytic
expressions for scattering lengths and phase shifts. In particular, we
consider the dimension, $d_{E}$, for infinite scattering length, where the
Efimov effect occurs by addition of a third particle. We give explicitly the
equivalent continuum wave functions in $d$ and ordinary three dimensional (3D)
space, and show that the phase shifts are the same in both methods.
Consequently the $d$-method can be used to obtain low-energy two-body elastic
cross sections in an external field.

###### pacs:

21.45.-v, 21.60.n, ?

##  I Introduction

The experimental possibilities with cold atomic gases allow both huge two-body
interaction variation as well as an overall confining deformed external field
koh06 ; blo08 ; chi10 ; den16 . Few-body physics quickly becomes complicated
or even impossible as the number of particles increase, and other techniques
have to be employed, see for example hov18 . One purpose of studying these
cold systems is the experimental availability, where laboratory simulation and
manipulation are possible. These systems may themselves be of practical use,
but also able to teach us how to control and create similar properties in
systems, which so far are outside of experimental reach. Comparing universal
behavior of systems from different subfields of physics, perhaps also
chemistry and mathematics, is then a tool to exchange knowledge between
science subfields efi70 ; sim76 ; lan77 ; nie01 ; jen04 ; fre12 ; zin13 ;
nai17 ; gar18 .

The advantage of few-body physics is that all relevant degrees-of-freedom can
be accurately treated. Increasing the number of particles from two to three is
known to increase substantially the complications, like the appearance of new
possible structures, but also the interest. The complications are accentuated
by use of overall external fields, where the description requires more than
relative degrees-of-freedom. It is then worth cashing in on a reduction of
degrees-of-freedom, which is very useful, by means of the use of an effective
dimension that changes continuously nie01 ; lev14 ; yam15 ; san18 ; ros18 .

We shall in this report concentrate on the $d$-method, introduced in Ref.
nie01 , and subsequently developed and applied for two and three particles
chr18 ; gar19a ; gar19b ; gar20 ; gar21 . The theoretical formulation assumes
non-integer dimensions, $d$, only relative degrees of freedom, but with a
$d$-dependent angular momentum barrier, and spherical calculations without
angular dependence corresponding to zero angular momentum for $d=3$. The
equivalence to the description with an external deformed oscillator potential
is available, such that the $d$-method results can be translated to ordinary
three-dimensional calculations, and in this way open for experimental
comparison gar21 .

Interesting features revealed and highlighted by this method can be understood
from the continuous change of dimension from $3$ to $2$. The properties of
quantum solutions for simple two-body systems differ enormously between these
two dimensions. For $d=3$, even for relative $s$-waves, a finite attraction is
necessary to bind, whereas for $d=2$ and relative $s$-waves, an infinitely
small attraction produces a bound state bru79 ; lim80 ; nie97 ; vol13 ; nai17
. For three particles the differences are much larger, as emphasized by the
existing Efimov effect for $d=3$, defined by no bound two-body subsystems, but
infinitely many bound three-body systems. In contrast, only a finite number of
bound states exist for $d=2$. Thus varying $d$ between $3$ and $2$ necessarily
implies that some states must change behavior from bound to continuum states
or vice versa. One especially spectacular phenomenon is that the Efimov effect
can be induced between dimensions $3$ and $2$, while absent in both ends of
the interval gar21 .

The $d$-variation amounts to varying an external field, which is a familiar
procedure. This is exemplified from the Feshbach tuning of a magnetic field
allowing huge changes of the effective two-body interaction vog97 ; cou98 .
This is the same principle as for variation of an external electric field as
first suggested in Ref. nie99 . The present use of the $d$-method can then be
viewed as a theoretical application of an external field without the necessary
additional degrees-of-freedom. However, these fields are unavoidable when
comparing with practical experiments.

Continuum or scattering calculations are usually more difficult than bound
state investigations sha18 . It is therefore tempting to extend the $d$-method
to the continuum, and first to learn from the simplest system, that is, a two-
body system. The task is then to perform calculations with the $d$-method, and
subsequently translate or interpret the results to three dimensions with an
external field. We shall study two particles interacting through simple short-
range potentials. The main properties can then be revealed by square-well
potentials, from which we can derive analytic solutions with characteristic
universal behavior. These square-well results may be compared to numerically
found solutions for more realistic Gaussian potentials.

The purpose of the present report is therefore to study continuum properties
of two-body systems with the $d$-method, and interpret as if obtained in an
external field with the possible experimental comparison. Special attention
will be paid to the dimension, $d=d_{E}$, for which the two-body binding
energy is equal to zero. This dimension is particularly interesting, since for
$d=d_{E}$ the Efimov conditions are fulfilled if a third particle is added.

After the introduction in Section I, we derive in Section II several
analytical expressions of interest assuming a square-well interaction. In the
different subsections we provide the energy of bound states as a function of
$d$ and of the strength of the potential, the critical dimension $d_{E}$ as a
function of the strength of the potential, and the phase shifts also as a
function of $d$ and of the strength of the potential. Finally, we also discuss
in this section how these phase shifts have to be treated in order to
construct the cross sections. The results and illustrations are provided in
Section III, where together with the direct applications of the expressions
derived in Section II, we focus on the universality of the results,
investigating how the values of $d_{E}$ and the phase shifts can be shown to
present a universal behavior, independent of the details of the potential. We
finish wit Section IV, which contains the summary and conclusions. Pertinent
properties of the Bessel functions are collected in an appendix.

##  II Two-body scattering

We consider the two-body problem in $d$ dimensions, where $2\leq d\leq 3$, and
where the two-body interaction is assumed to be of short range. Using a
schematic short-range square-well potential, we shall derive expressions for
bound- and continuum-state energies, threshold conditions, phase shifts,
scattering lengths, and cross sections, all as a function of $d$. In the end,
these results can be interpreted as obtained by an external deformed field and
full three-dimensional space.

The derived analytic properties, properly expressed, exhibit the highly
desired universal behavior. This universality means potential independence at
distances larger than the radius of the short-range interaction. In other
words, gross properties are sufficient to describe the results for relatively
small energies. Numerical calculations using a Gaussian potential will be used
as an illustration.

###  II.1 Theoretical formulation

For two-body systems and dimension $d$, the reduced, $R(r)$, and the total,
$\psi(r)$, radial relative wave functions are related by
$R(r)=r^{(d-1)/2}\psi(r)$, where $r$ is the relative distance coordinate. The
reduced wave function, $R(r)$, is obtained as a solution, with the proper
boundary conditions, of the $d$-dimensional radial Schrödinger equation for
the relative motion nie01 :

$\displaystyle\left[-\frac{\hbar^{2}}{2\mu}\bigg{(}\frac{\partial^{2}}{\partial
r^{2}}-\frac{\ell^{*}(\ell^{*}+1)}{r^{2}}\bigg{)}+V(r)-E\right]R(r)=0\;,$ (1)

where $V(r)$ is the spherical potential and $\mu$ the reduced mass. For two-
body systems the effective angular momentum $\ell^{*}$ takes the form

$\ell^{*}=\ell_{2b}+\frac{d-3}{2},$ (2)

where $\ell_{2b}$ is the relative orbital angular momentum between the two
particles.

