text:
\section{Introduction}

Study of electron and positron scattering off light atomic targets (like (anti)hydrogen atom and helium cation) is of  fundamental importance for atomic physics. These colliding systems represent genuine three-body Coulombic systems with variety of channels and with reach resonant structure of scattering cross sections.
Many calculations have been performed by now
for elastic and reactive scattering in these systems based on different computational platforms. Among others the variational methods \cite{bhatia74, humber97, humber01, gien99}, close coupling \cite{mitroy95, kadyr02} and
hyperspheric close coupling \cite{igara94, ward12} approaches and  methods of Faddeev-Merkuriev equations
 \cite{kvits95, hu99, Papp-Yak2001, yak07, lazau17}
 are in extensive use.
Other group of methods  exploit complex rotation of coordinates \cite{Res97} and complex rotation with potential splitting \cite{Volk09}, \cite{Yak10}, \cite{Volk11}, \cite{Volk15}, \cite{Yar15}, \cite{Yar17}. The latter is especially designed to treat the asymptotic Coulomb interaction and have been successfully applied for elastic electron (positron) hydrogen and electron (positron) helium cation collision.

Besides the features related to the long range character of the Coulomb interaction the collision in systems $e^+ - \mbox{H}$, $e^- - \bar{\mbox{H}}$ and $e^+ -\mbox{He}^+$ exhibits the fundamental rearrangement phenomenon of positronium
(electron-positron bound state) formation. For such a case the solution methods should be capable to efficiently represent the solution for all asymptotic fragmentations.
Highly accurate calculations of scattering processes and resonances in such systems of charged particles require {sophisticated}  methods from both theoretical and computational points of view.
The Faddeev equations \cite{fadd93} were designed especially to fulfill this requirement first for short-range interparticle interactions and then an extension on the long-range Coulomb case was made \cite{merkur80} . This extension, called the Faddeev-Merkuriev  (FM) equations, employs an idea of splitting the Coulomb potential into the interior and long range tail parts leading to the mathematically rigorous boundary value problem which solution is equivalent to the solution of the Schr\"odinger equation~\cite{fadd93}.
This approach perfectly suits  for the computationally difficult detailed low energy elastic and reactive scattering calculations in three-body Coulomb systems \cite{kvits92}, \cite{hu99}, \cite{Papp-Yak2001}.
Typically, splitting of the Coulomb potentials is done by an artificial functional parameter in the three-body configuration space, the Merkuriev cut-off function.
Recently, we showed in \cite{grad16}, that the splitting procedure can be made simpler and easier for implementing into FM equations.
 In this contribution we give practical advices for the choice of the cut-off function in the form of a smoothed Heaviside step function in the two-body configuration space, thus minimising the number of free parameters of the splitting procedure.
Following these advices allows us to produce more accurate results with smaller computational efforts.
Additionally, a special numerical technique for handling the matrices, known as ``tensor trick'', drastically reduces the computational complexity of the numerical solution of the FM equations~\cite{schelling89,roud00}.

Here, the formalism of FM equations has been used to calculate all possible S-wave cross sections in $e^-e^+\bar{p}$ and $e^+e^-\mbox{He}^{++}$ systems in the low-energy region.
Even though there are many calculations available in the literature, there is some lack of high-precision and detailed results.
A special emphasis is made on antihydrogen formation by antiproton impact of positronium which is currently used in experiments on antimatter at CERN~\cite{lazau18} (and references therein).

The paper is organised as follows. In section 2 we give necessary portion of the three-body FM equations formalism with splitting of the long-range Coulomb potential. Section 3 describes the solution technique of the resulting FM equations in the total angular momentum representation in 3D configuration space. Section 4 contains results of calculations of low-energy reactive scattering in $e^-e^+{\bar{p}}$ and $e^+e^-\mbox{He}^{++}$ systems. Last section 5 concludes the paper.

Throughout the paper we use atomic units.

\section{Theory}

We consider the system of three spinless nonrelativistic charged particles of masses $m_{\alpha}$ and charges $Z_\alpha$, $\alpha=1,2,3$. In what follows the set of indices $\{\alpha$, $\beta$, $\gamma\}$ runs over the set $\{1,2,3\}$ enumerating particles and it is also used for identifying the complementary pair of particles,
since
in the partition $\{\alpha(\beta \gamma)\}$ the pair of particles $\beta\gamma$ is uniquely determined by the particle $\alpha$.
The standard Jacobi coordinates are defined for a partition $\alpha(\beta\gamma)$ as relative position vectors between the particles of the pair $\beta\gamma$ and between their center of mass and the particle $\alpha$.
In applications it is convenient to use reduced Jacobi coordinates $\bm{x}_\alpha$, $\bm{y}_\alpha$ which are Jacobi vectors scaled by factors $\sqrt{2\mu_{\alpha}}$ and $\sqrt{2\mu_{\alpha(\beta\gamma)}}$, respectively, where the reduced masses are given by
\begin{equation}
\mu_{\alpha} = \frac{m_\beta m_\gamma}{m_\beta+m_\gamma},\quad \mu_{\alpha(\beta\gamma)}=\frac{m_\alpha (m_\beta+m_\gamma)}{m_\alpha+m_\beta+m_\gamma}.
\end{equation}
For different $\alpha's$ the reduced Jacobi vectors are related by an orthogonal transform
\begin{equation}
\bm{x}_\beta=c_{\beta\alpha}\bm{x}_\alpha + s_{\beta\alpha}\bm{y}_\alpha \ \
\bm{y}_\beta=-s_{\beta\alpha}\bm{x}_\alpha + c_{\beta\alpha}\bm{y}_\alpha,
\label{JacobiTrans}
\end{equation}
where
$$
c_{\beta\alpha}=-\left[\frac{m_\beta m_\alpha}{(M-m_\beta)(M-m_\alpha)}\right]^{1/2} \ \
s_{\beta\alpha}=(-1)^{\beta-\alpha}\mbox{sgn}(\alpha-\beta)(1-c^2_{\beta\alpha})^{1/2}
$$
and $M=\sum_\alpha m_\alpha$.
In what follows where it is due, it is assumed that $\beta$ Jacobi vectors are represented through $\alpha$ vectors via (\ref{JacobiTrans}).

In these reduced Jacobi coordinates FM equations for three charged particles \cite{fadd93} read
\begin{eqnarray}
\label{MFeq}
\{ T_\alpha  +  V_\alpha(x_\alpha) +
\sum_{\beta\ne\alpha}V_\beta^{(l)}(x_\beta,y_\beta) &-& E \} \psi_\alpha(\bm{x_\alpha}, \bm{y_\alpha}) = \nonumber \\
  &-&  V_\alpha^{(s)}(x_\alpha,y_\alpha) \sum_{\beta\ne\alpha}\psi_\beta(\bm{x_\beta}, \bm{y_\beta}).
\end{eqnarray}
Here $T_\alpha\equiv-\Delta_{\bm{x_\alpha}} - \Delta_{\bm{y_\alpha}}$ are the kinetic energy operators. Throughout the paper the magnitude of a vector $\bm{x}$ is denoted by $x$, i.e. $x=|\bm{x}|$, for a unit vector $\bm{x}/x$ the notation $\hat{\bm{x}}$ is used
The potentials $V_\alpha$ represent the pairwise Coulomb interaction $V_{\alpha}(x_\alpha)=\sqrt{2\mu_{\alpha}}Z_\beta Z_\gamma/x_\alpha$ ($\beta,\gamma\ne\alpha$), although a short-range (decreasing as $1/x^2_{\alpha}$ or faster as $x_\alpha \to \infty$) potential can also be included in the formalism. For definitiveness and in view of further application of the formalism to the concrete systems of this paper we assume that one of the Coulomb potentials is repulsive whereas the other two are attractive.
We chose assignments such that the inequalities $Z_1Z_2>0$, $Z_1Z_3<0$ and $Z_2Z_3<0$ hold,
thus $V_3$ is always  repulsive.
The potentials $V_\alpha$ are split into interior (short-range) $V^{(s)}_\alpha$ and tail (long-range) parts $V^{(l)}_\alpha$
\begin{equation}
\label{PotSplit}
V_\alpha(x_\alpha) = V^{(s)}_\alpha(x_\alpha,y_\alpha) + V^{(l)}_\alpha(x_\alpha, y_\alpha).
\end{equation}
The equations~(\ref{MFeq}) can be summed up leading to the Schr\"odinger equation for the wave-function
$\Psi=\sum_{\alpha}\psi_\alpha$, wherein functions $\psi_{\alpha}$ are called the components of the wave function.

