text:
\section{Introduction}

The last decade has provided unprecedented accurate techniques to tune
the effective interactions between pairs of ultracold atoms in
extremely controllable external fields.  The effective two-body
interactions can be varied, for a number of special atoms, from
strongly attractive to strongly repulsive by use of the Feshbach
resonance technique, see review in Ref. \cite{pethick}.  The two-body
interactions are experimentally determined functions of the applied
magnetic field strength.  The atomic gases, possibly with different
atoms, are confined by flexible deformed external fields.  Ultimately
this allows extreme asymmetry corresponding to spatial dimensions
lower than three \cite{modugnoPRA03,gunterPRL05}.  Structures of
different systems can then be simulated and studied.

Our interests in the present investigation are three-body structures
and two-dimensional universal properties.  First, few-body physics is
by definition accurately solvable without approximations in contrast
to many-body physics. But now experimental tests can be made for many
of the claims derived by theoretical calculations.  We focus here on
the absolutely simplest unsolved system of three particles in two
spatial dimensions (2D).  Second, the physics in 2D differ enormously
from the much more known properties in three dimensions (3D) and for
that matter any other spatial dimension
\cite{castinpra2012,bellottifbs2015}.  Third, the many universal
properties are arguably the most important results confirmed by cold
atomic gas experiments.  Properly formulated they are applicable
throughout physics as independent of scale and details of the
corresponding potentials \cite{jensenrmp2004,braatenpr2006}.

The third of these points refer to the concept of universality which
means that the related properties can be described by only a few
scale-parameters \cite{fre12}.  The interactions
between neutral atoms in dilute atomic gases are of very short range
compared to the two-body scattering length
\cite{kunitskiscience2015,cencekjcp2012,kolganovafbs2011}.  This
implies that low-energy observables are universal and only depend on
the scattering length which can be the same for disparate potentials.
Thus the universal regime can be defined as properties depending only
on lengths far larger than the range of the potential and far smaller
than the scattering length.  Thus, a zero-range interaction with its
strength proportional to the scattering length is automatically
focussing on universal properties.

The special interest in two dimensions is due to the properties
arising from the negative centrifugal barrier corresponding to the
relative coordinate between two particles.  This means that binding is
achieved with even an infinitesimal attraction between the two
particles.  Also three particles are bound in 2D with almost vanishing
attractive two-body potentials.  For identical bosons in the universal
regime all observables in 2D can be expressed as functions of only one
two-body scale parameter, e.g. the two-body scattering length
\cite{jensenrmp2004,braatenpr2006}.  In 3D, both a two- and a
three-body scale parameter are needed to describe universal observables
\cite{skorniakov1957}.  This difference is closely related to the
appearance of the pathological Efimov effect for three particles in 3D
\cite{efimov} and its absence in 2D \cite{nielsenpr2001}.  This
apparent discontinuity can now be studied with external traps varying
continuously from spherical to cylindrical geometry \cite{yama15}.

The status for weakly bound three-body systems in 2D is that energies
are well studied theoretically
\cite{bruchpra1979,adhikarijp1986,adhikaripra1993,nie99,bellottijpb2011,
  bellottipra2012,bellottijpb2013,khurifbs2002,levinsenprx2014,esry2014,esry2015}.
The simplest reference system we have is three identical bosons with
only two three-body bound states with energies, $E_3$, proportional to
the two-body energy, $E_2$, that is $E_3=16.52E_2$ and $E_3=1.27E_2$
\cite{bruchpra1979}.  The previous studies include energies of three
non-identical particles where different masses and scattering lengths
substantially complicate systematic characterization of the universal
properties \cite{nie99}.  The simplest asymmetric system, denoted
$AAB$, is formed by two identical bosons, $A$, and a distinguishable
particle, $B$.  With spin-independent interactions, the results also
apply to two identical fermions with symmetric spatial wave function.
The number of three-body bound states increases as the mass ratio,
${\cal A}=m_B/m_A$, between $B$ and $A$ decreases \cite{bellottipra2012}.
Then the $B$-particle can be exchanged more easily between the heavy
$A$-particles, which in turn generates an effective potential
eventually of infinite attraction for vanishing ${\cal A}$
\cite{limZPA80,bellottijpb2013,levinsenprx2014}.

In contrast to the energies, knowledge about structures of asymmetric
three-body systems in 2D is virtually not existing.  One reason is
that experiments have only now become feasible, but the lack of the
necessary operating theoretical techniques is probably also partly
responsible for this delay.  Most of the studies have employed
zero-range interactions treated in momentum space, and have calculated
energies without direct access to the corresponding wave
functions. The structures therefore require a substantially larger
additional effort.  Coordinate space calculations in 2D with direct
link to wave functions are also not available at the moment.

We shall continue to use the zero-range interaction with the easy
interpretation in terms of universal properties
\cite{yamashitapra2003}.  It is worth mentioning that it would be
interesting to extend the present calculation to more than three
atoms to see how the universal relations derived in \cite{hammerprl2004}
would change for a mass-imbalanced situation.  We use tedious mathematics
to express various radial moments of relative distances as derivatives of
momentum space integrals.  This is instead of the extensive numerical
computation of momentum-space wave functions with subsequent Fourier
transformation and derivation of structural properties.  We shall be
content with second radial moments, which are the simplest observable
quantities that carry structure information.  If necessary, higher
moments can also be computed by use of the same technique, but we
believe the second radial moments of various relative distances are
sufficient to extract the dominating underlying structure.

The paper is organized as follows. In section \ref{formalism} we give
first the appropriate Faddeev equations in momentum space for an $AAB$
system of one distinguishable and two identical particles.  Then we
construct the three-body wave functions and the form factors from
which the radii of interest are calculated.  In section \ref{mass} we
present and analyze numerical results of energy spectra and
mean-square radii, as functions of the mass ratio.  In particular, we
discuss the threshold behavior when a level is passing into the
three-body continuum of one particle and a bound pair.  Finally,
conclusions and perspectives are summarized in section
\ref{conclusions}.

\section{Theoretical formulation}
\label{formalism}

We consider a three-body system, $AAB$, consisting of two identical
bosons, $A$, of mass, $m_A$, and bound state energy, $E_{AA} < 0$, and
a third distinguishable particle, $B$, of mass, $m_B$, bound to each
$A$-particle with energy, $E_{AB} < 0$.  We denote the identical
particles by $A$ and $A'$ when it is necessary to distinguish their
coordinates.  The two-body interactions are assumed to be of very
short range, and in this paper they are parametrized and applied in the extreme
zero-range limit where only $s$-waves are important.  This rather
schematic interaction extracts universal properties due to the absence
of spatial regions inside the potential.

\subsection{Faddeev equations}

The preferred method in most investigations is the use of the Faddeev
decomposition in momentum space.  The theoretical formulation is a
little elaborate but available in details in the literature
\cite{braatenpr2006,fre12}.  We shall here only
specify our notation and present the Faddeev equations in momentum
space.  We shall then proceed to sketch the novel derivation of the
different mean-square radii expressed as derivatives of integrals over
the momentum-space solutions.  The three-body state, $ | \Psi_{AAB}
\rangle$, corresponds to the three-body momentum-space wave function,
$\Psi_{A'B,A}$, when it is expressed in terms of $\vec{p}_A$ and
$\vec{q}_A$, that are, respectively, the relative momenta between the $B$
and the $A'$-particle, and the center-of-mass of $A'+B$ and the
$A$-particle.  The Faddeev decomposition then amounts to
\begin{eqnarray}
\label{psiab}
&&\Psi_{AB,A}(\vec{q}_A,\vec{p}_A)\equiv \langle\vec{q}_A,\vec{p}_A\,|\Psi_{AAB}\rangle \\
\nonumber
&&=\frac{\chi_A(|\vec{q}_A|)+\chi_B
(|\vec{p}_A-\frac{{\cal A}}{{\cal A}+1}\vec{q}_A|)+\chi_A(|\vec{p}_A+
\frac{1}{{\cal A}+1}\vec{q}_A|)}{|E_3|+\frac{{\cal A}+1}{2{\cal A}}
\frac{p^2_A}{m_A}+\frac{{\cal A}+2}{2({\cal A}+1)}\frac{q^2_A}{m_A}},
\end{eqnarray}
where $|E_3|$ is the three-body binding energy, the mass ratio between
particles $B$ and $A$ is denoted ${\cal A}\equiv m_B/m_A$, and the
so-called spectator functions, the Faddeev components $\chi_A$ and
$\chi_B$, are given as functions of the relative momenta in the
corresponding Jacobi coordinates, that are, $\vec{q}_A$, $\vec{q}_B$ and
$\vec{q}_{A'}$, where
\begin{eqnarray}
\vec{q}_B = \vec{p}_A-\frac{{\cal A}}{{\cal A}+1}\vec{q}_A \;\;,\;\;
\vec{q}_{A'} = \vec{p}_A+ \frac{1}{{\cal A}+1}\vec{q}_A \; .
\end{eqnarray}