All along the squeezing process, the momentum $\ell_{2b}$ is a good quantum
number, which for $d=3$ becomes the usual orbital angular momentum. Also, when
confining from three to two dimensions by means of an external squeezing
potential, it is known that the angular momentum projection quantum number,
$m$, is conserved gar19b , and, furthermore, $m$ is the angular momentum in
the two-dimension limit nie01 . Therefore, for $d=2$ we have $m=\ell_{2b}$,
which, due to the conservation of $m$, implies that the same equality holds
for the initial non-squeezed 3D-state as well.

Le us assume $V(r)$ is a spherical square-well potential, that is

$\displaystyle V(r)=\begin{cases}-V_{0},&\ r<r_{0}\\\ \ 0&\
r>r_{0}\end{cases},$ (3)

where $V_{0}$ ($>0$) is the depth and $r_{0}$ the radius.

After introducing the dimensionless variable $x=r/r_{0}$, the equation
_inside_ the box, $r<r_{0}\;(x<1)$, then reads

$\displaystyle\left[-\frac{\partial^{2}}{\partial
x^{2}}+\frac{\ell^{*}(\ell^{*}+1)}{x^{2}}-k^{2}\right]R_{in}(x)=0,$ (4)

and similarly outside the box, $r>r_{0}\;(x>1)$,

$\displaystyle\left[-\frac{\partial^{2}}{\partial
x^{2}}+\frac{\ell^{*}(\ell^{*}+1)}{x^{2}}-\kappa^{2}\right]R_{out}(x)=0,$ (5)

where the wave numbers are given by

$\displaystyle k=\sqrt{\frac{2\mu
r_{0}^{2}(E+V_{0})}{\hbar^{2}}}\;,\;\kappa=\sqrt{\frac{2\mu
r_{0}^{2}E}{\hbar^{2}}}.$ (6)

Note that $\kappa$ is imaginary for bound states, but real and positive for
continuum states.

The solution inside the box, $R_{in}$, is simply given by

$\displaystyle R_{in}(x)=A_{k}xj_{\ell^{*}}(kx)\;,$ (7)

where $j_{\ell^{*}}$ is the spherical Bessel function of first kind and
$A_{k}$ is a normalization constant. The other Bessel function,
$\eta_{\ell^{*}}$, also a solution to Eq.(4), diverges at $x=0$ and is
therefore removed from Eq.(7), since the physical solution must satisfy that
$R_{in}(x)\rightarrow 0$ for $x\rightarrow 0$. Note that $\ell^{*}$ is in
general non-integer, and even negative for $s$-waves ($\ell_{2b}=0$) when
$d<3$ ($\ell^{*}=(d-3)/2$).

###  II.2 Bound states

For a bound state ($E<0$ and $\kappa=i|\kappa|$), the solution outside the
box, $R_{out}(x)$, takes the form

$\displaystyle R_{out}(x)=B_{\kappa}xh^{(+)}_{\ell^{*}}(\kappa x)\;,$ (8)

where $h^{(+)}_{\ell^{*}}$ is the spherical Hankel function of first kind and
$B_{\kappa}$ is a normalization constant. The other Hankel function solution
to Eq.(5), $h^{(-)}_{\ell^{*}}(\kappa x)$, is removed, since the bound state
solution must fall off exponentially at large distance.

Matching the logarithmic derivatives of Eqs.(7) and (8) at the box radius,
$x=1$, we obtain the eigenvalue equation for bound states. The normalization
factors disappear and we find, by use of the two identities Eqs.(36) and (37),
that

$\displaystyle
1+\ell^{*}-k\frac{j_{\ell^{*}+1}(k)}{j_{\ell^{*}}(k)}=1+\ell^{*}-\kappa\frac{h^{(+)}_{\ell^{*}+1}(\kappa)}{h^{(+)}_{\ell^{*}}(\kappa)}\;,$
(9)

where the left and right hand sides in Eq.(9) are the logarithmic derivatives
of the solutions inside and outside the box, respectively.

Eq.(9) is an equation relating the square-well parameter, $S_{0}$, and
$\kappa$ in Eq.(6) through

$\displaystyle S_{0}^{2}\equiv\frac{2\mu
r_{0}^{2}V_{0}}{\hbar^{2}}\;,\;k^{2}=S_{0}^{2}+\kappa^{2}\;.$ (10)

In other words, the bound state energy, $E$, in units of $\hbar^{2}/(\mu
r_{0}^{2})$, is a unique function of the square-well parameter, $S_{0}$. The
oscillatory behavior of the Bessel functions provides more and more discrete
energy solutions with increasing $S_{0}$. All this is in complete analogy to
the usual square-well relative two-body problem, but now depending on
$\ell^{*}$, and therefore on $d$.

###  II.3 Efimov condition

Let us consider now a moderate square-well parameter, $S_{0}$, which is too
small to support any bound state for $d=3$. However, if the system is bound in
two dimensions, we then know that, when decreasing $d$ from $3$ to $2$, a
bound state of zero energy must appear at some value, $d=d_{E}$. This always
happens for $s$-waves ($\ell_{2b}=0$), since for $d=2$ a bound state is always
present, even for an infinitesimal attraction. We label this dimension,
$d_{E}$, because if an additional particle is added, the resulting three-body
system would exhibit Efimov properties, if at least two of the two-body
subsystems, in a relative $s$-wave, are bound by precisely zero energy. In
particular, if dealing with identical particles, this critical value, $d_{E}$,
is therefore the non-integer dimension where the Efimov effect occurs on the
path towards two dimensions gar20 .

To find an expression for $d_{E}$, we take the limit $E\rightarrow 0^{-}$ in
the eigenvalue equation, Eq.(9). This means that $\kappa\rightarrow 0$, and
$k\rightarrow\sqrt{2\mu r_{0}^{2}V_{0}/\hbar^{2}}=S_{0}$. Using the low-energy
expansion of the Bessel-functions, Eq.(38), we easily get:

$\displaystyle\kappa\frac{h^{(+)}_{\ell^{*}+1}(\kappa)}{h^{(+)}_{\ell^{*}}(\kappa)}\rightarrow
2\ell^{*}+1\;.$ (11)

The eigenvalue equation, Eq.(9), reduces then to

$\displaystyle
S_{0}\frac{j_{\ell^{*}+1}(S_{0})}{j_{\ell^{*}}(S_{0})}=2\ell^{*}+1=2\ell_{2b}+d_{E}-2\;,$
(12)

which is the condition for a zero-energy solution, and therefore gives
$d=d_{E}$ as a function of the square-well parameter, $S_{0}$. We emphasize
that the index, $\ell^{*}$, on the Bessel function depends on $d$ as expressed
in Eq.(2).