Splitting (\ref{PotSplit}) of the potentials in general case is done in the three-body configuration space by the  Merkuriev cut-off function $\chi_\alpha$ \cite{fadd93}
\begin{equation}
V_\alpha^{(s)}(x_\alpha,y_\alpha) = \chi_\alpha(x_\alpha, y_\alpha) V_\alpha(x_\alpha).
\label{split3b}
\end{equation}

 This splitting  confines  the short-range part of the potential to the regions in the three-body configuration space corresponding to the three-body collision point (particles are close to each other) and the binary configuration ($x_\alpha \ll y_\alpha$, when $y_\alpha \to \infty$). The form of the cut-off function can be rather arbitrary within some general requirements~\cite{merkur80}. Traditionally, the following form of the cut-off function proposed in~\cite{kvits92} is in use

\begin{equation}
\label{Mcutoff0}
\chi_\alpha(x_\alpha, y_\alpha) =
2/\left\{1+\exp[ (x_\alpha/x_{0\alpha})^{\nu_\alpha}/(1+y_\alpha/y_{0\alpha}) ]\right\}.
\end{equation}
The parameters $x_{0\alpha}$ and $y_{0\alpha}$ can in principle be chosen arbitrarily, but their choice changes the properties of components $\psi_\alpha$ important from both the theoretical and computational points of view \cite{YakPapp2010}.
The splitting procedure (\ref{PotSplit},\ref{Mcutoff0}) is done in the three-body configuration space and is suitable for energies as below the three-body breakup threshold as well as above this threshold.
In the paper \cite{grad16} we have shown, that for energies below the disintegration threshold splitting
can be vastly simplified by confining a cut-off function $\chi_\alpha(x_\alpha)$ on the two-body configuration space.
Formally it is obtained by chousing $y_{0\alpha}=\infty$,  which leads to
\begin{equation}
\label{Mcutoff}
\chi_\alpha(x_\alpha) = 2/\left\{1+\exp[ (x_\alpha/x_{0\alpha})^{\nu_\alpha} ]\right\}.
\end{equation}
With this smoothed Heaviside step function with $\nu_\alpha=2.01$ which is used in this paper for actual calculations, the splitted potentials $V^{(s,l)}_\alpha$
become the two-body quantities $V^{(s,l)}_\alpha=V^{(s,l)}_\alpha({x}_\alpha)$.

The splitting procedure makes the properties of the FM equations for Coulomb potentials for treating the scattering problem as appropriate as standard Faddeev equations in the case of short-range potentials \cite{Papp-Yak2001}. The key one of these properties  of FM equations~(\ref{MFeq})
is that the right-hand side of each equation is localized in the region of configuration space corresponding to three-body collision point \cite{YakPapp2010}. It results in the asymptotic uncoupling of the
set of FM equations and, accordingly, the asymptote of each component $\psi_\alpha$ for energies below the breakup threshold contains only terms corresponding to binary configurations of pairing $\alpha$  \cite{YakPapp2010, Papp-Yak2001}.
For the total energy $E$ of the system below the three-body ionization threshold it reads
\begin{eqnarray}
\label{asympt3D}
\psi_\alpha(\bm{x_\alpha},\bm{y_\alpha}) & = & \Phi^{\mathrm{in}}_{A_0}(\bm{x}_\alpha,\bm{y}_\alpha;\bm{p}_{n_0}) \delta_{\alpha,\alpha_0} \nonumber \\
& + &\sum\limits_{A}\frac{\phi_{n\ell}(x_\alpha)}{x_\alpha}Y_{\ell m}(\hat{\bm{x}}_\alpha)\sqrt{\frac{p_{n_0}}{p_n}}
\mathcal{A}_{A,A_0}(\hat{\bm{y}}_\alpha)
\frac{e^{i(p_n y_\alpha-\eta_n\ln(2p_n y_\alpha))}}{y_\alpha},
\end{eqnarray}
where the multi-index $A=\{n\ell m\}$ specifies various two-body Coulombic bound states of particles of pairing $\alpha$ (that is, binary scattering channels $\{\alpha;A\}$) with wave function $\phi_{n\ell}(x_\alpha)Y_{\ell m}(\hat{\bm{x}}_\alpha)/x_\alpha$ and energy $\varepsilon_n$. Here and in what follows by $Y_{\ell m}(\hat{\bm x})$ the standard spherical harmonic is denoted.  The momentum $p_n$ of the outgoing particle
is determined by the energy conservation condition $E=p_n^2+\varepsilon_n$ and the Sommerfeld parameter is defined as  $\eta_n\equiv Z_\alpha(\sum_{\beta\ne\alpha}Z_\beta)\sqrt{2m_{\alpha(\beta\gamma)}}/(2p_n)$. The binary scattering amplitude $\mathcal{A}_{A,A_0}(\hat{\bm{y_\alpha}})$ corresponds to transition from the initial binary channel $\{\alpha_0;A_0\}$ to the binary channel $\{\alpha;A\}$. The initial channel is specified by the incoming wave, generated
by the asymptotic Coulomb interaction between the projectile and the target
 $V_{\alpha}^{\mathrm{eff}}(y_\alpha)=2p_n\eta_n/y_\alpha$ ($\beta,\gamma\ne\alpha$)
\begin{eqnarray}
\Phi^{\mathrm{in}}_{A_0}(\bm{x}_\alpha,\bm{y}_\alpha;\bm{p}_{n_0}) & = & \frac{\phi_{n_0\ell_0}(x_\alpha)}{x_\alpha}Y_{\ell_0m_0}(\hat{\bm{x}}_\alpha)
e^{i(\bm{p_{n_0}},\bm{y_\alpha})} e^{-\pi\eta_{n_0}/2}\Gamma(1+i\eta_{n_0}) \nonumber \\
& \times & {}_1F_1(-i\eta_{n_0}, 1, i(p_{n_0}y_\alpha-(\bm{p_{n_0}},\bm{y_\alpha}))),
\end{eqnarray}
where ${}_1F_1$ is the confluent hypergeometric function \cite{abram86}.