The spectator functions, $\chi_A$ and $\chi_B$, obey coupled integral
equations obtained after $s$-wave projection.  The detailed standard
derivation of these equations in 2D can be found in
\cite{bellottijpb2011}, with the result
\begin{eqnarray}
\label{chib}
\nonumber
&&\chi_B(q)=2\tau_{AA}\left(|E_3|+\frac{{\cal A}+2}{4{\cal A}}\frac{q^2}{m_A}\right)\\
&&\times \int d^2p G_1(p,q;E_3)\chi_A(p)\\
\label{chia}
&&\chi_A(q)=\tau_{AB}\left(|E_3|+\frac{{\cal A}+2}{2({\cal A}+1)}\frac{q^2}{m_A}\right)\\
\nonumber
&&\times\int d^2p
\Bigg[G_1(q,p;E_3)\chi_B(p)+G_2(q,p;E_3)\chi_A(p)\Bigg]. \\
&&G_1(q,p;E_3)=\frac{1}{|E_3|+\frac{q^2}{m_A}+\frac{{\cal A}+1}{2{\cal A}}\frac{p^2}{m_A}
+\frac{\vec{q}\cdot\vec{p}}{m_A}}  ,\\
&&G_2(q,p;E_3)=\frac{1}{|E_3|+\frac{{\cal A}+1}{2{\cal A}}\frac{(q^2+p^2)}{m_A}+\frac{1}{{\cal A}}
\frac{\vec{q}\cdot\vec{p}}{m_A}} ,
\end{eqnarray}
where the two-body T-matrices, $\tau_{A\alpha}$, of the interacting pair are
given by
\begin{equation}
\label{2btmatrix}
\tau_{A\alpha}(E)=\left[-4\pi\frac{m_Am_\alpha}{m_A+m_\alpha}
\ln\left(\sqrt{\frac{|E|}{|E_{A\alpha}|}}\right)\right]^{-1},
\end{equation}
where $E_{A\alpha}$ is the two-body binding energy of the pair
$A\alpha$, with $\alpha=A,B$.

\begin{figure}[htb!]
\epsfig{file=Newradischeme,width=\columnwidth}
\caption{Schematic figure showing the three-body system of two
  identical bosons, $A$, and one distinguishable particle, $B$.  The
  vectors $\vec{r}_{AB}$ and $\vec{r}_{AA}$ are relative distances
  between the respective pairs of particles, whereas $\vec{r}_{A}$ and
  $\vec{r}_{B}$ denote distances between the specified particle and
  the center-of-mass of the remaining pairs.  The point marked
  C.M. means center-of-mass of the three-body system.  The notation
  used in the paper for the relevant distances are also shown.  }
\label{figradii}
\end{figure}

\subsection{Mean-square radii}

The distances in the three-body system can be seen in
Fig.\ref{figradii}, where $\vec{r}_{AB}$, $\vec{r}_{AA'}$ and
$\vec{r}_{A'B}$, are relative distances between pairs of particles,
and $\vec{r}_{A'}$, $\vec{r}_{A'}$ and $\vec{r}_B$ denote distances
between the center-of-mass of such pairs and the remaining last
particle.  The momenta, $\vec{q}_A$ and $\vec{p}_A$, are repectively
the canonically conjugated to the radii, $\vec{r}_A$ and
$\vec{r}_{AB}$.

The mean-square radii can be computed from the momentum-space wave
function by Fourier transform and subsequent calculation of the
corresponding matrix element.  To be precise we first define one-body
densities related to one set of Jacobi coordinates, $\vec{r}_A$ and
$\vec{r}_{AB}$, that is
\begin{eqnarray}
 \rho(\vec{r}_A)=\int d^2r_{AB} |\langle\vec{r}_A,\vec{r}_{AB}|
 \Psi_{AAB}\rangle|^2  \;,  \label{rhoa} \\  \label{rhoab}
 {\bar \rho}(\vec{r}_{AB})=\int d^2r_A |\langle\vec{r}_A,\vec{r}_{AB}|
 \Psi_{AAB}\rangle|^2    \; ,
\end{eqnarray}
where $\langle\vec{r}_A,\vec{r}_{AB}| \Psi_{AAB}\rangle$ is the
coordinate-space wave function expressed in terms of $\vec{r}_A$ and
$\vec{r}_{AB}$.  We first expand the Fourier transform, $F_A$ of
$\rho(\vec{r}_A)$ to second order in the corresponding momentum, that
is
\begin{eqnarray}
F_A(Q^2) &=& \int d^2r_A e^{\frac{i}{\hbar}\vec{Q}\cdot\vec{r}_A}\rho(r_A)
\approx  1-\frac{1}{4\hbar^2}Q^2 \langle r_A^2\rangle \;, \label{fourierff} \\
\langle r_A^2\rangle &=&   \int d^2r_A  r_A^2 \rho(r_A) \; ,
\end{eqnarray}
where we assumed a normalized one-body density and used that all odd
powers of $\vec{Q}$ vanish due to spherical symmetry.  The first order
derivative with respect to $Q^2$ then gives the desired mean-square
radius.  In the same way we obtain
\begin{eqnarray}
 \langle r_{AB}^2\rangle  &=& \int d^2r_{AB}  r_{AB}^2 {\bar \rho}(r_{AB}) \; .
\end{eqnarray}
from the Fourier transform, $F_{AB}(Q^2)$ of ${\bar
  \rho}(\vec{r}_{AB})$.  These Fourier transforms are related to the
momentum-space wave function in Eq.(\ref{psiab}) by
\begin{eqnarray}
\label{fa}
\nonumber
F_A(Q^2)&=&\int d^2qd^2p\Psi_{AB,A}\left(\vec{q}+\frac{\vec{Q}}{2},\vec{p}\right)\\
&&\times\Psi_{AB,A}\left(\vec{q}-\frac{\vec{Q}}{2},\vec{p}\right),\\
\nonumber
F_{AB}(Q^2)&=&\int d^2qd^2p\Psi_{AB,A}\left(\vec{q},\vec{p}+\frac{\vec{Q}}{2}\right)\\
&&\times\Psi_{AB,A}\left(\vec{q},\vec{p}-\frac{\vec{Q}}{2}\right),
\label{fab}
\end{eqnarray}
which can be verified by directly inserting the definitions of
$\rho(r_A)$ and $\rho(r_{AB})$ from Eqs.(\ref{rhoa}) and (\ref{rhoab})
into Eq.(\ref{fourierff}) and the corresponding definition of
$F_{AB}(Q^2)$.

To calculate the remaining mean-square radii we express the three-body
wave function in the set of Jacobi momenta of the $B$-particle,
$\Psi_{AA,B}(\vec{q}_B,\vec{p}_B)\equiv
\langle\vec{q}_B,\vec{p}_B\,|\Psi\rangle$.  The mean-square radii,
$\langle r_B^2 \rangle$ and $\langle r_{AA}^2 \rangle$, are then
obtained as first order derivatives with respect to $Q^2$ of the
Fourier transforms, $F_B(Q^2)$ and $F_{AA}(Q^2)$, defined by replacing
$\Psi_{AB,A}$ with $\Psi_{AA,B}$ in Eqs.(\ref{fa}) and (\ref{fab}).

The procedure is now to solve the three-body equations in
Eqs.(\ref{chib}) and (\ref{chia}) and obtain three-body energy,
spectator functions, and the total wave function in Eq.(\ref{psiab}).
The next steps are to calculate numerically $F_A$, $F_B$, $F_{AA}$ and
$F_{AB}$ and the necessary first order derivatives of these functions
of the squared momenta.

\section{Mass dependence}
\label{mass}

The three-body wave function is completely determined from the two
ratios of particle masses, ${\cal A} = m_B/m_A$, and two-body
bound-state energies, $E_{AA}/E_{AB}$, both negative.  To extract
meaningful structure properties it is crucial to know all independent
distances in a given system, as illustrated in Fig.\ref{figradii}.  We
calculate energies and radii of the lowest-lying excited states as
functions of ${\cal A}$ for an energy ratio of $E_{AA}/E_{AB} = 1$.
Each of the two-body energies results from the related reduced mass,
which therefore is varied in our investigations.  The calculated
results therefore illustrate typical behavior of mean-square radii and
structure in general.  We confirm this expectation with test
calculations of varying energy ratio.

\subsection{Three-body energies}
\label{ener}

\begin{figure}[htb!]
\epsfig{file=energyspectrum.eps,width=\columnwidth}
\caption{Low-energy spectrum of an $AAB$ system as a function of the
  mass ratio ${\cal A}=m_B/m_A$.  The two two-body energies are equal,
  $E_{AB}=E_{AA}$, and the three-body energy is $E_3$.  The
  energies on the $y$-axis are given relative to the two-body bound
  state energies, $E_{AB}$.  The vertical lines
  indicate the mass ratios ${\cal A}=6/133$ and ${\cal A}=6/87$
  corresponding to the systems $^6$Li-$^{133}$Cs-$^{133}$Cs and
  $^6$Li-$^{87}$Rb-$^{87}$Rb. }
\label{figenergy}
\end{figure}

The energy is for most quantum states the all-important characterizing
quantity which is necessary to understand first.  Universal structure
of weakly bound systems often relates spatial extension and binding
energy \cite{nie99}.  We therefore first focus on the energy
dependence of the system, $AAB$, sketched in Fig. \ref{figradii}.  We
solve the coupled set of integral equations, Eqs.(\ref{chib}) and
(\ref{chia}), and show the calculated energy spectrum in
Fig. \ref{figenergy}.  For ${\cal A}=1$ we recover the well known
results that only ground and first excited states are bound with
universally given energies, $E_3=16.52E_{AB}$ and $E_3=1.27E_{AB}$.
These energies are predicted by a number of entirely different
calculations \cite{bruchpra1979,nie99,bellottipra2012}.