It is important to note here that the philosophy of studying scattering
properties through an extension of the angular momentum to non-integer values,
$\ell^{*}$, is akin to the concept of Regge poles, where the angular momentum
is generalized to a continuum, and even complex, variable, and the analytical
behavior of the $S$-matrix is subsequently studied. In fact, Eq.(12) can be
simplified into:

$J_{\ell^{*}-\frac{1}{2}}(S_{0})=0,$ (13)

which, as shown in bol62 , determines the zero-energy position of the Regge
poles for the square-well potential. Therefore, the critical dimension,
$d_{E}$, for a given $S_{0}$ value corresponds to the zero-energy Regge pole
of the potential.

###  II.4 Scattering length

The zero-energy condition is closely related to the scattering length, which
also must depend on the dimension $d$. We derive an expression by first noting
that the zero-energy reduced radial wave function must by definition have the
form nie01

$\displaystyle R_{out}(x)\propto
x^{\ell^{*}+1}-\frac{(a_{d}^{(\ell^{*})}/r_{0})^{2\ell^{*}+1}}{x^{\ell^{*}}},$
(14)

expressed in terms of the scattering length, $a_{d}^{(\ell^{*})}$ divided by
the length unit, $r_{0}$. We used that the reduced, $R$, and total, $\psi$,
radial wave functions are related by $R(x)=x^{(d-1)/2}\psi(x)$.

Matching the logarithmic derivatives of Eqs.(7) and (14) at $x=1$, we get

$\displaystyle
1+\ell^{*}-S_{0}\frac{j_{\ell^{*}+1}(S_{0})}{j_{\ell^{*}}(S_{0})}=\frac{1+\ell^{*}+\ell^{*}\big{(}\frac{a_{d}^{(\ell^{*})}}{r_{0}}\big{)}^{2\ell^{*}+1}}{1-\big{(}\frac{a_{d}^{(\ell^{*})}}{r_{0}}\big{)}^{2\ell^{*}+1}}\;,$
(15)

which immediately leads to

$\displaystyle\frac{a_{d}^{(\ell^{*})}}{r_{0}}=\Big{(}{1-\frac{(2\ell^{*}+1)j_{\ell^{*}}(S_{0})}{S_{0}j_{\ell^{*}+1}(S_{0})}}\Big{)}^{-1/(2\ell^{*}+1)}\;.$
(16)

The scattering length depends on $S_{0}$ and $\ell^{*}$, or $S_{0}$, $d$ and
$\ell_{2b}$, in units of the square-well radius.

We see from Eq.(12) that this scattering length, $a_{d}^{(\ell^{*})}$, is
infinitely large exactly when $d=d_{E}$. Another consistency check is for
$d=3$ and $\ell_{2b}=0$ ($\ell^{*}=0$), where Eq.(16) reduces to the well-
known expression:

$\displaystyle\frac{a_{d=3}^{(\ell^{*}=0)}}{r_{0}}=1-\frac{\tan(S_{0})}{S_{0}}\;.$
(17)

###  II.5 Phase-shifts

Let us consider now the case of continuum states ($E>0$). The large-distance
solution in Eq.(8) should be replaced by

$\displaystyle R_{out}(x)=x(\cos\delta_{\ell^{*}}j_{\ell^{*}}(\kappa
x)-\sin\delta_{\ell^{*}}\eta_{\ell^{*}}(\kappa x))\;,$ (18)

where $\eta_{\ell^{*}}$ is the irregular Bessel function, and
$\delta_{\ell^{*}}$ is the phase-shift corresponding to this continuum state
of generalized angular momentum $\ell^{*}$ and energy $E$.

The solution of $\delta_{\ell^{*}}$ as function of $E$ and $\ell^{*}$ is found
by matching the logarithmic derivatives of Eqs.(7) and (18), that is

$\displaystyle\frac{kj^{\prime}_{\ell^{*}}(k)}{\kappa
j_{\ell^{*}}(k)}=\frac{\cot\delta_{\ell^{*}}j^{\prime}_{\ell^{*}}(\kappa)-\eta^{\prime}_{\ell^{*}}(\kappa)}{\cot\delta_{\ell^{*}}j_{\ell^{*}}(\kappa)-\eta_{\ell^{*}}(\kappa)}\;,$
(19)

where the prime denotes derivative with respect to the full argument of the
function. The phase-shift can then be calculated from Eq.(19) to give

$\displaystyle\cot\delta_{\ell^{*}}=\frac{\eta_{\ell^{*}}(\kappa)}{j_{\ell^{*}}(\kappa)}\frac{\kappa\frac{\eta_{\ell^{*}+1}(\kappa)}{\eta_{\ell^{*}}(\kappa)}-k\frac{j_{\ell^{*}+1}(k)}{j_{\ell^{*}}(k)}}{\kappa\frac{j_{\ell^{*}+1}(\kappa)}{j_{\ell^{*}}(\kappa)}-k\frac{j_{\ell^{*}+1}(k)}{j_{\ell^{*}}(k)}}\;,$
(20)

where we used the derivative expressions for both $j_{\ell^{*}}$ and
$\eta_{\ell^{*}}$ from Eqs.(36) and (37).

The low-energy limits, $\kappa\rightarrow 0$, $k\rightarrow S_{0}$, are then
to leading order obtained from Eq.(38),

$\displaystyle\cot\delta_{\ell^{*}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{-1}{(2\ell^{*}+1)\kappa^{2\ell^{*}+1}}$ (21)
$\displaystyle\times\bigg{(}\frac{\Gamma(2\ell^{*}+2)}{2^{\ell^{*}}\Gamma(\ell^{*}+1)}\bigg{)}^{2}$
$\displaystyle\times$
$\displaystyle\bigg{(}1-\frac{(2\ell^{*}+1)j_{\ell^{*}}(S_{0})}{S_{0}j_{\ell^{*}+1}(S_{0})}\bigg{)}\;,$

where $\Gamma$ is the usual Gamma function.

Note that for $d=d_{E}$, making use of Eq.(12) in the last bracket of Eq.(21),
we get that $\cot\delta_{\ell^{*}}=0$, or, in other words,
$\delta_{\ell^{*}}=\pi/2$.