The total angular momentum is an integral of motion for the three-particle system. This fact allows one to reduce
the set of FM equations by projecting (\ref{MFeq}) onto a subspace of a given total angular momentum
\cite{kvits95}.
In this article we consider the case of zero total angular momentum of the system.
The kinetic energy operator in the left-hand side of equations~(\ref{MFeq}) on the subspace of the zero
total orbital momentum have the form
\begin{eqnarray}
T_\alpha = & - & \frac{\partial^2}{\partial y_\alpha^2} - \frac{2}{y_\alpha}\frac{\partial}{\partial y_\alpha}
- \frac{\partial^2}{\partial x_\alpha^2} - \frac{2}{x_\alpha}\frac{\partial}{\partial x_\alpha} \nonumber \\
& - & \left(\frac{1}{y_\alpha^2}+\frac{1}{x_\alpha^2}\right)
\frac{\partial}{\partial z_\alpha}(1-z_\alpha^2)\frac{\partial}{\partial z_\alpha},
\end{eqnarray}
where $z_\alpha\equiv\cos(\hat{\bm{x}}_\alpha\cdot\hat{\bm{y}}_\alpha)$. The corresponding projection of the
component $\psi_\alpha$ depends only on the coordinates $X_\alpha=\{x_\alpha,y_\alpha,z_\alpha\}$ in the plane containing all three particles.
By choosing appropriately the coordinate system the projection of the asymptote~(\ref{asympt3D}) on the state with zero total angular momentum can be written as (the constant factor is omitted)
\begin{eqnarray}
\label{zAs}
\psi_\alpha(X_\alpha) & \sim & -\frac{\phi_{n_0\ell_0}(x_\alpha)}{x_\alpha y_\alpha}
Y_{\ell_0 0}(\theta_\alpha,0)
 e^{-i\vartheta_{\ell_0}(y_\alpha,p_{n_0})} \delta_{\alpha,\alpha_0}\nonumber\\
& + & \sum\limits_{n\ell}
\frac{\phi_{n\ell}(x_\alpha)}{x_\alpha y_\alpha}
Y_{\ell 0}(\theta_\alpha,0)
\sqrt{\frac{p_{n_0}}{p_n}}
S_{n\ell,n_0\ell_0}
e^{+i\vartheta_{\ell}(y_\alpha,p_n)},
\end{eqnarray}
where $\vartheta_{\ell}(y_\alpha,p_n)\equiv p_{n}y_\alpha-\eta_{n}\ln(2p_{n}y_\alpha)-\ell\pi/2+\sigma_{n}$, $\sigma_{n}=\arg\Gamma(1+i\eta_{n})$ and $S_{n\ell,n_0\ell_0}=
\delta_{n\ell,n_0\ell_0}
+2ip_{n_0}e^{-i(\sigma_n+\sigma_{n_0})}e^{i\pi(\ell-\ell_0)/2}A_{n\ell,n_0\ell_0}$ are the $S$-matrix elements.
The total amplitude $\mathcal{A}_{AA_0}$ is connected to partial amplitudes $A_{n\ell,n_0\ell_0}$ according to
\begin{equation}
\sum_{m=-\ell}^{\ell}\int\mathrm{d}\hat{\bm{y_\alpha}}|\mathcal{A}_{AA_0}(\hat{\bm{y}}_\alpha)|^2 = 4\pi|A_{n\ell,n_0\ell_0}|^2+\ldots,
\end{equation}
where the omitted terms in the right-hand side are contributions from higher total angular momenta.

\section{Numerical solution}

To reduce the computational cost of solving the system of FM equations~(\ref{MFeq}), few modifications are  made. At first, since the potential $V_3$ is repulsive and thus the two-body Hamiltonian with this potential does not support bound states, this potential is included in the left-hand side of equations~(\ref{MFeq}) thus reducing the number of equations from 3 to 2:
\begin{equation}
\label{2FM}
\{T_\alpha + V_\alpha(x_\alpha) +
V_\beta^{(l)}(x_\beta) + V_3(x_3) - E\} \psi_\alpha(X_\alpha) =
 -V_\alpha^{(s)}(x_\alpha) \psi_\beta(X_\beta),
\end{equation}
where $\beta\ne\alpha=1,2$.
Formally, it is done by setting  $\chi_3=0$.
Secondly, the asymptotic particle-atom Coulomb potential
$V_{\alpha}^{\mathrm{eff}}(y_\alpha)$
is introduced explicitly  in~(\ref{2FM}) for treating the asymptotic Coulomb singularity
\begin{eqnarray}
\{T_\alpha  +  V_\alpha(x_\alpha) & + &
V_\alpha^{\mathrm{eff}}(y_\alpha) - E\} \psi_\alpha(X_\alpha) =
  -  V_\alpha^{(s)}(x_\alpha) \psi_\beta(X_\beta) \nonumber \\
 & - & \left[ V_\beta^{(l)}(x_\beta) + V_3(x_3) - V_\alpha^{\mathrm{eff}}(y_\alpha) \right]\psi_\alpha(X_\alpha).
\end{eqnarray}
After that the Coulomb singularity can be effectively inverted as described below.
Thirdly, it is more convenient to work with the modified components
\begin{equation}
\widetilde{\psi_\alpha}(X_\alpha) = x_\alpha y_\alpha \psi_\alpha(X_\alpha)
\end{equation}
that satisfy the system of equations
\begin{eqnarray}
\label{MFeq2}
\{\widetilde{T_\alpha}  +  V_\alpha(x_\alpha) & + &
V_\alpha^{\mathrm{eff}}(y_\alpha) - E\} \widetilde{\psi_\alpha}(X_\alpha) =
  -  \frac{x_\alpha y_\alpha}{x_\beta y_\beta}V_\alpha^{(s)}(x_\alpha) \widetilde{\psi_\beta}(X_\beta) \nonumber \\
 & - & \left[ V_\beta^{(l)}(x_\beta) + V_3(x_3) - V_\alpha^{\mathrm{eff}}(y_\alpha) \right]\widetilde{\psi_\alpha}(X_\alpha),\ \ \alpha=1,2,
\end{eqnarray}
with $\widetilde{T_\alpha}=- \frac{\partial^2}{\partial y_\alpha^2} -
 \frac{\partial^2}{\partial x_\alpha^2}
-\left(\frac{1}{y_\alpha^2}+\frac{1}{x_\alpha^2}\right)
\frac{\partial}{\partial z_\alpha}(1-z_\alpha^2)\frac{\partial}{\partial z_\alpha}$.
The component $\widetilde{\psi_\alpha}$ satisfies zero boundary conditions on the lines $x_\alpha=0$, $y_\alpha=0$ and the asymptotic boundary condition obtained by multiplying formula~(\ref{zAs}) by $x_\alpha y_\alpha$ reads:
\begin{eqnarray}
\label{zAs2}
\widetilde{\psi_\alpha}(X_\alpha) & \sim & -\phi_{n_0\ell_0}(x_\alpha)
Y_{\ell_0 0}(\theta_\alpha,0)
 e^{-i\vartheta_{\ell_0}(y_\alpha,p_{n_0})} \delta_{\alpha,\alpha_0}\nonumber\\
& + & \sum\limits_{n\ell}
\phi_{n\ell}(x_\alpha)
Y_{\ell 0}(\theta_\alpha,0)
\sqrt{\frac{p_{n_0}}{p_n}}
S_{n\ell,n_0\ell_0}
e^{+i\vartheta_{\ell}(y_\alpha,p_n)}.
\end{eqnarray}
Another modification is done to make solutions of equations~(\ref{MFeq2}) real functions. We recast (\ref{zAs2}) into
 the form
\begin{eqnarray}
\label{rAs}
\widetilde{\psi_\alpha}(X_\alpha) & \sim & -\phi_{n_0\ell_0}(x_\alpha)
Y_{\ell_0 0}(\theta_\alpha,0)
 \sin\left(\vartheta_{\ell_0}(y_\alpha,p_{n_0})\right) \delta_{\alpha,\alpha_0}\nonumber\\
& + & \sum\limits_{n\ell}
\phi_{n\ell}(x_\alpha)
Y_{\ell 0}(\theta_\alpha,0)
\sqrt{\frac{p_{n_0}}{p_n}}
K_{n\ell,n_0\ell_0}
\cos\left(\vartheta_{\ell}(y_\alpha,p_n)\right),
\end{eqnarray}
where the real numbers $K_{n\ell,n_0\ell_0}$ form the $K$-matrix.
One can show that a solution of the system of equations~(\ref{MFeq2}) with complex valued asymptotic boundary conditions~(\ref{zAs2})
is a linear combination of its solutions with boundary conditions~(\ref{rAs}) (standing waves) with different initial channels $\{\alpha_0,A_0\}$.
It gives the standard matrix relation between the $S$- and $K$-matrices
\begin{equation}
S = -(K+iI)^{-1}\cdot(K-iI).
\label{SK}
\end{equation}
Here $I$ is the identity matrix with linear size equal to the number of open channels $N_{\mathrm{ch}}$.