Increasing the mass ratio above unity, ${\cal A} \geq 1$, the two
bound states remain with moderate relative energy variations.  This
may be understood from the limit of one very heavy particle surrounded
by two light masses moving around the center-of-mass in a roughly
mass-independent field.  Superficially this resembles a helium atom
with two electrons moving around the four bound nucleons.  However,
the interactions differ enormously from the long-range Coulomb
potential to the zero-range interaction.  The properties in 2D also
differ enormously from the 3D calculations of a helium atom.  These
differences are evident from the order-of-magnitude-larger ground-state
binding of three particles compared to two particles.

In the other limit of small ${\cal A}$ we find a tremendous increase
of bound states with decreasing mass ratio ${\cal A}$.  The smallest
mass ratio we consider is ${\cal A}=0.01$, where the system displays
eight bound states.  The excited three-body states disappear into the
continuum of positive energies as ${\cal A}$ increases from very
small values.  The rather dramatic mass dependence for small ${\cal
  A}$ can be understood as a consequence of the increase of the
effective interaction generated by the light particle.  This has been
known since 1979 \cite{fonsecaNPA78} and 1980 \cite{limZPA80} for
systems in three and two dimensions, respectively.

Recently, a different technique was used to derive an effective
two-body potential in two dimensions \cite{bellottijpb2013}.  This
derivation assumed the Born-Oppenheimer approximation whose validity
only is justified for ${\cal A} \ll 1$.  After integrating out the
coordinate of particle $B$, an effective hamiltonian is left for the
relative motion of the two heavy $A$-particles, that is
\begin{equation} \label{HBO}
 H_{BO} =  - \frac{\hbar^2}{2\mu_{AA}} \Delta_{\vec{r}_{AA}} + V_{AA}(r_{AA}) +
 V^{(BO)}(r_{AA}) \; ,
\end{equation}
where $\vec{r}_{AA}$ is the relative coordinate between the two
$A$-particles, $\mu_{AA}$ and $V_{AA}(r)$ are their reduced mass and
two-body potential, and $V^{(BO)}(r_{AA})$ is the strongly
${\cal A}$-depending Born-Oppenheimer potential resulting from the
light third particle, $B$.

The Born-Oppenheimer potential is obtained by standard derivation in
three-body physics, that is with two centers at a fixed distance
interacting with the light particle through zero-range interactions.
This quantum mechanical problem in 2D is solved in the center-of-mass
of the total system.  The result is in Ref. \cite{bellottijpb2013}
found to be
\begin{equation}
\ln \frac{\left\vert V^{(BO)}(R)\right\vert }{\left\vert E_{AB} \right\vert} =2 K_0 \left(\sqrt{\frac{2\mu_{AA,B}|V^{(BO)}(R)|}{\hbar^2}} r_{AA} \right)  \ , \label{eq.51}
\end{equation}
where $K_0$ is a Bessel function, $\mu_{AA,B}$ is the reduced mass of
$B$ versus the $AA$-system, and $R = r_{AA} \sqrt{m_A
  |E_{AB}|/\hbar^2}$ with $\mu_{AA,B}$.  This analytical result was in
Ref. \cite{bellottijpb2013} shown to be well approximated as a function
of ${\cal A}=m_B/m_A$ by simpler expressions in both limits of small
and large distances.  The ${\cal A}$-dependence is through the
combination
\begin{equation} \label{BOmass}
 m_{ef} = \bigg(\frac{4{\cal A}}{2+{\cal A}} \bigg)^{1/2} \; ,
\end{equation}
and the results from Ref.\cite{bellottijpb2013} are
\begin{equation} \label{BOpota}
V^{(BO)}(r_{AA}) \approx - \frac{2 |E_{AB}|\exp(-\gamma) }{Rm_{ef}} \;
\end{equation}
for $R m_{ef} \leq 1.15 $, where $\gamma = 0.5772156649$ is Euler's
constant, and
\begin{equation}
V^{(BO)}(r_{AA}) \approx  - |E_{AB}| \Big( 1 +
\frac{\sqrt{2\pi}\exp(-R m_{ef})}{\sqrt{R m_{ef}}}\Big)
\label{BOpotb}
\end{equation}
for $R m_{ef} \geq 1.15 $.  These approximations are accurate to
better than $10\%$, where the largest deviations found around $R
m_{ef} = 1.15 $ \cite{bellottijpb2013}.

The short-distance part exhibits an attractive Coulombic behavior in
Eq. (\ref{BOpota}).  The large-distance behavior in Eq. (\ref{BOpotb})
is exponentially convergent towards the negative two-body energy,
$E_{AB}$, with a length scale proportional to $1/m_{ef}$.  Thus, only
a finite number of bound states below $E_{AB}$ are possible, although
that number would increase without limit since the $1/m_{ef}$ diverges
with vanishing mass ratio ${\cal A}$.  However, we are mostly
interested in the lowest energy states which necessarily are located
in the strongly attractive region at smaller distances.  The
small-distance Coulomb-like behavior is controlled by an effective
charge squared, $Z_{ef}^2$, defined by
\begin{equation} \label{charge2}
  Z_{ef}^2 =
 \frac{ 2 \hbar \exp(-\gamma) \sqrt{|E_{AB}|}}{m_{ef} \sqrt{m_A}} \; .
\end{equation}
The kinetic energy operator in Eq. (\ref{HBO}) has a mass of $\mu_{AA}
= m_A/2 $, and in two dimensions the crucial negative centrifugal
barrier term corresponds to the angular momentum quantum number of
$\ell = -1/2$.  The Coulomb energy spectrum for this small-distance
behavior is therefore
\begin{eqnarray} \label{energy}
\nonumber
E^{(BO)}_3 &=& - \frac{Z_{ef}^4 \mu_{AA}}{2\hbar^2} \frac{1}{(n_r+\ell+1)^2}\\
\nonumber
  &=&  \frac{(1+{\cal A}/2)}{2{\cal A}}\frac{\exp(-2\gamma)|E_{AB}|}{(n_r+\ell+1)^2}\\
 &=&  \frac{0.630473504 (1+{\cal A}/2) |E_{AB}| }{{\cal A}(2n_r+1)^2}  \; .
\end{eqnarray}
The entire Coulomb spectrum is then obtained with $n_r =0,1,2,...$ and
$\ell=-1/2,1/2,3/2,5/2...$

\begin{table}
\begin{tabular}{@{}|c|c|c|c|c|}
\hline
& ${\cal A}$ & $E^{(BO)}_{3}/E_{AB}$ & $E^{(NI)}_{3}/E_{AB}$ & $E_{3}/E_{AB}$ \\
\hline
Ground & 0.01       & 63.04          & 53.07         & 157.56  \\
       & 0.02       & 31.52          & 27.76         & 90.75   \\ \hline
First  & 0.01       & 7.00           & 7.72          & 11.34   \\
       & 0.02       & 3.50           & 4.25          & 6.36    \\ \hline
Second & 0.01       & 2.52           & 3.21          & 3.94    \\
       & 0.02       & 1.26           & 1.92          & 2.34    \\ \hline
Third  & 0.01       & 1.28           & 1.92          & 2.18    \\
       & 0.02       & 0.64           & 1.28          & 1.42    \\
\hline
\end{tabular}
\caption{\label{I}Energies of ground and first excited states for the mass
  ratios ${\cal A} =$ 0.01, 0.02 for the analytic Born-Oppenheimer
  approximation, $E^{(BO)}_{3}/E_{AB}$ (in Eq. (\ref{energy})), the numerical
  results both without, $E^{(NI)}_{3}/E_{AB}$, and with, $E_{3}/E_{AB}$, interaction
  between the two heavy $A$-particles.}
\end{table}

We can use Eq. (\ref{energy}) as a useful reference spectrum for
comparison to the energies in the limit of ${\cal A} \ll 1$, shown in
Fig. \ref{figenergy}.  To do this we have to distinguish between the
present numerical calculations and the pure Coulomb spectrum obtained
by neglecting the short-range potential, $V_{AA}$, between the two
heavy $A$-particles.  We compare these results in the first two
columns of Table \ref{I}.  The pure Born-Oppenheimer Coulomb estimate
of the ground state binding energy is $15$~\% larger than the
numerically calculated value.  On the other hand, the Coulomb
estimates give too small binding for all the excited states.  They
exploit distances outside the Coulomb-like attractive region.

The ground state deviations were also found in
Ref. \cite{bellottijpb2013}.  They are necessarily due to the
approximations in the Born-Oppenheimer procedure, since the accuracy
of our numerical computations are fractions of permille.  The most
obvious reason for these differences is the neglect of the
non-vanishing term arising from the heavy-heavy kinetic energy
operator.  The Born-Oppenheimer wave function depends first of all on
the fast coordinate but there is still also a dependence on the slowly
varying coordinate, $r_{AA}$.  This term is not included in our
Born-Oppenheimer calculation, where the inherent two-step quantization
procedure in any case is an approximation.

It is illuminating to compare three-body calculations with the
hyperspherical adiabatic expansion method, where the hyperradial part
of the kinetic energy operator gives rise to an analogous term
\cite{nielsenpr2001}.  Inclusion or not of this term, in the adiabatic
expansion with only one adiabatic potential, provides an upper or
lower bound on the correct energy \cite{nielsenpr2001}.  Including
more and more of the higher-lying potentials in this method provides a
fully converged solution.  The Born-Oppenheimer potential is analogous
to the one-potential approximation, which overbinds in agreement with
the bounds of the mentioned hyperspherical adiabatic method.  The
simplicity of the Born-Oppenheimer results allows qualitative
understanding of spectra and structure, which is appealing even though
the Faddeev calculations could speak for themselves.  In any case the
present Born-Oppenheimer approximation suffice for our purpose.