Assuming that the low-energy limit of the phase-shift is proportional to
$\kappa^{2\ell^{*}+1}$, we can express this limit in terms of the scattering
length in Eq.(16), that is

$\displaystyle\delta_{\ell^{*}}\rightarrow-(2\ell^{*}+1)\frac{2^{2\ell^{*}}\Gamma^{2}(\ell^{*}+1)}{\Gamma^{2}(2\ell^{*}+2)}\bigg{(}2\mu
E[a_{d}^{(\ell^{*})}]^{2}/\hbar^{2}\bigg{)}^{\frac{2\ell^{*}+1}{2}}\;.$ (22)

###  II.6 Cross sections

As mentioned several times, the $d$-method has been introduced as a tool that
simplifies the description of systems squeezed due to the presence of an
external field. It is obvious that any observable will be measured in the
squeezed 3D space, and therefore, in order to compare the calculations with
whatever available experimental data, it is necessary to translate the
observable from the $d$-dimensional space into the 3D space.

This is in particular necessary when computing cross sections. It is not
obvious that the phase shifts derived in the previous subsection can be
directly employed to get the cross sections simply by using them in the usual
3D cross section expressions in terms of the phase shifts.

To understand how to proceed, we first interpret the $d$-dimensional wave
function as described in gar19a , that is, as a wave function in the ordinary
3D space where the relative coordinate in the $d$-space, $\bm{r}$, is now
understood as a 3D vector, denoted as $\tilde{\bm{r}}$, but where the third
coordinate (the $z$-axis is chosen along the squeezing direction) is squeezed
by means of a scale parameter $s$. In other words, the $d$-coordinate,
$\bm{r}$, is then understood as a 3D-coordinate as:

$\bm{r}\rightarrow\tilde{\bm{r}}=(x,y,\tilde{z})=(x,y,\frac{z}{s}),$ (23)

where $x$, $y$, and $z$ are the usual Cartesian coordinates in 3D of the
actual relative coordinate, $r$, in the 3D-squeezed space. In this way:

$\tilde{r}^{2}=x^{2}+y^{2}+\frac{z^{2}}{s^{2}}=r_{\perp}^{2}+\frac{z^{2}}{s^{2}}=r^{2}\left(\sin^{2}\theta+\frac{\cos^{2}\theta}{s^{2}}\right),$
(24)

where $\theta$ is the usual polar angle.

In the same way, the direction of the 3D-coordinate, $\tilde{\bm{r}}$, is
given by the polar and azimuthal angles
$\tilde{\theta}=\arctan(r_{\perp}/\tilde{z})$ and
$\tilde{\varphi}=\arctan(y/x)$. They can be easily related to the polar and
azimuthal angles of the actual relative coordinate,
$\theta=\arctan(r_{\perp}/z)$ and $\varphi=\arctan(y/x)$, leading to:

$\tan\theta=\frac{1}{s}\tan{\tilde{\theta}},\hskip
14.22636pt\varphi=\tilde{\varphi},$ (25)

which reflects the fact that the squeezing is taken along the $z$-axis, and in
the case of large squeezing (very small $s$) only values of $\theta$ very
close to $\theta=\pi/2$ are allowed, as it corresponds to a system moving in
the $xy$-plane.

To determine the value of the scale parameter $s$ we use the approximate
expression derived in gar20 :

$\frac{1}{s^{2}}=\left[1+\left(\frac{(3-d)(d-1)}{2(d-2)}\right)^{2}\right]^{1/2},$
(26)

which is based on the assumption of harmonic oscillator particle-particle
interactions. For identical particles, only the harmonic frequency of the
interaction enters and it can be adjusted to give the same $d=2$ binding
energy as the general short-range interaction. Numerical tests show that this
is a good approximation gar20 .

However, for continuum states this interpretation of the radial coordinates is
still not enough. Even for very small values of the scale parameter $s$ (which
imply a large squeezing along the $z$-direction) the asymptotic wave function
(18) is still oscillating, and not vanishing, for sufficiently large values of
$z$. Therefore, it is necessary to introduce an additional factor imposing the
confinement along the squeezing direction. In particular, assuming a harmonic
oscillator squeezing, we write the asymptotic form as:

$R(\tilde{r})\stackrel{{\scriptstyle
r\rightarrow\infty}}{{\longrightarrow}}\left[\tilde{r}\left(\cos\delta_{\ell^{*}}j_{\ell^{*}}(k\tilde{r})-\sin\delta_{\ell^{*}}\eta_{\ell^{*}}(k\tilde{r})\right)\right]e^{-\frac{z^{2}}{2b_{ho}^{2}}},$
(27)

where $z=r\cos\theta$ and $b_{ho}$ is the harmonic oscillator length
associated to the squeezing potential. According to our estimates gar20 , the
connection between $d$ and $b_{ho}$ is given by:

$b_{ho}=\sqrt{\frac{2(d-2)}{(d-1)(3-d)}},$ (28)

which is given in units of the range of the particle-particle interaction.

The idea now is to interpret the $d$-wave function as an ordinary wave
function in three dimensions whose radial part depends on $\tilde{r}$, and and
whose angular part is an ordinary spherical harmonic depending on
$\tilde{\theta}$ and $\tilde{\varphi}=\varphi$. Remembering that in 3D, as
discussed below Eq.(2), we must have that $m=\ell_{2b}$ we can then write:

$\frac{R(\tilde{r})}{\tilde{r}^{\frac{d-1}{2}}}Y_{\ell_{2b}\ell_{2b}}(\tilde{\theta},\varphi)=\sum_{\ell
m}\frac{u_{\ell}(r)}{r}Y_{\ell m}(\theta,\varphi),$ (29)

such that we expand the wave function in terms of the spherical harmonics
expressed as function of the polar and azimuthal angles of the 3D relative
coordinate.

From the equation above we can extract the projected radial wave functions as:

$\frac{u_{\ell}(r)}{r}=\int
d\Omega\frac{R(\tilde{r})}{\tilde{r}^{\frac{d-1}{2}}}Y_{\ell_{2b}\ell_{2b}}(\tilde{\theta},\varphi)Y_{\ell
m}^{*}(\theta,\varphi),$ (30)

which, after integrating over $\varphi$ (leading to $\delta_{m,\ell_{2b}}$)
and getting rid of the constants factors, becomes:

$u_{\ell}(r)\propto
r\int_{0}^{\pi}d\theta\sin\theta\frac{R(\tilde{r})}{\tilde{r}^{\frac{d-1}{2}}}P_{\ell_{2b}}^{(\ell_{2b})}(\cos\tilde{\theta})P_{\ell}^{(\ell_{2b})}(\cos\theta)$
(31)

whose asymptotic behavior can be computed by introducing Eq.(27). For a given
$r$, the integrand in the equation above is just a function of $\theta$
through Eqs.(24) and (25).