The boundary value problem~(\ref{MFeq2}),~(\ref{rAs}) is solved by the spline collocation method.
We solve equations in a box $[0,R_\alpha^x]\times[0,R_\alpha^y]\times[-1,1]$ for each component $\widetilde{\psi_\alpha}$.
As a basis set for expanding the component we use products of Hermite basis $S^3_5$ splines (splines of degree 5 with 2 continuous derivatives) in each coordinate.
Hermite basis splines form a basis in a space of $S^3_5$ splines on a given grid of knots. Each spline is a local function that is nonzero only on two adjoining intervals of the grid.
The boundary conditions~(\ref{rAs}) are implemented as follows.
For each open channel $\{\alpha_0,A_0\}$ we construct the driven system of equations by making substitutions
\begin{equation}
\widetilde{\psi_\alpha}(X_\alpha) = \widehat{\psi_\alpha}(X_\alpha) + \phi_{n_0\ell_0}(x_{\alpha})S(y_{\alpha})Y_{\ell_0 0}(\theta_\alpha,0)\delta_{\alpha, \alpha_0},
\end{equation}
where $S$ is the Hermite basis spline satisfying $S(R_{\alpha_0}^y)=1$:
\begin{eqnarray}
\label{MFfin}
\{\widetilde{T_\alpha} & + & V_\alpha(x_\alpha) +
V_\alpha^{\mathrm{eff}}(y_\alpha) - E\} \widehat{\psi_\alpha}(X_\alpha) =
  -  \frac{x_\alpha y_\alpha}{x_\beta y_\beta}V_\alpha^{(s)}(x_\alpha) \widehat{\psi_\beta}(X_\beta) \nonumber \\
 & - & \left[ V_\beta^{(l)}(x_\beta) + V_3(x_3) - V_\alpha^{\mathrm{eff}}(y_\alpha) \right]\widehat{\psi_\alpha}(X_\alpha) + f_\alpha(X_\alpha),\ \ \alpha=1,2.
\end{eqnarray}
Here the inhomogeneous term is given by
\begin{eqnarray}
f_\alpha(X_\alpha) = & - & \phi_{n_0\ell_0}(x_{\alpha_0}) Y_{\ell_0 0}(\theta_{\alpha_0},0) \Bigg\{ \left(-\frac{d^2}{d y_\alpha^2}+\frac{\ell_0(\ell_0+1)}{y_\alpha^2}+V_\beta^{(l)}+V_3-p_{n_0}^2\right) \delta_{\alpha,\alpha_0} \nonumber \\
& - & \frac{x_\alpha y_\alpha}{x_\beta y_\beta}V_\alpha^{(s)}(x_\alpha)\delta_{\beta,\alpha_0}\Bigg\}S(y_{\alpha_0}).
\end{eqnarray}
The system of equations~(\ref{MFfin}) is supplied  with zero boundary conditions $\widehat{\psi_\alpha}(X_\alpha)=0$ on the sides of the box $[0,R_\alpha^x]\times[0,R_\alpha^y]$.
The so obtained solution components $\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}$, being a linear combination of physical solutions, behave asymptotically as
\begin{eqnarray}
\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}(X_\alpha) & \sim & \sum\limits_{n\ell}
\phi_{n\ell}(x_\alpha)
Y_{\ell 0}(\theta_\alpha,0) \nonumber\\
& \times & \Big(
si_{n\ell,n_0\ell_0}
\sin\left(\vartheta_{\ell}(y_\alpha,p_n)\right)
+
co_{n\ell,n_0\ell_0}
\cos\left(\vartheta_{\ell}(y_\alpha,p_n)\right)
\Big),
\end{eqnarray}
with some coefficients $si_{n\ell,n_0\ell_0}$ and $co_{n\ell,n_0\ell_0}$.
These coefficients can be extracted by projecting the function $\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}$ at some distant point $y_\alpha$ on two-body bound states
\begin{eqnarray}
\int_0^{R_\alpha^x}\mathrm{d}x_\alpha\int_{-1}^{1}\mathrm{d}\cos\theta_\alpha\,\phi_{n\ell}(x_\alpha)Y_{\ell 0}(\theta_\alpha,0)\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}(X_\alpha)\nonumber\\
\qquad= si_{n\ell,n_0\ell_0}
\sin\left(\vartheta_{\ell}(y_\alpha,p_n)\right)
+
co_{n\ell,n_0\ell_0}
\cos\left(\vartheta_{\ell}(y_\alpha,p_n)\right)
\end{eqnarray}
and using extrapolation.
We obtain $N_{\mathrm{ch}}$ solutions $\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}$ of driven equations corresponding to different open channels $\{\alpha_0,A_0\}$ for a given total energy $E$.
These solutions are linearly independent due to conditions $\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}(X_\alpha)=\phi_{n_0\ell_0}(x_{\alpha})Y_{\ell 0}(\theta_\alpha,0)\delta_{\alpha, \alpha_0}$ at $X_\alpha=\{x_\alpha,R_\alpha^y,z_\alpha\}$.
The solution of interest $\widetilde{\psi_\alpha}$ is a linear combination of them.
The formula connecting $\widetilde{\psi_\alpha}$ and solutions $\widetilde{\psi_\alpha}^{(\alpha_0, A_0)}$ allows one to obtain all $K$-matrix elements from coefficients $si_{n\ell,n_0\ell_0}$ and $co_{n\ell,n_0\ell_0}$.