We now include the $AA$-interaction and focus on the results in
Fig. \ref{figenergy} where each level for small ${\cal A}$ roughly
follows the predictions in Eq. (\ref{energy}).  However, the
calculated ground state binding energies for both values, ${\cal A} =
0.01, 0.02$, are much larger than the pure Coulomb spectrum, see Table
\ref{I}.  The strong short-range $AA$-attraction has its largest effect on
the lowest-lying levels, that is more binding by factors of about $3$
and $1.5$ for ground and first excited states, respectively.  The
higher-lying excited states shown in Table \ref{I} and
Fig. \ref{figenergy} feel the long-range non-Coulombic
Born-Oppenheimer potential and exhibit a less systematic behavior.  As
${\cal A}$ increases, the bound states move up in energy, the number
decreases, and the Coulomb potential supports fewer and fewer bound
states.

\subsection{Sizes }

Equipped with an understanding of the energy spectrum as function of
the mass ratio we turn to the corresponding underlying structure.  We
show first in Fig. \ref{figraa} the calculated mean-square radial
distances, $<r^2_{AA}>$, between the two identical particles.  We
notice an almost reflected behavior compared to the three-body
energies.  The radii decrease while the binding energies increase.
The two lowest bound states are present for all mass ratios, whereas
the sizes for the higher-lying excited states diverge as their energies
reach their thresholds for binding.

\begin{figure}[htb!]
\epsfig{file=radiAA.eps,width=\columnwidth}
\caption{Dimensionless product $m_A\langle r_{AA}^2\rangle
  |E_{AB}|/\hbar^2$ ($E_{AA} = E_{AB}$) as a function of the mass
  ratio ${\cal A}$. As ${\cal A}$ is increased the radii diverge at
  the threshold where the excited states disappear. The remaining
  ground and first excited states assume a constant value as ${\cal
    A}\rightarrow\infty$.  Vertical lines are the mass ratios
  corresponding to the systems $^6$Li-$^{133}$Cs-$^{133}$Cs and
  $^6$Li-$^{87}$Rb-$^{87}$Rb.}
\label{figraa}
\end{figure}

These radii are for small mass ratios, ${\cal A} \ll 1$, related to the
Coulombic orbits discussed in connection with Fig. \ref{figenergy}.
The Coulomb radii are given analytically by
\begin{eqnarray}
\nonumber
 &&<r^2_{AA}> =  \frac{1}{2} \bigg(\frac{\hbar^2}{\mu_{AA} Z_{ef}^2 }\bigg)^2
 (n_r+\ell+1)^2\\
 &&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\bigg(5(n_r+\ell+1)^2 +1 -3\ell(\ell +1)\bigg)
  \nonumber \\ \label{radi}
 &&=\frac{3.1722\hbar^2{\cal A}(2n_r+1)^2 (5(2n_r+1)^2 + 7)}{16 m_A |E_{AB}|(1+{\cal A}/2)}
\end{eqnarray}
where we used Eq.(\ref{charge2}) and $\ell = -1/2$.

\begin{table}
\begin{tabular}{@{}|c|c|c|c|c|}
\hline
& ${\cal A}$ & $\langle\, r^{(BO)2}_{AA}\,\rangle/R_u^2$ & $\langle\, r^{(NI)2}_{AA}\,\rangle/R_u^2$ & $\langle r^{2}_{AA}\rangle/R_u^2$ \\
\hline
Ground & 0.01       & 0.023          & 0.029         & 0.006  \\
       & 0.02       & 0.047          & 0.056         & 0.011   \\ \hline
First  & 0.01       & 0.923           & 0.916          & 0.427   \\
       & 0.02       & 1.837           & 1.767          & 0.757    \\ \hline
Second & 0.01       & 6.510           & 6.268          & 3.911    \\
       & 0.02       & 12.955           & 12.254          & 7.201    \\ \hline
Third  & 0.01       & 24.359           & 23.381          & 16.594    \\
       &0.02        &48.478        & 48.534          & 32.219    \\
\hline
\end{tabular}
\caption{\label{II} Mean-square-radii in the Born-Oppenheimer
  approximation, $\langle r^{(BO)2}_{AA}\rangle$, from
  Eq. (\ref{radi}), and numerical results repectively with and without
  the interaction between the two $A$-particles, $\langle
  r^{2}_{AA}\rangle$ and $\langle r^{(NI)2}_{AA}\rangle$.  All these
  lengths are in units of $R_u \equiv \hbar/\sqrt{m_A |E_{AB}|}$.}
\end{table}

Let us first consider the radii of the states obtained for the
smallest mass ratios of ${\cal A}=$ 0.01, 0.02.  We compare with the
Born-Oppenheimer estimates precisely as we did for the energies in
Table \ref{II}.  As usual for universal structures the radii have the
opposite behavior to the energies, that is the mean square radii are
largest for the smallest binding energies.  The Born-Oppenheimer
results for the ground state are about $15$~\% smaller than obtained
from the full numerical calculation without an $AA$ interaction.
These deviations are again probably mostly due to the neglect of the
heavy-heavy kinetic energy operator on the light particle
Born-Oppenheimer wave function.  For the excited states the same
comparison show larger Born-Oppenheimer radii.  The observed opposite
tendencies of energy and radii are reflected in the formula in
Eqs.~(\ref{energy}) and (\ref{radi}) where the product is state
independent apart from the last factor in Eq. (\ref{radi}).

We now turn to the full calculation with sizable two-body energies
between all three pairs of particles, shown in the last column of
Table \ref{II}.  The ground state mean-square radius is a factor of
about $3.5$ smaller than derived from the Coulomb estimate in
Eq. (\ref{radi}).  This deviation is again consistent with the similar
larger binding energy obtained for that state with the same
interaction.  The first excited state is only smaller than the Coulomb
estimate by a factor of $2.2$.  The following two higher-lying states
have radii rather similar to the Coulomb estimates.  These states
extend beyond the Coulomb region and into the region of the more
confining large-distance Born-Oppenheimer potential from
Eq. (\ref{BOpotb}).  The effect is a comparably smaller spatial
extension which is very close to the results from Eq. (\ref{radi}).

\begin{figure}[htb!]
\epsfig{file=radiAB.eps,width=\columnwidth}
\caption{The same as Fig. \ref{figraa} when $\langle r_{AB}^2\rangle$
  replaces $\langle r_{AA}^2\rangle$. }
\label{figrab}
\end{figure}

At least one more distance is required to characterize the geometric
structure of the three-body system.  We first choose the distance,
$r_{AB}$, between unequal pairs of particles, $A$ and $B$.  The
mean-square radius is a quantum mechanical expectation value where the
two identical particles cannot be distinguished.  The results shown in
Fig. \ref{figrab} are therefore averages over distances between
particle $B$ and the two $A$-particles.  We notice first that the $AB$
mean-square distance is a flat or slightly increasing function with
decreasing small mass ratios for both ground and first excited states.
The higher-lying states show the opposite tendency of marginal
decrease.  Furthermore the $r_{AB}$-value in Fig. \ref{figrab} is much
larger than the $r_{AA}$ radius in Fig. \ref{figraa}, although with a
difference decreasing with excitation energy.  To understand this we
cannot turn to the Born-Oppenheimer calculations which only provides
information about the $AA$ system while the $B$-coordinate is
integrated out.  In the limit of small mass ratios the light
$B$-particle is very little concerned with the slow relative $AA$
motion.  The $B$-particle moves much faster and almost independently
in an orbit of much larger radius.

The behavior changes drastically with increasing mass ratios.
First the radii of the excited states diverge at their thresholds of
binding.  Second, the distance $r_{AB}$ decreases with increasing
${\cal A}$ for the two bound states.  The results for ${\cal A} = 1$
agree with the known mean-square ratio of about $70$ between the two
bound states \cite{nielsenpr2001}.  As ${\cal A}$ increases above $1$
we see that the radii only changes very little precisely consistent
with the same behavior as shown by the binding energies.

\subsection{Structure}

The sizes reflect the structures described by the corresponding wave
functions.  We would like, schematically and optimistically, to
reproduce the relative as well as the absolute sizes in
figs.~\ref{figraa} and \ref{figrab} by interpretation with appropriate
wave functions and/or geometric configurations.  We focus on a large
mass ratio of ${\cal A} = 100$ where the ratio between mean-square
distances of $AA$ and $AB$-particles is $1.67$ and $1.3$ for ground
and excited states, respectively.  A fully symmetric wave function
in hyperspherical coordinates corresponds to
$\rho^{-3/2} \exp(-\kappa \rho)$ where $\rho$ in the present case is
defined by
\begin{eqnarray} \label{rho}
\nonumber
(2m_A +m_B) \rho^2 &=&   m_A (\vec{r}_{A}- \vec{r}_{A'})^2
  +  m_B (\vec{r}_{A}- \vec{r}_{B})^2\\
  &&+  m_B (\vec{r}_{A'}- \vec{r}_{B})^2 \; ,
\end{eqnarray}
where the $\kappa$-value is given by $\hbar^2 \kappa^2 = 2 m_A|E_3|$.
This wave function has the correct large-distance asymptotic behavior.
The mean-square radius between a pair of particles, $i$ and $k$, is
then from this very simplified symmetric wave function found by
straightforward calculations to give
\begin{equation} \label{radsym}
  \langle r_{ik}^2\rangle = \hbar^2/(8\mu_{ik}|E_3|) \; ,
\end{equation}
where $\mu_{ik}$ is the reduced mass of particles $i$ and $k$
($i,k$=A,A',B).  We shall refer to it as a symmetric estimate as it is
obtained from the fully symmetric wave function.  This schematic
estimate from Eq. (\ref{radsym}) for the $AA$ mean-square radii of
ground and excited states is smaller by a factor of $2.6$ and $8.8$,
respectively.  The corresponding ratios for the $AB$ mean-square radii
are $3.1$ and $14$ which first of all reflect the different reduced
mass dependence.  These numbers are for squared radii, and the linear
distances, obtained by taking the square root, then only deviate by
factors varying from $1.5$ to about $4$.