By fitting the $u_{\ell}$ functions to the expected asymptotic behavior,

$u_{\ell}(r)\stackrel{{\scriptstyle
r\rightarrow\infty}}{{\longrightarrow}}r\left(\cos\delta_{\ell}j_{\ell}(kr)-\sin\delta_{\ell}\eta_{\ell}(kr)\right)$
(32)

we can extract the phase shift for each partial wave $\ell$.

However, it is important to keep in mind that Eq.(32) gives the asymptotic
behavior of the continuum wave functions in the ordinary, non-squeezed, 3D
space. The effect of the confining external potential is not present in
Eq.(32). In fact, in the case of no interaction between the particles, the
$d$-phase shift, $\delta_{\ell^{*}}$, obtained from the asymptotic behavior
(18) is trivially equal to zero, and the radial wave function is simply given
by $xj_{\ell^{*}}(kx)$. When introducing this function into Eq.(31), and
making use of Eq.(32), we can easily observe that the corresponding computed
3D phase shift is not zero. This non-zero phase shift, which we denote as
$\delta_{\mbox{\scriptsize free}}$, reflects the effect of the confining
external potential.

By definition, the phase shift is just the shift of the asymptotic wave
function when compared to the free wave function. It is then clear that the
phase shift in the confined 3D space, $\delta_{\mbox{\scriptsize
conf}}^{(\ell)}$, is then given by:

$\delta_{\mbox{\scriptsize
conf}}^{(\ell)}=\delta_{\ell}-\delta_{\mbox{\scriptsize free}},$ (33)

where $\delta_{\ell}$ and $\delta_{\mbox{\scriptsize free}}$ are then the
phase shifts obtained from Eqs.(31) and (32) for interacting and free
particles, respectively.

We shall later show that $\delta_{\mbox{\scriptsize conf}}^{(\ell)}$ is
actually independent of $\ell$, which permits to associate a single phase
shift $\delta_{\mbox{\scriptsize conf}}\equiv\delta_{\mbox{\scriptsize
conf}}^{(\ell)}$, to the two-body continuum state in the relative
$\ell_{2b}$-wave in the squeezed 3D space. Furthermore, we shall also show the
nice result that $\delta_{\ell^{*}}=\delta_{\mbox{\scriptsize conf}}$, which
in fact means that the phase shifts obtained with the $d$-method are actually
the phase shifts in the squeezed 3D space, and therefore they can be directly
used to compute the cross section in the 3D space, which is given by the
simple well-known formula:

$\sigma=\frac{4\pi}{k^{2}}(2\ell_{2b}+1)\sin^{2}\delta_{\ell^{*}}.$ (34)

##  III Results

In this section we show the results for the key quantities analyzed in the
previous section for two-body continuum states, i.e., critical dimension,
scattering length, and phase-shifts. The derivations have been carried out
assuming a square-well potential, and therefore the derived formulae, which of
course depend on the dimension $d$, will depend as well on the square-well
parameter, $S_{0}$. This dependence has been obtained in a very specific
model, and provides results that, in principle, are not universal. However, as
we shall show, it is possible to extract a universal behavior, such that the
square-well results previously derived can be exported to any short-range
potential.

###  III.1 Critical dimension

Figure 1: The critical dimension, $d_{E}$, for $s$-waves, as a function of the
potential parameter, $S_{0}$, defined in Eq.(10).

The critical dimension, $d_{E}$, is obtained by solving Eq.(12) as function of
$S_{0}$. This dimension always exists for relative $s$-waves ($\ell_{2b}=0$),
which is particularly relevant, since this is the dimension for which the
Efimov effect arises in the three-body case.

The resulting dependence, shown in Fig. 1 for $\ell_{2b}=0$, is a smoothly
increasing function of $S_{0}$, changing from $d_{E}=2$ at $S_{0}=0$, to
$d_{E}=3$ for $S_{0}=\pi/2$. This reflects the fact that an infinitely small
attractive potential is, for relative $s$-waves, sufficient to bind for $d=2$,
but a finite potential is necessary to support bound states as the dimension
approaches $d=3$. The value of $S_{0}=\pi/2$ for $d=3$ is the well-known
critical size for binding in a square-well. Note also that, as shown in
Eq.(13), for $d=3$ the values of the critical dimension, $d_{E}$, appear for
the $S_{0}$ values matching with the zeros of $J_{-1/2}(S_{0})$, which are
given by $S_{0}=(2n+1)\pi/2$, with $n=0,1,2,\cdots$.

Increasing the attraction of the potential still smoothly increases $d_{E}$
above $3$, as shown in the inset of Fig. 1. This behavior is abruptly broken
by a sudden downwards jump at $S_{0}=3.83171$, where a second bound state
appears at zero energy. The following increase is the beginning of repeating
this behavior reflecting the discrete increase of bound states. In fact,
again, we can from Eq.(13) see that for $d=2$ the $d_{E}$ values for which new
bound states appear are determined by the zeros of
$J_{-1}(S_{0})=-J_{1}(S_{0})$, whose first three values are
$S_{0}=3.83171,7.01559,10.17347$, and which give the position of the jumps in
the curve shown in the inset of Fig. 1.

The square-well parameter is by definition not a universal quantity that could
be used for other short-range potentials. Instead, in order to pursue such a
universality, we turn to the scattering length for $d=3$, which is a unique
characteristic for any short-range potential, as well as definable,
measurable, and reflecting a gross property independent of potential details.
It determines rather accurately all low-energy scattering properties, and as
such is a universal quantity.

Although in principle possible as well for relative angular momenta larger
than zero, universality features are specially relevant for $s$-waves, for
which the wave function can more easily extend beyond the range of the
interaction, becoming then less sensitive to the details of the potential. As
shown in Eq.(22), for small energy the phase shifts behave as
$E^{(2\ell_{2b}+d-2)/2}$, meaning that the $s$-wave scattering clearly
dominates in the low-energy limit, and for this reason we shall focus here on
universal behavior in the $\ell_{2b}=0$ case (an exception could be the case,
not considered here, of two identical fermions, for which a relative $s$-wave
is not possible bra06 ).

Figure 2: The critical dimension, $d_{E}$, as a function of the scattering
length, $a^{(\ell_{2b}=0)}_{d=3}/r_{\mbox{\scriptsize eff}}$, for the square-
well (blue thick-solid) and the Gaussian potentials (green thick-dashed). The
dimension $d_{E}$ corresponds to the appearance of the first bound state. The
green thin-dashed curve shows the $d_{E}$-values corresponding to the
appearance of the second bound state with the Gaussian potential.