Equations~(\ref{MFfin}) are the final equations that are solved numerically.
The components are expanded in terms of products of Hermite basis splines:
\begin{equation}
\label{ser}
\widehat{\psi_\alpha}(X_\alpha) = \sum_{ijk}c_\alpha^{ijk}S_i(x_\alpha)S_j(y_\alpha)S_k(z_\alpha),\quad\alpha=1,2.
\end{equation}
Substituting the series~(\ref{ser}) into equations~(\ref{MFfin}) and requiring the resulting equalities to be satisfied in a number of points $\left(x_\alpha^\xi,\ y_\alpha^\eta,\ z_\alpha^\zeta\right)$ of a rectangular grid one gets the matrix equation on coefficients $c_\alpha^{ijk}$ of the form:
\hspace{-1cm}\begin{equation}
\label{mEq}
\left(
\begin{array}{cc}
[\bm{H}_1-E \bm{S}_1] & 0 \\
0 & [\bm{H}_2-E \bm{S}_2]
\end{array}
\right)
\left(
\begin{array}{c}
\bm{c_1} \\
\bm{c_2}
\end{array}
\right)
 =
\bm{R}
\left(
\begin{array}{c}
\bm{c_1} \\
\bm{c_2}
\end{array}
\right)
+
\left(
\begin{array}{c}
\bm{f_1} \\
\bm{f_2}
\end{array}
\right),
\end{equation}
where the matrices $\bm{H}_\alpha=\widetilde{\bm{T}_\alpha}+\bm{V}_\alpha+\bm{V}_\alpha^{\mathrm{eff}}$ and
\hspace{-2cm}\begin{equation}
\bm{R} =
-\left(
\begin{array}{cc}
0 & \bm{V}_1^{(s)} \\
\bm{V}_2^{(s)} & 0
\end{array}
\right)
-
\left(
\begin{array}{cc}
\bm{V}_2^{(l)}+\bm{V}_3-\bm{V}_1^{eff} & 0 \\
0 & \bm{V}_1^{(l)}+\bm{V}_3-\bm{V}_2^{eff}
\end{array}
\right)
\end{equation}
are discretized versions of the operators of the left- and right-hand sides of the system~(\ref{MFfin}).
Here for example the matrix $\bm{V}_\alpha$ has elements $V(x_\alpha^\xi)S_i(x_\alpha^\xi)S_j(y_\alpha^\eta)S_k(z_\alpha^\zeta)$ with rows enumerated by point numbers $\xi\eta\zeta$ and columns by basis function numbers $ijk$.
Right preconditioning by its left-hand side matrix turns the system of linear equations~(\ref{mEq}) into
\hspace{-4cm}{\begin{equation}
\label{SLAUfin}
\left\{
\bm{I}-\bm{R}
\left(
\begin{array}{cc}
[\bm{S}_1^{-1}\bm{H}_1-E \bm{I}]^{-1}\bm{S}_1^{-1} & 0 \\
0 & [\bm{S}_2^{-1}\bm{H}_2-E \bm{I}]^{-1}\bm{S}_2^{-1}
\end{array}
\right)
\right\}
\left(
\begin{array}{c}
\widetilde{\bm{c_1}} \\
\widetilde{\bm{c_2}}
\end{array}
\right)
=
\left(
\begin{array}{c}
\bm{f_1} \\
\bm{f_2}
\end{array}
\right).
\end{equation}}
This system of linear equations is solved by Arnoldi iterations in GMRES variant~\cite{saad03}.
On each iteration of the algorithm the most computationally expensive operation is a multiplication of the matrix of~(\ref{SLAUfin}) by a vector.
The matrix $\bm{R}$ is sparse, it has maximum of $2\times6\times6\times6=432$ nonzero elements in a row due to the locality property of Hermite basis splines.
To invert the matrix $[\bm{S}_\alpha^{-1}\bm{H}_\alpha-E\bm{I}]$ we use the algorithm which is known as ``tensor trick'' or matrix decomposition method (\cite{schelling89},\cite{bial93},\cite{roud00}). It provides a fast diagonalization of a matrix using its tensor product structure
\hspace{-3cm}\begin{equation}
\begin{array}{l}
[\bm{S}_\alpha^{-1}\bm{H}_\alpha-E\bm{I}]=(\bm{S}_{\alpha}^{x})^{-1}\bm{D}_{\alpha}^{x}\otimes \bm{I}\otimes \bm{I} + \bm{I}\otimes (\bm{S}_{\alpha}^{y})^{-1}\bm{D}_{\alpha}^{y}\otimes \bm{I} \\
+\Big(
(\bm{S}_{\alpha}^{x})^{-1}\bm{X}_{r2}\bm{S}_{\alpha}^{x}\otimes \bm{I}\otimes \bm{I} + \bm{I}\otimes (\bm{S}_{\alpha}^{y})^{-1}\bm{Y}_{r2}\bm{S}_{\alpha}^{y}\otimes \bm{I}
\Big)\cdot
(\bm{I}\otimes \bm{I}\otimes (\bm{S}_{\alpha}^{z})^{-1}\bm{D}_{\alpha}^{z})\\
+(\bm{S}_{\alpha}^{x})^{-1}\bm{V}_\alpha \bm{S}_{\alpha}^{x}\otimes \bm{I}\otimes \bm{I}+
\bm{I}\otimes (\bm{S}_{\alpha}^{y})^{-1}\bm{V}_\alpha^{eff}\bm{S}_{\alpha}^{y} \otimes \bm{I} - E\,\bm{I}\otimes \bm{I}\otimes \bm{I},
\end{array}
\end{equation}
where the matrices $\bm{S}^{x}_{\alpha}$, $\bm{D}^{x}_{\alpha}$, $\bm{S}^{y}_{\alpha}$, \ldots represent the ``one-dimensional'' matrices of basis splines and their second derivatives values at points of the grid in respective coordinates.
For example, $\bm{D}_\alpha^{x}$ is a matrix with elements $ S_i''(x_\alpha^\xi)$ enumerated by indices $i$ and $\xi$.
The matrices $\bm{V}_\alpha$, $\bm{X}_{r2}$, $\bm{Y}_{r2}$, \ldots are the diagonal matrices of pair and centrifugal potentials values.
For a more detailed description of a method we refer the reader to~\cite{roud00}.

\begin{figure}[t]

\center{\includegraphics[width=0.8\textwidth]{vl.pdf}}

\caption{The long-range tail part of the potential $V_\beta=-1/x_\beta$ obtained by using the cut-off function~(\ref{Mcutoff}) with parameter $x_{0\beta}=3.0$. The dotted line denotes the ground state energy level of two particles interacting by this potential.}
\label{vl}

\end{figure}

At the end of this section we discuss the choice of the Merkuriev cut-off function~(\ref{Mcutoff}) parameter $x_{0\alpha}$.
We have found that for a given total energy $E$ there exists a lower bound for its values that has a simple physical interpretation.
Remember that the long-range part $V_\beta^{(l)}$ of the potential between particles of pair $\beta\ne\alpha$ is included in the left-hand side operator of equation $\alpha$ in~(\ref{MFeq}).
The potential $V_\beta^{(l)}$ is plotted in Figure~\ref{vl}, the ground state (g.s.) energy of two particles interacting via this potential is marked with a horizontal line.
The value of the g.s. energy is governed by the parameter $x_{0\beta}$.
The lower the parameter is, the deeper is the potential and the lower is the g.s. energy.
If the total energy of the three-body system $E$ is higher than this g.s. energy the operator in the left-hand side of equation $\alpha$ of~(\ref{MFeq})
\begin{equation}
T_\alpha + V_\alpha(x_\alpha) +
\sum_{\beta\ne\alpha}V_\beta^{(l)}(x_\beta)=
T_\beta +
\sum_{\beta\ne\alpha}V_\beta^{(l)}(x_\beta) + V_\alpha(x_\alpha)
\end{equation}
supports open channels in both arrangements $\alpha$ and $\beta$.
Subsequently, the outgoing waves corresponding to binary channels of pairing $\beta$ appear in the asymptote of $\psi_\alpha$.
It cancels one of the main advantages of Faddeev-Merkuriev equations approach --- the asymptotic uncoupling of components --- and should be avoided.
Thus, for a given total energy $E$ and a given $\beta$ the cut-off function parameter $x_{0\beta}$ should be bigger than some critical value $x_{c}$ that is defined as follows:
the ground state energy of a two-body Hamiltonian with the potential $V^{(l)}_\beta$ for $x_{0\beta}=x_{c}$ should be equal to $E$.

There is no upper bound on the values of $x_{0\beta}$, but we observed that the bigger is the value of the parameter, the bigger basis in angular coordinate $z_\alpha$ sizes are to be taken to obtain the desired accuracy.
Summarizing, we propose the following algorithm of choosing the cut-off function parameter $x_{0\beta}$:
\begin{enumerate}
\item
\label{vvv}
For a total energy interval $[E_{\min},E_{\max}]$ of interest find the critical values of parameters $x_{0\beta}$ for the total energy value $E_{\max}$ for each $\beta$. The lowest energy levels of the two-body Hamiltonian with the potential $V_\beta^{(l)}$ can be calculated by any simple two-body code.
\item Take the value of the parameter $x_{0\beta}$ only slightly bigger than the critical value calculated at step~(\ref{vvv}), since bigger values of $x_{0\beta}$ require employing larger grids.
\end{enumerate}

\section{Results}
\subsection{$e^-e^+\bar{p}$ and $e^+e^-p$ scattering}
Positron--hydrogen atom scattering is the simplest example of positron-atom scattering process.
Many calculations are presented in the literature, among them are detailed calculations using the FM equations for both low energies~\cite{lazau18, hu99,yak07} and wide energy region including that above the ionisation threshold~\cite{lazau17}.
Other methods include hyperspherical calculations~\cite{igara94,ward12}, variational~\cite{humber97,gien99}, close coupling calculations using two center basis functions expansion~\cite{mitroy95,kadyr02}.
The renewed interest in studying the reactions involving positron, electron and (anti)proton is motivated by experiments on antimatter that take place at CERN~\cite{lazau18}.
The key role in antimatter formation plays the reaction of antihydrogen formation via antiproton ($\bar{p}$) impact of positronium (Ps, the bound state of $e^+$ and $e^-$) atom.
Due to the symmetry in particle charges, the cross sections in $e^+e^-p$ and $e^-e^+\bar{p}$ systems are identical.
Further in this section we refer to the $e^+e^-\bar{p}$ system.