Extreme asymmetric structures may be viewed with two particles in a bound
state at small distance and the third particle more loosely bound at a
larger distance.  To produce a finite $AA$-distance the configuration
must correspond to a structure like $A-(AB)$ (see Eq. \ref{radi}).
The mean-square radius is then $\hbar^2/(4\mu_{A,AB}|E_{A,AB}|)$ which
again is inversely proportional to the two-body reduced mass of the distant
$A$-particle relative to the stronger bound $AB$-entity.  However, the absolute
value of this estimate lies between those of ground and excited states
for a reasonable choice of $E_{A,AB} \approx 2 \times E_{AB}$.

Both extreme structures of total symmetry and extreme asymmetry fail
to reproduce the calculated moments.  Furthermore, the large
difference between ground and excited states indicate a sizable
structure variation.  The ground state is within a factor of two from
the symmetric estimate, whereas the excited state extends in size
substantially beyond the inverse three-body energy relation in
Eq. (\ref{radsym}).  The latter is more reminiscent of the huge
increase of radii found in the Efimov states which in three dimensions
are coherent superpositions of asymmetric geometric structure.

\begin{figure}[htb!]
\epsfig{file=radiA.eps,width=\columnwidth}
\caption{Same as Fig. \ref{figraa} when the mean-square distance,
  $\langle r_{A_{CM}}^2\rangle$, from C.M to particle $A$ replaces
  $\langle r_{AA}^2\rangle$, as ${\cal A}\rightarrow\infty$ $\langle
  r_{A_{CM}}^2\rangle\rightarrow\langle r_{AB}^2\rangle$.}
\label{figracm}
\end{figure}

Let us finally consider asymmetric structures where a finite average
radius is assumed between the strongest bound $AB$ two-body
substructure.  We first assume a given distance, $r_{AA'}$, between
the two $A$-particles, where the $B$-particle moves in a uniform
circular orbit of size $r_{A'B}$
around one of the $A'$-particles. Then we estimate the
mean-square distance $\langle r_{AB}^2 \rangle \approx (\langle
r_{AA'}^2 \rangle + \langle r_{A'B}^2 \rangle)/2$, where $1/2$ is from
adding the two symmetrized configurations. To reproduce the calculated
mean-square radii we need $\langle r_{AB}^2 \rangle / \langle r_{AA}^2
\rangle \approx 0.2, 0.5$ for ground and excited states.  These values
show a strong tendency towards symmetric configurations.

The sizes are often measured as distances relative to the total
center-of-mass. This only provides little additional information but
it is very illustrative besides serving as a consistency check on our
understanding of the underlying structures. We show in
Fig. \ref{figracm} the mean-square radial distance of particles $A$
from the center-of-mass.  The two limits of ${\cal A}$ differ very
much from each other.  Varying ${\cal A}$ from small to large values
causes the three-body center-of-mass to move from the center-of-mass of the
$AA$ system to the center of particle $B$.  This implies by geometric
reasoning that $\sqrt{\langle r_{A}^2 \rangle}$ approaches
$\sqrt{\langle r_{AA}^2\rangle}/2$ and $\sqrt{\langle r_{AB}^2\rangle}$
in these two limits, respectively.  These predictions are confirmed by comparing
Fig. \ref{figracm} with the results in Figs. \ref{figraa} and
\ref{figrab}, that is observing that $\langle r_{A}^2\rangle \rightarrow
\langle r_{AA}^2\rangle/4$ and $\langle r_{A}^2\rangle \rightarrow \langle r_{AB}^2\rangle$
for small and large ${\cal A}$.

The different behavior of $\langle r_{B}^2 \rangle$ is seen in
Fig. \ref{figrbcm}.  The geometry now predicts that $r_{B}$ should
vanish for large ${\cal A}$ since the total center-of-mass coincides
with the center of particle $B$.  In the other limit of small ${\cal
  A}$ we know that $\langle r_{AA}^2 \rangle \ll \langle r_{AB}^2
\rangle$ and consequently $\langle r_{B}^2 \rangle$ should approach
$\langle r_{AB}^2 \rangle$, since the $AA$ system looks like an entity from
far away where particle $B$ is located.  Again these predictions are
confirmed numerically by comparing Figs. \ref{figrab} and
\ref{figrbcm}.

\begin{figure}[htb!]
\epsfig{file=radiB.eps,width=\columnwidth}
\caption{Same as Fig. \ref{figracm}, but when the mean-square distance,
  $\langle r_{B_{CM}}^2\rangle$, from C.M to particle $B$ replaces
  $\langle r_{A_{CM}}^2\rangle$, as ${\cal A}\rightarrow\infty$
  $\langle r_{B_{CM}}^2\rangle\rightarrow0$.}
\label{figrbcm}
\end{figure}

There is a lack of experimental information concerning the sizes of
these systems in two dimensions. However, we are able to compare our
results for ${\cal A}=1$ with those from Ref. \cite{nielsenpr2001} as
follows. The spatial extension of a system can be measured by radial
moments. Often the second moment, the root-mean-square radius, is used
as an average measure.  Different distributions can be of interest as
exemplified by average charge- or mass-radii.  For a self-supported
system of bound particles the center-of-mass is conserved and it is
natural to measure all distances with respect to this point,
$\vec{R}_{CM}$.  The corresponding expectation value is related to the
root-mean-square radius, $R_{rms}$, of the mass distribution as
defined for our three-body system by
\begin{eqnarray}
\nonumber
 &&R_{rms}^2 \sum_{k=A,A',B} m_k = \sum_{k=A,A',B} m_k \langle (\vec{r}_{k}  - \vec{R}_{CM})^2\rangle \\   \label{rms}
 &&=\sum_{k=A,A',B} m_k \langle \vec{r}^2_{k_{CM}}\rangle  \; ,
\end{eqnarray}
where the vectors, $\vec{r}_{k_{CM}}$, by definition connects particle
$k$ and the center-of-mass of the three-body system.  The three-body
center-of-mass lies on the line connecting particle $A$ with the
center-of-mass of the remaining pair of $AB$-particles, and
analogousluy for particle $B$.  We therefore have the relations
\begin{eqnarray}
\langle r_{A_{CM}}^2 \rangle = \left(\frac{1+{\cal A}}{2+{\cal A}}\right)^2 \langle r_{A}^2 \rangle \;\; ,\\
\label{racm}  \;\; \langle  r_{B_{CM}}^2 \rangle = \left(\frac{2}{2+{\cal A}}\right)^2 \langle r_B^2\rangle \;,
\end{eqnarray}
which can be used to express $R_{rms}^2$ from Eq. (\ref{rms}) in terms
of different mean-square radii, that is
\begin{eqnarray}
\nonumber
 R_{rms}^2 &=& \frac{2 \langle r_{A_{CM}}^2 \rangle  +
 {\cal A}  \langle  r_{B_{CM}}^2 \rangle  } {2 + {\cal A}}\\
 &=& 2 \frac{(1 + {\cal A})^2 }{(2 + {\cal A})^3} \langle r_{A}^2 \rangle +
 \frac{ 4 {\cal A} }{(2 + {\cal A})^3} \langle r_{B}^2 \rangle \;.
\end{eqnarray}

For three identical particles, where ${\cal A} = 1$ and $\langle
r_{A}^2 \rangle = \langle r_{B}^2 \rangle$, this relation reduces to
\begin{eqnarray}
R_{rms} &=&\frac{2}{3} \sqrt{ \langle r_{A}^2 \rangle } \;,
\end{eqnarray}
with $\langle r_{A}^2 \rangle$ plotted in Fig. \ref{figracm}.

The two bound three-body states labeled $0$ and $1$ are in
\cite{nielsenpr2001} found to have root-mean-square radii given by
\begin{eqnarray}
 R_{rms}^{(0)} = 0.111 a = 0.125  \sqrt{\frac{\hbar^2}{m_A |E_{AB}|}}
\; \; , \; \;\\
 R_{rms}^{(1)} = 0.927 a  = 1.041  \sqrt{\frac{\hbar^2}{m_A |E_{AB}|}} \, ,
\end{eqnarray}
where the two-dimensional scattering length, $a$, is related to the
two-body energy $E_{AA} = E_{AB} = -4 e^{-2 \gamma} \hbar^2 /(m_A
a^2)$. These values match exactly our numerical results.

\subsection{Threshold behavior}

\begin{figure}[htb!]
\epsfig{file=DivergenceSecond.eps,width=\columnwidth}
\caption{Threshold behavior of different mean-square radii for the
  second excited state as a function of mass ratio.  The divergent
  mean-square radii multiplied by $E_3-E_{AB}$ reach constant values
  at threshold.  Vertical lines are the mass ratios corresponding to
  the systems $^6$Li-$^{133}$Cs-$^{133}$Cs and
  $^6$Li-$^{87}$Rb-$^{87}$Rb.}
\label{figdiv}
\end{figure}

We want now to investigate in more detail how the structure varies
with mass ratio when the energy of a state approaches its threshold
for binding.  We know that in three dimensions the sizes of bosonic
three-body systems diverge logarithmically at their three-body energy
thresholds. In contrast, the sizes remain finite for brunnian systems
(all pairs are unbound) with more than three bosons \cite{brunnian},
even at their corresponding thresholds of binding.  In these cases,
the thresholds are reached from the bound side by decreasing the
two-body attraction.  In the present study we
are decreasing the mass ratio, and in practice decreasing the
effective interaction between the constituents of the system.