In Fig. 2 we show the critical dimension, $d_{E}$, as a function of
$a^{(\ell_{2b}=0)}_{d=3}$ for both square-well and Gaussian potentials.
Different length units have been considered, and we have found that clearly
the length scale with the effective range, $r_{\mbox{\scriptsize eff}}$,
provides the most satisfactory universal behavior. This is in accordance with
the findings for three-body systems nai14 . The results for the two potentials
are pretty close to each other, and we therefore have, to a very large extent,
the desired, accurate universal function, $d_{E}$, as function of scattering
length in units of the effective radius, both obtained in the ordinary three
dimensions. This is what shown by the two thick curves in the figure, solid
(blue) and dashed (green), which have been obtained with potential strengths
such that $d_{E}$ corresponds to the appearance of the first bound state,
i.e., to potential strengths unable to bind the system in three dimensions.

In principle one could still increase $S_{0}$ such that a second bound state
appears (as indicated by the jump in the curve shown in the inset of Fig. 1).
As exemplified in previous works, the depth of the two-body potential could be
an important factor for universality nai14 ; mes19 . Although this interesting
relationship goes beyond the scope of this work, we show in Fig. 2, as an
indication, the curve (thin-dashed) obtained for the critical dimension
$d_{E}$ corresponding to the appearance of the second bound state with the
Gaussian potential. As we can see, this curve does not follow the universal
curve obtained when $d_{E}$ indicates the appearance of the first bound state.

The scattering length and effective range ratio for $\ell_{2b}=0$ and $d=3$ is
therefore adopted as the easy accessible universal quantity. This has the
advantage of requiring only one adjustment, the
$a_{d=3}^{(\ell_{2b}=0)}/r_{\mbox{\scriptsize eff}}$ ratio, when two
potentials should be compared. Another possibility could perhaps be to use the
same $a^{(\ell^{*})}_{d}$ for different potentials as defined in Eq.(16), or
may be the same ratio between $a^{(\ell^{*})}_{d}$ and the corresponding
$d$-dependent effective range. However, this would require calculation with
different potentials for any $d$. The simplicity of the method would be lost,
and we therefore we maintain the simple procedure.

###  III.2 Scattering length

In any case, even if not used in the universal curve shown in Fig. 2, a given
two-body potential, has associated a different scattering length for different
values of the dimension $d$, as shown in Eq.(16) for the case of the square-
well potential.

Figure 3: (a) Scattering length, $a^{(\ell_{2b}=0)}_{d}$, as function of $d$
for $S_{0}\simeq 4/3$, such that $d_{E}=2.75$. (b) Scattering length as
function of $S_{0}$ for $d=2.75$.

This $d$-dependent scattering length is a function of $S_{0}$, as shown in
Fig. 3 for the square-well potential and $\ell_{2b}=0$. The left and right
parts show the dependence on $d$ for a given $S_{0}$ ($S_{0}\simeq 4/3$ such
that $d_{E}=2.75$), and on $S_{0}$ for a given $d$ ($d=2.75$), respectively.
If desired $S_{0}$ can, through Fig. 1, be substituted by
$a^{(\ell_{2b}=0)}_{d=3}$. The qualitative features are precisely the same.

The most pronounced feature is that the scattering length diverges at the
critical dimension, $d_{E}$, see Fig. 3a. For $S_{0}\simeq 4/3$ we can see
from Fig. 1 that $d_{E}\approx 2.75$, which is the $d$-value where the
divergence of the scattering length shows up. This demonstrates the existence
of a zero-energy bound state for this dimension. The picture would be very
similar for other values of $S_{0}$, where the zero-energy would appear at
another $d$. For illustration, we show in Fig. 3b the other dependence of the
scattering length as a function of $S_{0}$ for given $d$ ($d=2.75$). Again we
observe a divergence of $a^{(\ell_{2b}=0)}_{d}$ for a specific $S_{0}$,
$S_{0}\approx 4/3$, which agrees with the value observed in Fig. 1 for
$d_{E}=2.75$, and defines the parameter set producing a zero-energy bound
state.

###  III.3 Phase-shifts

Figure 4: Phase shifts, $\delta_{\ell^{*}}$, for $\ell_{2b}=0$ as function of
$d$ for different values of $\kappa/S_{0}$. The thick-solid-black curves are
square-well results from Eq.(22), the thick-dashed-red curves are the square-
well low-energy approximation from Eq.(21), the thin-solid-blue curves are for
a Gaussian interaction comparing results for equal
$a_{d=3}^{(\ell_{2b}=0)}/r_{0}$ (scattering length divided by the Gaussian or
square-well range) in 3D, and the thin-dashed-green curves are for a Gaussian
interaction comparing results for equal
$a_{d=3}^{(\ell_{2b}=0)}/r_{\mbox{\scriptsize eff}}$ (scattering length
divided by effective range) in 3D. The potential parameters produce
$d_{E}=2.50$ for the square-well interaction.

The phase-shifts contain the crucial information about the continuum states.
They provide the information about the asymptotic behavior of the wave
function, and they are the key ingredient to obtain an important observable
like the cross sections.

In Fig. 4 we show the dependence of the $s$-wave phase shifts on the dimension
$d$ and on the energy (through $\kappa$, Eq.(6)). Each sub-figure compares
four computed phase-shifts as function of the $d$ for given values of
$\kappa/S_{0}$.

The first two curves on each panel, thick-solid-black and thick-dashed-red,
are the analytic square-well results in Eq.(20) and the related low-energy
approximation in Eq.(22), respectively. Not surprisingly, we find the
approximation improves with decreasing energy (decreasing $\kappa$), but
apparently also with increasing $d$. The mathematical reason can be found in
the size of the $d$-dependent expansion parameter leading to Eq.(22). The
parameters for the square-well potential have been chosen such that
$d_{E}=2.50$. The consequence is that for this value of $d$, as mentioned
below Eq.(21), the low energy curves (thick-dashed-red) always take the value
$\delta_{\ell^{*}}=\pi/2$.

For $d=2$ the low-energy phase-shifts are always $\pi$ corresponding to a
bound state. The exact model results are also close to $\pi$ for $d\approx 2$
and sufficiently small energies. The validity range of the low-energy
expansion seem to be energies less than $\kappa/S_{0}\lesssim 0.001$. For all
these lower energies we observe a very sharp transition around $d_{E}$ from
phase-shifts of $\pi$ to zero moving towards larger $d$-values. This behavior
is connected to the general property of at least one bound state for any
attraction for $d=2$ and in our case for $d<d_{E}$. The phase shift is passing
$\pi/2$ and a bound state of zero energy appears for $d=d_{E}$. For $d>d_{E}$
there is no bound state, but a negative centrifugal barrier leading to phase-
shifts close to zero, whereas there is at least one bound state for $d<d_{E}$,
perhaps also conditions corresponding to a resonance with phase shifts $\pi$.