We have calculated K-matrix elements of all possible scattering processes in $e^+e^-\bar{p}$ system in the total energy range from $-0.49973\ a.u.$ to  $-0.05553\ a.u.$ with the energy step of calculation $0.0007\ a.u.$
In this interval elastic, excitations and rearrangement processes leading to $\bar{\mbox{H}}$($n=1,2$) and Ps($n=1,2$) atom states are possible.
The maximum linear size of K-matrix equals to 6.
We proceed as follows: the whole energy region is divided into subintervals associated with energy thresholds.
For every interval we choose appropriate discretisation and cut-off function parameters according to the rules described in the previous section.
This is done to minimize the computational cost, since generally  higher energy calculations require
\begin{itemize}
\item bigger box sizes in $x_\alpha$, $y_\alpha$ and more basis functions involved
\item bigger cut-off parameters $x_{0\alpha}$, which in turn increases the number of functions in the angular coordinate basis.
\end{itemize}
The energy intervals and corresponding discretisation and cut-off function parameters are given in table~\ref{H_discr}.
In this table and in further applications we use shortcuts $\bar{\mbox{H}}(n)$ and $\bar{\mbox{H}}(n, \ell)$
for the atom states with principal quantum number $n$ and angular momentum $\ell$.
\begin{table}[t!]
\centering
\begin{tabular}{c|c|c|c}
\hline
 & $\bar{\mbox{H}}$(1)--Ps(1) & Ps(1)--$\bar{\mbox{H}}$(2) & $\bar{\mbox{H}}$(2)--$\bar{\mbox{H}}$(3) \\
\hline
First component\\
\hline
$x_{\max}$ & 17.7 & 28.3 & 70.7 \\
$y_{\max}$ & 42.4 & 84.9 & 141 \\
$n_x$ & 45 & 60 & 75 \\
$n_y$ & 120 & 300 & 510 \\
$n_z$ & 24 & 24 & 24 \\
$x_0$ & 1.4 & 3.5 & 8.5 \\
\hline
Second component\\
\hline
$x_{\max}$ & 35 & 70  & 100 \\
$y_{\max}$ & 17.5 & 35 & 115 \\
$n_x$ & 60 & 75 & 75 \\
$n_y$ & 60 & 105  & 600 \\
$n_z$ & 18 & 30  & 45 \\
$x_0$ & 1 & 4 & 8 \\
\end{tabular}
\caption{Discretisation and cut-off function parameters used in calculations of $e^+e^-\bar{p}$ system cross sections. The column headings denote the energy intervals associated with atoms excitations thresholds.}
\label{H_discr}
\end{table}