Again we use the mean-square radii as indicative measures.  We focus
on the second excited state which is the lowest-lying state with
diverging size.  Its behavior is repeated by the other excited
states, which we therefore do not need to discuss in this paper.  We
show in Fig. \ref{figdiv} the results as functions of ${\cal A}$ for
the different radii of the second excited state.  We multiply the
radii by the energy deviation from the two-body threshold,
$|E_3-E_{AB}|$, in order to extract the behavior of the divergence.  The
striking result is that this product converges towards a constant as
the threshold is approached. Thus, the divergence of the mean-square
radius is inversely proportional to $E_3-E_{AB}$.  This behavior is
well-known for weakly bound two-body halos in three dimensions
\cite{jensenrmp2004}, and shown in Ref. \cite{nielsenpr2001} to be
valid also in two dimensions.  The implication is that the threshold
structure must correspond to one particle moving away from a bound
and spatially confined subsystem, which in our case only can be a bound
dimer.

Two structures are possible at threshold, that is, either particle $A$
or particle $B$ is ejected while the remaining pair settles in their
ground state.  In both cases the threshold structure resembles
two-body systems, that is, $(AA)-B$ or $(AB)-A$ with corresponding
reduced masses $\mu_{AA,B} \approx 0.18 $ or $\mu_{AB,A} \approx
0.54$, where the numbers are obtained with a threshold value of
${\cal A} \approx 0.2$.

These structures would according to Ref. \cite{nielsenpr2001} lead to
the two-body divergence, $<r^2_{ik}> = \hbar^2/(3\mu_{ik}|E_{th}|)$,
where $\mu_{ik}$ and $|E_{th}|$ are two-body reduced mass and the
vanishing threshold energy, respectively.  This means that the curves
in Fig. \ref{figdiv} for $m_A<r^2_{ik}> |E_{th}|/\hbar^2 =
m_A/(3\mu_{ik})$ can be either $1.85$ or $0.62$.  Thus, $(m_A
<r^2_{AA}> |E_{th}|/\hbar^2, m_A<r^2_{AB}> |E_{th}|/\hbar^2)
\rightarrow (0.0,1.85)$ or $(m_A<r^2_{AA}> |E_{th}|/\hbar^2,
m_A<r^2_{AB}> |E_{th}|/\hbar^2) \rightarrow (0.62,0.62)$ are the two
corresponding two-body threshold structures, where either $B$ or $A$
are ejected.  The two identical finite values in the latter case are
due to the same distance between the ejected $A$-particle and both the
remaining $A$ and $B$ particles in a bound state.

Obviously the best match to the results displayed in
Fig. \ref{figdiv} is ejection of one of the $A$-particles.  In the
calculated expectation value is contained an average over the
distances between the two $A$-particles and particle $B$.  One of
these distances remains finite and does not contribute except through a
reduction of the probability by a factor of $2$.  This accounts for the
value of $\frac{m_A<r^2_{AB}>}{\hbar^2}(E_{3}-E_{AB})=0.6$, being only about two thirds
of $\frac{m_A<r^2_{AA}>}{\hbar^2}(E_{3}-E_{AB})=0.9$, at threshold.

\section{Summary, Conclusions and Perspectives}
\label{conclusions}

We studied the structures of three-body systems in two spatial
dimensions. We concentrate on asymmetric systems formed by two
identical bosons and a third particle.  We assume spin-independent
interactions and all results are therefore also valid for two
identical fermions with spatially symmetric wave functions.  We use
two-body zero-range interactions and the Faddeev equations in momentum
space to solve the three-body problem.  Universal properties are then
emphasized due to the vanishing interaction range.  The three-body
system is characterized by two masses and two zero-range strength
parameters chosen to reproduce specified two-body energies.  The
equations only depend on two parameters, that is mass and energy
ratios.  In this paper we essentially only investigate the mass
dependence for equal two-body energies.

Our aim is to obtain information about universal structures.  We
calculate second order radial moments corresponding to two independent
distances within the three-body system.  Correlations between these
quantities are very indicative of the underlying dominating structure.
Relatively low orders of radial moments are substantially easier to
compute than the density distributions, especially when the starting
point is in momentum space.  We therefore first derive suitable
expressions for the desired radial moments expressed in terms of
single first-order derivatives of integrals over momentum-space wave
functions.

We first calculate and analyze the energy spectrum as the primary
characterizing quantity of any quantum system.  We keep equal two-body
energies. The number of bound states are always finite but varies from
2 and upwards, diverging as the relative mass of the distinguishable
particle decreases from very large to approaching zero.  Only
precisely two bound states are present as for equal masses until the
mass ratio has decreased to about $0.2$.  Then a third state appears
followed by numerous other states as the mass ratio decreases towards
zero.  This agrees qualitatively with similar previous observations,
as well as quantitatively for the known case of equal masses.

We analyze the spectra for small mass ratio by use of the
Born-Oppenheimer approximation of the potential.  The effective
interaction generated by the light particle has Coulomb behavior for
small distances between particles of the heavier pair.  Combining with
the kinetic energy operator in two dimensions the Coulomb energy
spectrum appears for an angular momentum quantum number equal to
$-1/2$.  The small-distance region is the most attractive region,
where the short-range interaction between the two heavy particles also
is strongest.  Consequently this is where the lowest states are
located.

Considering this Coulomb potential calculated for small mass ratios
from the Born-Oppenheimer approximation, the three-body energies and
heavy-heavy mean-square radii are derived and compared to the full
calculations.  We see that the numerically obtained ground state is
more bound with a smaller radius than arising from the Coulomb
estimate.  This is due to the additional attractive heavy-heavy
zero-range interaction.  The first few of the following higher-lying
states still feel the heavy-heavy short-range interaction but they
have energies and radii closer to the Coulomb estimates. This reflects
that the effect of the zero-range attraction quickly decreases as the
states move to larger distances.  The higher-lying excited states
extend spatially beyond the Coulomb-like potential, but all bound
states are eventually confined within the exponentially decaying
Born-Oppenheimer potential at large distance.

The mean-square radii are to a large degree a reflection of the
energies, that is, small energy corresponds to a large radius and vice
versa.  This is the general observation for universal properties
when all angular momenta are zero or vanishingly small.  By comparing
two independent lengths in the same three-body system we can generate
a geometric picture of an underlying schematic structure.  The
variation with mass ratio is substantial. The two remaining states for
moderate and large mass ratios appears to be consistent with the light
particles at roughly twice the distance between light and heavy
particles.  For small mass ratios we find for both ground and excited
states that the light particle is located at distances much larger
than the distance between the two heavy particles

The disappearance of excited states into the continuum for threshold
values of the mass ratio is related to specific bound-state structures
close to the thresholds.  The mean-square radii here prove themselves
to be efficient measures of the corresponding structures.  We show
numerically that these mean-square radii diverge inversely
proportional to the deviation of the three-body energy from the
two-body threshold energy.  Comparing distances between identical and
unequal particles we conclude that the threshold structure corresponds
to one of the identical particles far away from a bound dimer of the
other two particles.  Thus, this far-away particle is being ejected
into the continuum.  It should be possible to confirm these predictions
by cold atom experiments where three-body atomic systems of
$^6$Li-$^{133}$Cs-$^{133}$Cs and $^6$Li-$^{87}$Rb-$^{87}$Rb among
others, are already produced in laboratories.

The technique we developed is based on the schematic, but hugely
popular, zero-range interaction treated in momentum space.  This
interaction is the extreme limit of a short-range potential, and
therefore in general the simplest tool to provide quantitative
information about universal properties.  The present method to
calculate mean-square radii is a special application to derive
structure information through expectation values.  The method is
simpler and faster than using brute force to obtain wave functions
which afterwards supply the desired observables. An interesting
continuation of the present study is to extend it to more than
three atoms and see how the universal relations derived in
\cite{hammerprl2004} would change for a mass-imbalanced system.

The method can be used directly to investigate the dependence of radii
on the strengths of the two-body interactions.  The formalism can be
relatively easily generalized to apply to three-body systems where one
or more two-body subsystems are unbound, whereas the three-body system
still remains bound.  All applications may be extended to three
distinguishable particles with their different masses and interaction
parameters.  The application to two spatial dimensions is interesting
because this is a rigorous limit with unique properties compared to
other geometries.  However, also both one and three dimensions are equally
accessible, and a continuous variation between dimensions also becomes
increasingly interesting.