In Fig. 4 the results arising from the use of a Gaussian potential are also
shown. To investigate the proper comparison between different potentials we
first consider the square-well and the Gaussian potentials having the same
scattering length ratio to the interaction range, $r_{0}$, for $d=3$. The
results for the Gaussian case are given by the thin-solid-blue curves. When
compared to the square-well results (thick-solid-black) we see that both
curves are quantitatively very similar, although the agreement between them is
not perfect. In fact, the Gaussian potential constructed as mentioned above
leads to a critical dimension $d_{E}=2.43$, which implies that the crossing
through $\delta_{\ell^{*}}=\pi/2$ at low energies can be easily distinguished
from the one of the square-well potentials, which happens at $d_{E}=2.50$.

However, when the Gaussian potential is constructed having the same ratio
between the scattering length and the effective range as in the square-well
potential for $d=3$, we then observe a clearly better agreement between both
cases, as seen when comparing the thin-dashed-green and the thick-solid-black
curves in Fig. 4 (the Gaussian potential has now $d_{E}=2.53$). The universal
function describing the critical dimension as function of the scattering
length divided by the effective range is therefore also valid for continuum
states. This result is consistent with the universal curve shown in Fig. 2,
also a function of $a_{d=3}^{(\ell_{2b}=0)}/r_{\mbox{\scriptsize eff}}$.

This is very reassuring, because we can then compare potentials with the same
gross properties like scattering length and effective range, and claim the
computed universal relation between phase shifts and dimension. In other words
one can use the analytic square-well results and translate to any other
potential by use of the potential-independent parameters, scattering length
and effective range.

###  III.4 Cross sections

As described in Section II.6, the $d$-dimensional wave function is interpreted
as a wave function in the squeezed 3D space. As shown in Eq.(29), this wave
function can be expanded in terms of the usual spherical harmonics, which
permits to extract the projected radial wave functions $u_{\ell}(r)$ as shown
in Eq.(31).

Figure 5: Projected radial wave functions $u_{\ell}(r)$ for $\ell_{2b}=0$,
$d=2.9$, and $\ell=0$ (black), 10 (red), and 40 (blue) and the Gaussian
potential in Ref. gar21 . The incident energy (in units of $\hbar^{2}/(\mu
b^{2})$) is equal to 1 (a), 0.1 (b), and 0.01 (c). The solid curves are the
results from Eq.(31), and the dots are the asymptotic behavior given in
Eq.(32). The wave functions, $u_{\ell}(r)$, and the radial coordinate, $r$,
are multiplied by $b^{1/2}$ and divided by $b$, respectively, to make them
dimensionless ($b$ is the range of the potential).

In Fig. 5 we show, for $\ell_{2b}=0$, the large distance part of the
$u_{\ell}(r)$ functions after the calculation in Eq.(31) for $d=2.9$, $\ell=$
0 (black), 10 (red), and 40 (blue), and three different energies, $E=1$ (a),
$E=0.1$ (b), and $E=0.01$ (c) in units of $\hbar^{2}/(\mu b^{2})$, where $b$
is the range of the Gaussian two-body interaction used in the calculation. To
be precise, we have used the Gaussian two-body potential given in Ref. gar21 ,
for which $d_{E}=2.75$. In the figure we have multiplied $u_{\ell}(r)$ by
$b^{1/2}$ and divided $r$ by $b$ in order to make both quantities
dimensionless.

We can see that the larger $\ell$, the smaller the weight of the corresponding
$u_{\ell}(r)$ wave function. Therefore, one would expect that for sufficiently
high values of $\ell$, the contribution of the $u_{\ell}$ wave functions to
the total wave function (29) can be neglected. In general, the larger the
squeezing, the larger the number of $\ell$-values contributing. In fact, for
$d=3$ only the $u_{\ell=0}(r)$ function enters.

After matching the $u_{\ell}$ functions with the analytic asymptotic behavior
in Eq.(32) we can extract the $\delta_{\ell}$ phase shifts. The result of this
matching is shown in the figure by the dots, which we can see lie very much on
top of the solid curves. We can also see that, as expected, the smaller the
energy, the slower the oscillations of the wave functions. An important point
in the figure is that, for each energy $E$, all the wave functions,
independently of $\ell$, have the zeros at the same $r$-values. Although not
shown in the figure, this happens no matter the value chosen for the dimension
$d$. Therefore, as one could expect, it is possible to assign a single phase
shift, $\ell$-independent, to the 3D wave function introduced in Eq.(29).

Figure 6: Value of $\sin^{2}\delta_{\ell}$, as a function of the angular
momentum $\ell$, for the $E=1$ case in Fig. 5. The different curves correspond
to different values of the large $r$-values used to extract $\delta_{\ell}$.
The larger the $r$-values, the closer the curve to a constant value of
$\delta_{\ell}$.

We illustrate this same result in Fig. 6, where we show
$\sin^{2}\delta_{\ell}$ as a function of $\ell$ for the case in Fig. 5a, i.e.,
$\ell_{2b}=0$, $d=2.9$, and $E=1$. The different curves in the figure
correspond to different values of the radial coordinate $r$ used to match the
computed $u_{\ell}(r)$ function and the asymptotic behavior in Eq.(32). The
result is that the larger the value of $r$ chosen for the matching, the closer
the curve to a horizontal line, corresponding to a constant value of
$\delta_{\ell}$. This is due to the fact that the larger the value of $\ell$,
the farther one has to go in the wave function in order to get the correct
asymptotic behavior. Again, the results shown in Fig. 6 are qualitatively the
same as the ones obtained for a different value of $d$ and a different energy.

Finally, as mentioned in Eq.(33), the phase shift, $\delta_{\mbox{\scriptsize
conf}}$, in the squeezed 3D space is not given just by $\delta_{\ell}$, but by
the shift of the wave function for two interacting particles compared to the
wave function corresponding to two free particles. The phase shift obtained
from Eq.(32) for two non-interacting particles is what in Eq.(33) is denoted
by $\delta_{\mbox{\scriptsize free}}$.

Figure 7: For $\ell_{2b}=0$, values, as a function of $d$, for the phase
shifts $\delta_{\ell}$ (solid-black), $\delta_{\mbox{\scriptsize free}}$
(dashed-red), $\delta_{\mbox{\scriptsize conf}}$ (dot-green), and
$\delta_{\ell^{*}}$ (long-dashed-blue), for the same potential and energies as
in Fig. 5.

In Fig. 7 we show $\delta_{\ell}$, $\delta_{\mbox{\scriptsize free}}$,
$\delta_{\mbox{\scriptsize conf}}$, and $\delta_{\ell^{*}}$ as a function of
$d$ for the same Gaussian potential as in Fig. 5, and the same three energies.
The phase shifts are given by the solid-black, dashed-red, dot-green, and
long-dashed-blue curves, respectively.