In the calculations we hold the accuracy level such that the maximum error does not exceed 1\%.
Tables~\ref{conv_H} and~\ref{conv_H_2} illustrate the convergence of different cross sections at energy $E=-0.0572\  a.u.$ with respect to discretisation and cut-off function parameters.
To control our calculations, we have also used the value
\[
\mathfrak{K}=||K-K^T||/||K||
\]
with the Frobenius norm of a matrix, as a measure of K-matrix asymmetry.
The values of $\mathfrak{K}$ are shown in tables~\ref{conv_H} and~\ref{conv_H_2}.
\begin{landscape}
\begin{table}
{\tiny
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}
 & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(1)}$ & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(2,s)}$ & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(2,p)}$ & $\sigma_{\bar{\mbox{H}}(1)- \mbox{Ps}(1)}$ & $\sigma_{\bar{\mbox{H}}(2,s)- \bar{\mbox{H}}(1)}$ & $\sigma_{\bar{\mbox{H}}(2,s)- \mbox{Ps}(1)}$ & $\sigma_{\mbox{Ps}(1)- \bar{\mbox{H}}(1)}$ & $\sigma_{\mbox{Ps}(1)- \bar{\mbox{H}}(2)}$ & $\sigma_{\mbox{Ps}(1)- \mbox{Ps}(1)}$ & $\sigma_{\mbox{Ps}(2)- \mbox{Ps}(2)}$ & $\sigma_{\mbox{Ps}(2)- \bar{\mbox{H}}(n\le2)}$ & $\mathfrak{K}$ \\
\hline
$x_{max}$ & & & & & & & & & & & & \\
70.0 & 0.11392 & 0.0023206 & 0.0018759 & 0.005665 & 0.015184 & 0.37833 & 0.0064832 & 0.10521 & 4.8348 & 270.21 & 16.104 & 0.42704 \\
80.0 & 0.11392 & 0.0023966 & 0.0019211 & 0.0056769 & 0.015648 & 0.39138 & 0.0065249 & 0.1099 & 4.833 & 194.17 & 12.235 & 0.24687 \\
90.0 & 0.11392 & 0.0024092 & 0.0019205 & 0.0056744 & 0.015725 & 0.39225 & 0.0065278 & 0.11046 & 4.8326 & 184.23 & 12.122 & 0.11033 \\
100.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
110.0 & 0.11392 & 0.0024063 & 0.0019188 & 0.005674 & 0.015705 & 0.39142 & 0.0065287 & 0.11029 & 4.8329 & 183.81 & 12.105 & 0.10578 \\
$y_{max}$ & & & & & & & & & & & & \\
170.0 & 0.11402 & 0.0023717 & 0.0019656 & 0.0056777 & 0.015481 & 0.39408 & 0.0065316 & 0.11075 & 4.832 & 183.37 & 12.052 & 0.098401 \\
180.0 & 0.114 & 0.0023633 & 0.0019241 & 0.0056804 & 0.015461 & 0.39297 & 0.0065341 & 0.11053 & 4.8324 & 183.37 & 12.091 & 0.097098 \\
190.0 & 0.11395 & 0.0023846 & 0.0019089 & 0.0056846 & 0.015583 & 0.39205 & 0.0065375 & 0.11037 & 4.8327 & 183.39 & 12.095 & 0.096431 \\
200.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
210.0 & 0.11385 & 0.0023946 & 0.0018732 & 0.0056825 & 0.01568 & 0.3918 & 0.006535 & 0.11042 & 4.8325 & 183.37 & 12.124 & 0.095862 \\
$n_{x}$ & & & & & & & & & & & & \\
30 & 0.11315 & 0.0024155 & 0.0019283 & 0.0057331 & 0.015828 & 0.39656 & 0.0066087 & 0.11167 & 4.8343 & 186.54 & 11.755 & 0.10776 \\
45 & 0.11385 & 0.002411 & 0.0019194 & 0.0056766 & 0.015732 & 0.39186 & 0.0065335 & 0.11038 & 4.8317 & 183.67 & 12.078 & 0.097658 \\
60 & 0.11391 & 0.0024083 & 0.0019188 & 0.0056751 & 0.015717 & 0.39176 & 0.0065298 & 0.11037 & 4.8326 & 183.41 & 12.105 & 0.096165 \\
75 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
90 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056736 & 0.015714 & 0.39162 & 0.0065282 & 0.11035 & 4.8328 & 183.38 & 12.109 & 0.096093 \\
$n_{y}$ & & & & & & & & & & & & \\
420 & 0.11354 & 0.0024063 & 0.0019183 & 0.0056739 & 0.015703 & 0.39167 & 0.0065284 & 0.11036 & 4.8327 & 183.38 & 12.11 & 0.096054 \\
450 & 0.11373 & 0.002407 & 0.0019188 & 0.0056739 & 0.015708 & 0.39166 & 0.0065284 & 0.11035 & 4.8327 & 183.38 & 12.11 & 0.096055 \\
480 & 0.11384 & 0.0024074 & 0.0019189 & 0.005674 & 0.015712 & 0.39165 & 0.0065284 & 0.11035 & 4.8327 & 183.38 & 12.109 & 0.096056 \\
510 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
540 & 0.11397 & 0.0024078 & 0.0019188 & 0.0056739 & 0.015716 & 0.39164 & 0.006528 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096058 \\
$n_{z}$ & & & & & & & & & & & & \\
15 & 0.11405 & 0.0024317 & 0.001936 & 0.005723 & 0.015875 & 0.41145 & 0.0066155 & 0.11639 & 4.7913 & 211.2 & 13.616 & 0.35515 \\
18 & 0.11398 & 0.0024236 & 0.0019208 & 0.0056932 & 0.015823 & 0.40079 & 0.006563 & 0.11321 & 4.8127 & 194.27 & 12.942 & 0.26182 \\
21 & 0.11394 & 0.002413 & 0.0019171 & 0.00568 & 0.01575 & 0.39465 & 0.0065396 & 0.11132 & 4.8252 & 186.65 & 12.389 & 0.17454 \\
24 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
27 & 0.1139 & 0.0024067 & 0.0019217 & 0.0056701 & 0.015709 & 0.39027 & 0.0065214 & 0.10987 & 4.8375 & 182 & 12.005 & 0.049207 \\
$x_{0}$ & & & & & & & & & & & & \\
8.0 & 0.11404 & 0.0033197 & 0.0076791 & 0.0046552 & 0.0042209 & 0.32523 & 0.0073201 & 0.30122 & 5.0731 & 444.54 & 23.429 & 1.3065 \\
8.5 & 0.11377 & 0.0024429 & 0.0026296 & 0.0058196 & 0.016796 & 0.44711 & 0.006609 & 0.1328 & 4.8479 & 199.52 & 14.579 & 0.65552 \\
9.0 & 0.11394 & 0.0024398 & 0.0017586 & 0.0056412 & 0.015703 & 0.37821 & 0.0064939 & 0.10462 & 4.8426 & 177.33 & 11.52 & 0.19912 \\
9.5 & 0.11395 & 0.0023476 & 0.0020437 & 0.0056189 & 0.014731 & 0.38564 & 0.0065479 & 0.11389 & 4.85 & 183.59 & 12.32 & 0.13869 \\
10.0 & 0.11395 & 0.0024016 & 0.0018837 & 0.0056341 & 0.015287 & 0.38313 & 0.0065232 & 0.10921 & 4.8434 & 180.02 & 11.895 & 0.041614 \\
10.5 & 0.11391 & 0.0024128 & 0.0019318 & 0.0056781 & 0.015796 & 0.39302 & 0.0065243 & 0.11053 & 4.8388 & 180.75 & 12.068 & 0.0089676 \\
11.0 & 0.11392 & 0.0024119 & 0.0019273 & 0.0056761 & 0.015759 & 0.39263 & 0.0065264 & 0.11051 & 4.837 & 180.45 & 12.042 & 0.011675 \\
11.5 & 0.11392 & 0.0024124 & 0.0019288 & 0.0056777 & 0.015759 & 0.39309 & 0.0065294 & 0.11064 & 4.8349 & 179.8 & 12.045 & 0.048489 \\
12.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
12.5 & 0.11392 & 0.0024087 & 0.0019224 & 0.0056761 & 0.015725 & 0.39223 & 0.0065314 & 0.11048 & 4.8302 & 181.47 & 12.025 & 0.034421 \\
13.0 & 0.11392 & 0.0024085 & 0.0019231 & 0.0056768 & 0.015725 & 0.39229 & 0.0065336 & 0.11051 & 4.8274 & 181.18 & 12.007 & 0.023126 \\
13.5 & 0.11392 & 0.0024083 & 0.0019235 & 0.0056772 & 0.015723 & 0.39225 & 0.0065356 & 0.1105 & 4.8244 & 181.05 & 11.993 & 0.018097 \\
14.0 & 0.11392 & 0.002408 & 0.0019238 & 0.0056774 & 0.015721 & 0.39215 & 0.0065375 & 0.11049 & 4.821 & 180.98 & 11.981 & 0.0151 \\
\end{tabular}
}
\caption{Convergence of cross sections for scattering in $e^+e^-\bar{\mbox{p}}$ system with respect to discretization and cut-off function parameters related to the first component $\psi_1$.}
\label{conv_H}
\end{table}
\begin{table}
{\tiny
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}
 & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(1)}$ & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(2,s)}$ & $\sigma_{\bar{\mbox{H}}(1)- \bar{\mbox{H}}(2,p)}$ & $\sigma_{\bar{\mbox{H}}(1)- \mbox{Ps}(1)}$ & $\sigma_{\bar{\mbox{H}}(2,s)- \bar{\mbox{H}}(1)}$ & $\sigma_{\bar{\mbox{H}}(2,s)- \mbox{Ps}(1)}$ & $\sigma_{\mbox{Ps}(1)- \bar{\mbox{H}}(1)}$ & $\sigma_{\mbox{Ps}(1)- \bar{\mbox{H}}(2)}$ & $\sigma_{\mbox{Ps}(1)- \mbox{Ps}(1)}$ & $\sigma_{\mbox{Ps}(2)- \mbox{Ps}(2)}$ & $\sigma_{\mbox{Ps}(2)- \bar{\mbox{H}}(n\le2)}$ & $\mathfrak{K}$ \\
\hline
\hline
$x_{max}$ & & & & & & & & & & & & \\
70.0 & 0.11394 & 0.0023579 & 0.0019599 & 0.0056999 & 0.015557 & 0.38978 & 0.006538 & 0.11023 & 4.8343 & 183.46 & 11.705 & 0.088877 \\
80.0 & 0.11395 & 0.0024123 & 0.0019042 & 0.0056629 & 0.015677 & 0.39072 & 0.0065238 & 0.10995 & 4.8331 & 183.45 & 11.986 & 0.093915 \\