\acknowledgments
This work was partly supported by funds provided by the Brazilian agencies
FAPESP (2016/01816-2), CNPq and CAPES (88881.030363/2013-01).

redpajama_set_name: RedPajamaArXiv


text:
Complete Sun Rcaa & Monument Releases 1956-62
Von Roy Orbison
Theme Grammy Winning Artist
Roy Orbison was one of the most talented and distinctive singers and songwriters of the rock 'n' roll era and beyond, with a unique and original approach to writing songs and a remarkable and unmistakeable voice. He was signed to the Sun label alongside Elvis Presley and Carl Perkins, but only had one minor hit during that period despite making some fine rockabilly records, and it was not until joining the Monument label after recording briefly for RCA during 1958 that he hit the big time with the landmark No. 1 "Only The Lonely", following that with a string of memorable Top 10 classics like "Blue Angel", "Running Scared", "Crying" and "Dream Baby", all included here along with his other chart entries from that time. This great-value 58-track 2-CD set provides a comprehensive overview of the formative years of his career, comprising the A and B sides of all his singles on Sun, RCA and Monument through to the end of 1962, plus all the tracks from his three albums from that period, "At The Rock House", "Lonely And Blue" & "Crying", which were not otherwise released on singles. It charts his progress towards establishing the style and formula which made him one of the most popular and respected artists of his generation and is a fine showcase for an exceptional talent.
Künstler: Roy Orbison
Titel: Complete Sun Rcaa & Monument Releases 1956-62

redpajama_set_name: RedPajamaCommonCrawl


Whether you want to get the most out of the bulb or the fronds, these 53 recipes for fresh fennel are guaranteed to steal your hearts.
Move over, celery. Another bright, crunchy stalk's stolen our hearts—and this one tastes like licorice. Whether you want to get the most out of the bulb or the fronds, these 53 recipes will totally make you fall for fresh fennel all over again.
On a warm, sunny day, pasta calls for fresh seafood, a big handful of herbs, and punchy lemon. All that's missing is the beach house.
This rich and juicy lamb burger is bliss.
Be choosy when picking your produce, and look for a new-harvest olive oil for this, which will lend a bright, robust flavor.
It's all in the details with this fennel salad recipe. Taste after each step to reach the perfect balance between salt and acid.
Want to build big flavor fast? Jarred marinara gives you a head start. Be sure to doctor it a bit for a homemade taste. Here we use fennel seeds and sliced fennel to serve with flounder.
Don't bother trying to divide this fillet into tidy portions. Instead, use a spoon to break it into perfectly imperfect pieces.
Drying out and toasting the cornbread in the first step guarantees a crisp, flavorful dressing.
This impressive-looking roast is easy to carve once you get started.
The type of fish you use is less important than its quality. Black bass, striped bass, and fluke all translate well; ask your fish guy for what's freshest.
The surface scoring used here helps to create a cutlet that will be evenly thin when pounded out. Try it on beef or veal, too.
You know what most chicken salads are missing? Crunch. Not this one!
This sandwich gets better and better the longer it sits (okay, to a point); assemble it after breakfast and eat it when you've worked up an appetite.
Poaching the shrimp with their shells on locks in their flavor.
Crunchy, bright, and hearty in all the right places, this salad just earned its spot in your weekly meal rotation. We're sure of that.
Add the vegetables partway through braising to ensure they will have enough time to be infused with meaty flavor without overcooking. Serve with a loaf of your favorite crusty bread for sopping up the juices.
Think of the crunchy, granola-ish sesame clusters as seedy croutons for this juicy and bracing salad.
The same technique and proportions will work for a pork chop, lamb chops, or any other steak you like.
A lighter, smoother, more grown-up guacamole.
Intentionally burning vegetables—it's a thing! The burnt onion powder adds a smoky, slightly bitter finish to the rich dip, and is responsible for the black part of this tuxedo-inspired crudite.
To make addictively crispy chicken skin, turn it often in the skillet, until it looks like bacon.
No need to prep everything ahead: Get your chopping in while the fennel cooks. Just keep an eye on the pot!
Opposites attract: This bright, sprightly salad is just the thing to cut through the richness of succulent pan-fried pork.
Okay, fine, you've never heard of baobab—but that's about to change. (Remember when no one knew how to pronounce quinoa?) You'll find the powder from the tangy superfruit stocked at the health food store.
Any type of lentil will work in this weeknight soup. Or, substitute a couple cans of cannellini beans in their place.
Consider the crunch: You want to choose a vegetable that's naturally firm, like fennel. The fresher it is, the crisper it will stay.
These hot and sour pickles make a perfect garnish for Bloody Beers (click for recipe).
This caponata uses a technique called a "cold fry." Instead of frying the vegetables in preheated, hot oil, you combine the vegetables and room temperature oil in a pot and heat them up together. The benefit of this method is that you use less oil than you would normally use when you "hot fry" and there is very little splatter because the food and oil are always at the same temperature. And, like "hot frying," you still get a golden crispy exterior and soft, creamy interior but with no mess.
We've said it before and we'll say it again: When seafood gets top billing, it has to be top quality. Don't even attempt this stew from Rose's Luxury with less-than-stellar stuff.
This riff on traditional tabbouleh brings home all of the flavors you want, but adds the crunch you deserve.
Red rice is a short-grain, nutty-tasting type; you can use short-grain brown rice in its place.
Make sure to have fresh bread on hand to soak up any extra dressing in the bowl.


text:
Q: Does printf() convert its arguments to string like cout? I start to learn C++ from C. Recently, I have just read a tutorial book about C++. In section Introduce to streams, the book has noted:

The  << operator is overloaded so that the operand on the right can be
  a string or any primitive value. If this operand is not a string, the
  << operator converts it to string form before sending it to the output
  stream.

So I wonder whether printf() function in C has the same effect. And if it doesn't, please tell me about the differences between both of them.

A: Well, of course it has to somehow generate a string representation of each argument, that is needed in order to have something to print. Printing involves sending streams of characters to an output device after all, you can't print unless you have a sequence of characters.
The printf() function uses the formatting string to control how to interpret each argument in order to create the character representation, and also how to format that representation when output.
Note that no "conversion" of the arguments happens that is visible externally, of course. There's no way
printf("%d\n", 47);

can make that 47 into a string in place; C uses call by value so the function only gets a copy of the value, and it then uses the type information implicit in the %d conversion specifier to figure out how to generate the two characters '4' and '7' that make up the printed representation.

A:
So I wonder whether printf() function in C has the same effect.

Both C and C++ uses streams for output. In C it is stdout and in C++ it is cout.
Though it is not evident from  the statement printf writes to standard output(stdout) say a terminal.
In case of cout, it is evident from a statement itself, where the output is going.
Some subtle differences
With cout you might need to include an additional header - say iomanip - and use some functions - say setw() - to have fine formatting where as in printf you  rely on format string.
Performance - Each has its own advantage depending on what you print and where you print. I borrowed this point from here.
Another similarity
Both C++ and C standards mention nothing about the order of evaluation of function arguments. So you must not try fancy stuff with functions. For example neither should you do
printf(%d%d",++i,i++); // The behaviour is undefined.

nor should you do
cout<<++i<<++i; // The behaviour is undefined.

Note:
Remember that the c streams are available in C++ if you include the necessary headers.