Except for $d=3$, where by definition they are the same, we can see that
$\delta_{\ell}$ and the phase shift computed directly with the $d$-method,
$\delta_{\ell^{*}}$, are very different. Also, when the two particles do not
interact, the phase shift, $\delta_{\mbox{\scriptsize free}}$, obtained from
Eq.(32) is clearly different from zero (except of course for $d=3$, where the
asymptotics in Eq.(32) is exact). The important result is that, as shown by
the dotted curves in the figure, the difference between $\delta_{\ell}$ and
$\delta_{\mbox{\scriptsize free}}$, that is, $\delta_{\mbox{\scriptsize
conf}}$, coincides perfectly with $\delta_{\ell^{*}}$, i.e., with the phase
shift obtained directly from the $d$-method.

Therefore, as anticipated in Section II.6, the phase shifts obtained with the
$d$-method are the phase shifts in the squeezed 3D space, and they can be used
to compute the cross section in the 3D space.

Figure 8: The same as Fig.7 for $\ell_{2b}=1$.

The result shown in Fig. 7 is actually general, not restricted to the
$\ell_{2b}=0$ case. As an illustration we show in Fig. 8 the same as in Fig. 7
but for $\ell_{2b}=1$. As we can see, the equality $\delta_{\mbox{\scriptsize
conf}}=\delta_{\ell^{*}}$ is also perfectly obtained, even in the case of very
small energy, Fig. 8c, where $\delta_{\ell}$ and $\delta_{\mbox{\scriptsize
free}}$, are very similar, leading to very small values of the true phase
shift.

##  IV Summary and conclusions

In this work we have used the $d$-method in order to investigate two-body
continuum states in a squeezed scenario. The method describes the system in a
non-integer dimensional space, which can be later translated to the usual 3D
space with an external confining potential. The $d$-method is technically
simpler, since the degrees of freedom associated to the external field do not
enter in the formalism.

We focus on two-body properties for short-range potentials in non-integer
geometry by use of a square-well potential. The details are independent of
this practical choice of the potential.

We first derive conditions for bound states expressed by the square-well
parameter. The depth-radius combination has to be large enough to support at
least one bound state. We show how the threshold value of the dimension for
binding is uniquely determined by the square-well parameter. This is in close
analogy to three dimensions, but for smaller dimensions less attraction is
necessary, and for $d=2$ even infinitesimally small attraction provides
binding.

The point of zero energy binding defines the Efimov dimension, $d_{E}$, since
for this dimension the Efimov effect may become effective by adding another
particle. We have derived the equation determining the value of $d_{E}$ as a
function of the strength of the square-well potential. This value of $d_{E}$
is shown to correspond to infinite scattering length, which in turn has also
been derived as a function of the same strength for each dimension $d$.

We have closed the theoretical part of this paper deriving the expressions for
the $d$-dependent phase shifts as a function of the potential strength. These
phase shifts have also been related to the ones corresponding to the physical
squeezed 3D space.

The main purpose of the illustrations shown in this work is to show that, even
if the derived expressions, obtained for a square-well two-body interaction,
are not necessarily universal, it is however possible, using the proper length
and energy units, to obtain a universal behavior, independent of the details
of the potential, for the Efimov dimension $d_{E}$ as well as for the phase
shifts.

In particular, focusing on the dominant $s$-wave case, we have shown that the
critical dimension $d_{E}$ follows a universal path when plotted as a function
of $a_{d=3}^{(\ell=0)}/r_{\mbox{\scriptsize eff}}$, showing that the effective
range in three dimensions is the appropriate length unit. Given the value of
$a_{d=3}^{(\ell=0)}/r_{\mbox{\scriptsize eff}}$ for any two-body short-range
potential it is then possible, by means of the universal curve, to obtain the
corresponding dimension $d_{E}$.

It is reassuring that potentials having the same value of the
$a_{d=3}^{(\ell=0)}/r_{\mbox{\scriptsize eff}}$ ratio follow as well a pretty
much universal curve for the phase shifts as a function of the dimension and
for given values of $\kappa/S_{0}$. In other words, for the universal curve to
show up, the incident momentum has to be taken in units of $S_{0}$, which
contains the dependence on the strength of the potential.

Finally, we have related the computed phase shifts in the $d$-method with the
ones corresponding to a squeezed 3D space. This is done by considering the
$d$-dimensional two-body wave function as an ordinary wave function in 3D, but
squeezing the coordinate along the direction of the external field after
inclusion of a scale parameter. When this is done, we have shown that, in case
of non-interacting particles, the usual asymptotic form of the continuum wave
functions in three dimensions gives rise to non-zero phase shifts,
$\delta_{\mbox{\scriptsize free}}$. This is simply reflecting the presence of
the squeezing potential. When the interaction is introduced, the relevant
phase shift is the difference between the new computed phase shift and
$\delta_{\mbox{\scriptsize free}}$. We have shown that this difference,
$\delta_{\mbox{\scriptsize conf}}$, is precisely the same as the phase shift
obtained directly from the $d$-method. This result opens the door to using the
$d$-computed phase shifts directly in order to obtain the cross sections in a
squeezed scenario.

In conclusion, we have investigated and illustrated two-body scattering
processes in an analytic schematic model. The cross section behavior and the
insight obtained are very hard to imagine found in any other way. Such results
are, to a large extent, universal, that is independent of the details of the
employed short-range potentials. The translation from $d$ to external field is
necessary, available and at least a semi-accurate description. The perspective
in our investigations is that scattering between particles confined by
deformed external fields may be useful tools in investigations of for example
structures related to Efimov physics. Transitions between other dimensions may
also be of interest.

###### Acknowledgements.

This work has been partially supported by the Spanish Ministry of Science,
Innovation and University MCIU/AEI/FEDER,UE (Spain) under Contract No.
PGC2018-093636-B-I00.

##  Appendix A Bessel function properties

The two regular spherical Bessel functions with indices $0$ and $1$ are

$\displaystyle j_{0}(z)=\frac{\sin z}{z}\;\;,\;j_{1}(z)=\frac{\sin
z}{z}-\frac{\cos z}{z^{2}}\;.$ (35)

We have two useful identities between any, $B_{l}$, of the regular, $j_{l}$,
and irregular, $\eta_{l}$, Bessel and Hankel, $h^{(\pm)}_{l}=\eta_{l}\pm
ij_{l}$ functions, that is

$\displaystyle z\frac{dB_{l}(z)}{dz}=lB_{l}(z)-zB_{l+1}(z)\;,$ (36)
$\displaystyle(2l+1)B_{l}(z)=z(B_{l+1}(z)+B_{l-1}(z))\;.$ (37)

Limits for zero arguments are useful. For $z\rightarrow 0$ we have

$\displaystyle
j_{l}(z)\rightarrow\frac{z^{l}}{(2l+1)!!}\;\;,\;\eta_{l}(z)\rightarrow\frac{(2l+1)!!}{(2l+1)}\frac{1}{z^{l+1}}\;.$
(38)

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