redpajama_set_name: RedPajamaStackExchange


text:
Revivals in the New Testament
The historical records of the New Testament cover approximately 60 years. Half of these cover the ministry of Jesus and half present the growth of the early church. It is not too difficult to find authentic revival within these records because it is everywhere.
For 400 years the Old Testament saints had been waiting for the coming of their Messiah King and His kingdom. First, when John the Baptist, then Jesus. burst on the scene they declared that the Kingdom of God was at hand. At last God's manifest, dynamic, observable and powerful kingdom had come. This kingdom rule was now available and accessible and was suddenly invading the world of men. What we call 'revival' was simply the multi-facetted in-breaking of the Kingdom of God. It brought a sense of God's presence, the conviction of sins, acts of awesome power that swept multitudes into the new, vibrant faith called Christianity. After Jesus' ministry the early church carried this breathtaking ministry into the known world. The Gospel was preached, signs and wonders were exhibited, conversions were realized and Christian churches began to fill the landscape. This is the stuff of revival is made of!
The entire historical record is really an account of authentic revival and because it is unquestionably Biblical, today's church would do well to re-examine its pages, discern the process and return to the pattern of revival it portrays. It depicts a new era of the Spirit, prophesied in the Old Testament, and personified in the Person of Jesus. It was the Spirit that fell upon those early disciples that thrust them into the world to blaze a trail of revival wherever they went.
The Revival under John the Baptist
The first discernable revival in the New Testament occurred in the ministry of John the Baptist. It is as if the Holy Spirit had gone underground for 400 years since Malachi, the last Old Testament prophet, had uttered the words of God. Now He re-emerged in the ministry of this strange looking revivalist. The only explanation for the awakened, spiritual fervour of the great crowds that attended his wilderness meetings was that God was on the move. Mark says, `The whole Judean countryside and all the people of Jerusalem went out to him' (Mk. 1:5). He was clothing and diet was atypical and his message was aggressively incisive and convicting. He addressed his hearers, of whom many were respected religious leaders, as `you brood of vipers', comparing them to snakes fleeing from a forest fire (Mt. 3:7; Lk. 3:7), and called for confession of sins and repentance before baptising them in water. Even this was an affront to them as baptism was usually offered to the 'unclean' Gentiles who requested inclusion in the Jewish community.
In this powerful movement of spiritual awakening produced by the prophetic preaching of a man of God, multitudes had their spiritual appetites whetted as they caught the first wind of the Spirit.
Though this was a genuine move of God in reviving power John made it clear that he was only the 'fore-runner,' whose ministry was to 'prepare the way of the Lord.' When He arrived He would dispense the Spirit with power for He was both the Lord and the Lamb who would take away the sins of the world. What would soon become familiar emphases in revival ministry we see in seminal form in John's ministry: the Lordship of Christ, the call for repentance, the promise of the Spirit and the supernatural attraction of multitudes to a new spirituality.
Revival in the ministry of Jesus
It must be said that revival manifestations and features accompanied Jesus throughout His three and a half year ministry. The Holy Spirit was upon Him 'without limit' (John 3:34). Multitudes were attracted to Him and sometimes thousands gathered to hear his life-changing, hope-producing words and to experience the hand of God upon their lives. Revival scenes accompanied Him everywhere. News spread rapidly the great crowds that flocked to Him were met with supernatural power which healed their diseases and delivered their enslaved. Clearly the Kingdom of God was present and revealed itself in revival power.
The ministry of the 12 and the 70
One gets the idea that Jesus could have gathered such a massive following that could have swept the Romans from Palestine. But Jesus concentrated His time and His ministry on His workers, His disciples. These He trained in revival ministry in preparation for the future, international church planting role they were destined to play. Revival phenomena accompanied them as they took their first tentative steps into the ministry of the Spirit. (Mt 10:1-8; Mk 6:7-13; Lk 10:1-20). Preaching the Kingdom of God, healing the sick and casting out demons which prefigured the explosive expansion of the church and the fall of Satan are true marks of Biblical revival.
Pentecost – the model revival – Acts 2
The events that occurred on the day Pentecost are often seen as the biblical pattern by which all subsequent revivals may be judged. Charles Finney said, 'The antecedents, accompaniments and results of revivals are always substantially the same as in the case of Pentecost.' Albert Barnes added, 'That day which shall convince the great body of confessing Christians of the reality and desirableness of revivals will constitute a new era in the history of religion and will precede manifestations of power like that of Pentecost.' Arthur Wallace in his 'In the Day of Thy Power' p.57 adds, 'As Pentecost was the first distinctive outpouring of the Spirit, a careful examination of that great event will reveal the distinctive features of every subsequent outpouring. Let Acts 2 be the textbook.'
All the marks of authentic revival are there: Initiated by the Spirit, prepared for by consecration and prayer, the surprising suddenness of His visitation, an awesome God-consciousness, anointed and Christ-centred preaching, supernatural manifestations, terrifying conviction and divine magnetism.
The results of the revival are equally magnificent. 'The infant Church ... was characterised by vigorous life, sustained growth, new accessions of spiritual power through new infillings of the Holy Spirit, and the presence of God experienced in an unusual way in miracles of both blessing and judgment. Problems arose, mainly caused by the large numbers of new converts from varying backgrounds; opposition and persecution occurred, but it failed to dampen the enthusiasm of the Christians.' (Davies, I Will Pour Out my Spirit. p.50)
Revival in Samaria – Acts 8
The first disciples in Jerusalem faced dreadful persecution but, as they moved out of their zone of fear, the revival went with them `Those who had been scattered preached the word wherever they went' (Acts 8:4). There was no holding back what they had seen, heard and experienced of God's power. When the deacon, Philip, arrived in Samaria it was as if he had lighted the blue touch-paper of revival. Many miraculous signs occurred and great numbers of people were converted as He preached Christ there. Evil spirits were expelled from victims, many experienced physical healing and the whole city was filled with the joy of salvation.
When the apostles arrived from Jerusalem there was a further outbreak of the reviving Spirit when they laid hands on the new converts. It is interesting to note that they 'heard that Samaria had accepted the word of God.' (Acts 8:14) conveying the idea that a large proportion of the city had come to Christ.
Revival at Caesarea – Acts 10-11
Peter's short visit to a single household in Caesarea brought a revival to a single extended family, which is probably the type of revival we should all be seeking God for! This time the event was clearly initiated by God. It had to be. Whoever would have volunteered to go into the 'forbidden territory' of an 'unclean' Gentile household and deliver what had been up to this point, a Jewish message? So God took the initiative and sent an angel to a Roman centurion called Cornelius and gave an apparently grotesque vision to Peter, the prince of preachers. The unusual tactics worked and Peter arrived to speak to a prepared group of God-fearing but unconverted people.
Peter almost reluctantly recounted his vision and then briefly spoke out a bare outline of the Gospel. Before he could conclude or make an appeal, the Holy Spirit fell on the hearers. They were instantaneously converted, spoke in tongues and worshipped God! This was revival!
Peter's subsequent defence of these events reveal that it was the arrival of the outpoured Spirit which bore such close resemblance to his own experience at Pentecost, that convinced him that this was an act of God. He also concluded, therefore, that 'God had granted repentance unto life' to the Gentiles there (Acts 11:15-18)
The distinct elements of this revival were:
God initiated the event by an angelic visitation and a vision.
Cornelius gathered a crowd and prepared them to receive something from God.
Peter arrived and preached the gospel
The crowd repented and were converted
The Holy Spirit was poured out on them
They spoke in tongues and praised God.
Antioch – Acts 11:19-30
The significance of the church in Antioch is important to grasp in order to understand the expansion of the early church. The persecution had pushed out many new and zealous converts beyond Samaria and Caesarea, to this strategic town. Josephus called it the third city of the Roman Empire, second to Alexandria in North Africa and Rome itself, the centre of the Roman world. Antioch housed at least 500,000 cosmopolitan people, having a large number of Jews because they were offered equal citizenship there. There were also Orientals from Persia, India and even China, but mostly they were Greeks.
It was a very strategic place to establish the Gospel and the Holy Spirit earmarked this place to launch the missionary thrust of the church, which would invade the nations. From here the great Apostle Paul was sent out on the first of three powerful missionary tours, winning souls and planting churches as he went. The centre of Christianity effectively shifted from Jerusalem to Antioch.
The gospel was first preached in Antioch initially to Jews only. Then some Christians from Cyprus and Cyrene broke the Gentile barrier and witnessed to Greeks as well. Great numbers of Gentiles believed and came to Christ. (Acts 11: 21)
Although there are no details recorded an amazing event had occurred – a beachhead had been captured by God from which Paul and others could invade more Gentile territories. The work continued to grow through the arrival of Barnabas from Jerusalem, as `a great number of people were brought to the Lord' through his ministry (Acts 11:24). It was like a second wave of revival power as another ministry came in to fan the flames further. It was out of this spiritual awakening that the first planned missionary thrust began (Acts 13:1-3).
Paul Carries the Revival Fire on the First Missionary Journey – Acts:13-14
Born again in revival, then immersed in Antioch's revival clearly had an effect on Paul. His first missionary journey with Barnabas was more than a successful evangelistic trip. It was attended by signs and wonders, apostolic preaching, deep conviction and innumerable conversions. Paul's influence was astonishing. Government officials, Jewish synagogue congregations and 'great numbers' of ordinary people, responded to his message, so much so that in Pisidian Antioch 'almost the whole city gathered to hear the word of the Lord' (Acts 13:44) and 'the word of the Lord spread though the whole region.' (Acts 13:49) Churches were planted and the disciples added 'were filled with joy and with the Holy Spirit.' Acts 13:52)
In Iconium `they spoke so effectively that a great number of Jews and Gentiles believed' (Acts 14:1); in Lystra, hostilities rose and there was an attempt to kill the powerful preachers (Acts 14:8-19); in Derbe `they preached the good news ... and won a large number of disciples' (Acts 14:21). When they returned to Syrian Antioch they had an amazing story to tell as they `reported all that God had done through them'. (Acts 14:27)
Paul's Second Missionary Trip Continues in Revival Power – Acts 15:36-18:22
Along with the first missionary journey the second one reads like a revival report. After visiting a few earlier church plants in Syria and Cilicia, Derbe and Lystra, Paul and Timothy made their way to Troas where Paul had a remarkable vision of a Macedonian man beckoning to him to come over the water to help him. They sailed immediately to Philippi, the capital city of Macedonia and soon found a group of praying people, amongst whom was a woman called Lydia. Her conversion led to her whole household being converted and baptized. The deliverance of a demonised girl caused the authorities to arrest Paul but a supernatural visitation in the night which caused an earthquake, opened the prison doors and loosened their chains eventually resulted in the jailers conversion together with his household too!
These events carry the signs of authentic revival: Divine initiation, supernatural guidance, effective proclamation of the gospel, acts of power and divine conviction and conversions.
In Thessalonica, Paul preached to them `not simply with words, but also with power, with the Holy Spirit and with deep conviction', (1 Thess 1:4-5) as a result of which a dynamic church was established which `became a model to all the believers in Macedonia and Achaia' and from which `the Lord's message rang out ... in Macedonia and Achaia (and their) faith in God has become known everywhere' (1 Thess 1:4-8). People, churches and movements that have their roots in revival often have a wide and lasting effect on their environments.
In Corinth (Acts 18:1-17) Paul spent over a year and a half regularly teaching and preaching until 'many of the Corinthians who heard him believed and were baptised.' (Acts 18:8) Establishing a powerful church in this strategic, if worldly, Greek port was nothing less than a supernatural act of God – 'a grande finalé' to the second missionary journey!
Paul's Third Missionary Trip Further Displays Revival Power – Acts 18:23-20:38
Paul's ministry in Ephesus was the high point of this mission and was outstanding in revival power (Acts 19:1-41). An awakening took place in this centre of pagan worship and occult practices, which resulted in great numbers coming to faith. The work of God was so vast and effective that 'all the Jews and Greeks who lived in the province of Asia heard the word of the Lord. (Acts 19:10) Many miracles and exorcisms occurred (Acts 19:11-20) resulting in extraordinary growth. Acts 19:20 'In this way the word of the Lord spread widely and grew in power.'
There is therefore undisputable evidence of widespread spiritual awakening throughout the New Testament period and which should be used to inspire faith and authenticate any claimed 'revival' in our world today.
R. E. Davies, I Will Pour Out My Spirit, 1992
Stuart Piggin, Firestorm of the Lord, 2000
Tony Cauchi 2009