90.0 & 0.11392 & 0.0024073 & 0.0019258 & 0.0056769 & 0.015732 & 0.39213 & 0.0065278 & 0.11047 & 4.8327 & 183.48 & 12.113 & 0.09549 \\
100.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
110.0 & 0.11391 & 0.002409 & 0.00192 & 0.0056742 & 0.015723 & 0.39184 & 0.0065281 & 0.11038 & 4.8327 & 183.41 & 12.117 & 0.096168 \\
$y_{max}$ & & & & & & & & & & & & \\
90.0 & 0.11389 & 0.00247 & 0.0020719 & 0.0057501 & 0.014244 & 0.36754 & 0.007186 & 0.16077 & 4.844 & 186.28 & 8.731 & 0.5278 \\
100.0 & 0.11387 & 0.002293 & 0.0020065 & 0.005741 & 0.015615 & 0.39253 & 0.006386 & 0.10129 & 4.8324 & 184.87 & 8.8342 & 0.35352 \\
110.0 & 0.11395 & 0.0024066 & 0.0018722 & 0.0056544 & 0.015669 & 0.38917 & 0.0064704 & 0.10832 & 4.8314 & 181.46 & 12.038 & 0.18133 \\
120.0 & 0.11393 & 0.0023884 & 0.001923 & 0.00567 & 0.015393 & 0.38183 & 0.0065911 & 0.1099 & 4.8393 & 183.51 & 13.255 & 0.16197 \\
130.0 & 0.11392 & 0.0023832 & 0.0019517 & 0.0056963 & 0.015605 & 0.39306 & 0.0065423 & 0.10991 & 4.8353 & 186.18 & 11.365 & 0.076749 \\
140.0 & 0.11394 & 0.0024358 & 0.0019393 & 0.0056783 & 0.015893 & 0.39998 & 0.0065331 & 0.1127 & 4.8298 & 184.25 & 11.062 & 0.083529 \\
150.0 & 0.11393 & 0.0024153 & 0.0019091 & 0.0056699 & 0.015761 & 0.38884 & 0.0065236 & 0.10948 & 4.8343 & 181.65 & 12.537 & 0.094786 \\
160.0 & 0.11391 & 0.0023749 & 0.0019151 & 0.0056802 & 0.0155 & 0.38659 & 0.0065357 & 0.10871 & 4.8354 & 182.8 & 12.705 & 0.101 \\
170.0 & 0.11391 & 0.0023921 & 0.0019395 & 0.0056847 & 0.015616 & 0.38949 & 0.0065397 & 0.10934 & 4.8356 & 185.15 & 11.564 & 0.095714 \\
180.0 & 0.11394 & 0.002419 & 0.0019339 & 0.0056795 & 0.015789 & 0.39555 & 0.0065347 & 0.11131 & 4.8325 & 184.81 & 11.401 & 0.092771 \\
190.0 & 0.11395 & 0.0024091 & 0.0019131 & 0.0056671 & 0.015721 & 0.39086 & 0.0065199 & 0.11024 & 4.832 & 182.97 & 12.303 & 0.094655 \\
200.0 & 0.11393 & 0.0023862 & 0.001916 & 0.0056784 & 0.015571 & 0.38777 & 0.0065339 & 0.109 & 4.8354 & 182.79 & 12.515 & 0.097142 \\
210.0 & 0.11391 & 0.0023951 & 0.0019344 & 0.0056847 & 0.015632 & 0.39134 & 0.0065404 & 0.10999 & 4.8344 & 183.84 & 11.764 & 0.097132 \\
220.0 & 0.11391 & 0.0024129 & 0.0019341 & 0.0056815 & 0.01575 & 0.39237 & 0.0065364 & 0.11027 & 4.8342 & 184.12 & 11.563 & 0.096512 \\
230.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
240.0 & 0.11393 & 0.0023906 & 0.0019161 & 0.0056734 & 0.015601 & 0.38804 & 0.0065273 & 0.10922 & 4.8342 & 182.89 & 12.398 & 0.095718 \\
$n_{x}$ & & & & & & & & & & & & \\
30 & 0.1156 & 0.00227 & 0.0018007 & 0.0056725 & 0.014761 & 0.39132 & 0.0065013 & 0.11016 & 4.833 & 183.34 & 12.072 & 0.095884 \\
45 & 0.11495 & 0.0024034 & 0.0019084 & 0.0056662 & 0.015692 & 0.39168 & 0.0065242 & 0.11034 & 4.8328 & 183.39 & 12.107 & 0.095996 \\
60 & 0.11395 & 0.0024074 & 0.0019179 & 0.0056726 & 0.015713 & 0.39165 & 0.0065278 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096045 \\
75 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
90 & 0.11393 & 0.0024077 & 0.0019185 & 0.0056724 & 0.015712 & 0.39164 & 0.0065263 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.09606 \\
$n_{y}$ & & & & & & & & & & & & \\
510 & 0.11399 & 0.0024076 & 0.001919 & 0.0056694 & 0.015714 & 0.39182 & 0.0065224 & 0.11042 & 4.8326 & 183.4 & 12.087 & 0.096081 \\
540 & 0.11399 & 0.0024072 & 0.0019188 & 0.0056685 & 0.015711 & 0.39175 & 0.0065215 & 0.11039 & 4.8327 & 183.4 & 12.095 & 0.09607 \\
570 & 0.11394 & 0.0024076 & 0.0019189 & 0.0056723 & 0.015714 & 0.3917 & 0.0065263 & 0.11037 & 4.8327 & 183.39 & 12.102 & 0.096062 \\
600 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
630 & 0.11392 & 0.0024075 & 0.0019188 & 0.0056739 & 0.015713 & 0.3916 & 0.0065282 & 0.11032 & 4.8328 & 183.37 & 12.114 & 0.096053 \\
$n_{z}$ & & & & & & & & & & & & \\
36 & 0.11441 & 0.002408 & 0.0019187 & 0.005678 & 0.015717 & 0.39161 & 0.0065335 & 0.11035 & 4.8327 & 183.38 & 12.111 & 0.09623 \\
39 & 0.11418 & 0.0024079 & 0.0019188 & 0.0056762 & 0.015716 & 0.39162 & 0.0065312 & 0.11035 & 4.8327 & 183.38 & 12.109 & 0.096127 \\
42 & 0.11404 & 0.0024077 & 0.0019188 & 0.0056748 & 0.015715 & 0.39164 & 0.0065295 & 0.11034 & 4.8327 & 183.38 & 12.109 & 0.096099 \\
45 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
48 & 0.11382 & 0.0024075 & 0.0019189 & 0.0056729 & 0.015714 & 0.39165 & 0.006527 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096018 \\
51 & 0.11374 & 0.0024075 & 0.001919 & 0.0056723 & 0.015713 & 0.39166 & 0.0065261 & 0.11034 & 4.8328 & 183.38 & 12.107 & 0.095978 \\
54 & 0.11368 & 0.0024074 & 0.001919 & 0.0056717 & 0.015713 & 0.39166 & 0.0065253 & 0.11034 & 4.8328 & 183.38 & 12.106 & 0.095942 \\
57 & 0.11419 & 0.0024067 & 0.0019185 & 0.0056749 & 0.015709 & 0.39158 & 0.006529 & 0.11033 & 4.8327 & 183.38 & 12.107 & 0.096171 \\
$x_{0}$ & & & & & & & & & & & & \\
6.0 & 0.11314 & 0.0025699 & 0.0024345 & 0.0059803 & 0.014396 & 0.38186 & 0.0069249 & 0.14078 & 4.8302 & 13263 & 907.81 & 0.37535 \\
6.5 & 0.11339 & 0.0025983 & 0.0021622 & 0.0057896 & 0.016454 & 0.4292 & 0.006689 & 0.12736 & 4.8258 & 228.04 & 5.2034 & 0.18133 \\
7.0 & 0.11358 & 0.0024108 & 0.0019345 & 0.0056821 & 0.015736 & 0.39353 & 0.0065382 & 0.11078 & 4.8332 & 179.76 & 11.774 & 0.086407 \\
7.5 & 0.11373 & 0.0024112 & 0.0019325 & 0.0056808 & 0.015745 & 0.39351 & 0.0065356 & 0.11073 & 4.833 & 179.59 & 11.984 & 0.10141 \\
8.0 & 0.11392 & 0.0024076 & 0.0019189 & 0.0056738 & 0.015714 & 0.39165 & 0.0065281 & 0.11034 & 4.8327 & 183.38 & 12.108 & 0.096057 \\
8.5 & 0.11414 & 0.0024084 & 0.0019229 & 0.0056756 & 0.015723 & 0.39229 & 0.0065296 & 0.11047 & 4.8327 & 180.96 & 12.016 & 0.013574 \\
9.0 & 0.1144 & 0.0024081 & 0.0019231 & 0.0056754 & 0.015722 & 0.39238 & 0.0065291 & 0.11049 & 4.8327 & 180.72 & 12.023 & 0.0042565 \\
9.5 & 0.11471 & 0.0024078 & 0.0019229 & 0.0056749 & 0.01572 & 0.39239 & 0.0065285 & 0.1105 & 4.8326 & 180.65 & 12.029 & 0.0018316 \\
10.0 & 0.11506 & 0.0024074 & 0.0019226 & 0.0056744 & 0.015718 & 0.39239 & 0.0065278 & 0.1105 & 4.8326 & 180.62 & 12.034 & 0.0011176 \\
\end{tabular}
}
\caption{Convergence of cross sections for scattering in $e^+e^-\bar{\mbox{H}}$ system with respect to discretization and cut-off function parameters related to the second component $\psi_2$.}
\label{conv_H_2}
\end{table}
\end{landscape}

In table~\ref{others} we compare our results with tabulated values of other authors and give the values of some additional cross sections for further references.
The agreement between our results and that of other calculations is quite good.
The only noticeable  disagreement is in some values of cross sections for $\bar{\mbox{H}}(1)$ formation, our results being a few percent lower than that of other authors.
We note that  the calculation of $\bar{\mbox{H}}(1)$ formation amplitudes should be done with extra accuracy otherwise
it may be prone to errors.
The reason is that the wave function of $\bar{\mbox{H}}(1)$ formation process is highly oscillatory in the asymptotic region of positron flying off the ground state of antihydrogen due to the large positron momentum $p_{1}=\sqrt{E-\varepsilon_{\bar{\mbox{{\tiny H}}}(1)}}$ in~(\ref{asympt3D}).
Therefore the extra precision of numerical solution is required to guaranty the extraction of the amplitude with controlled accuracy.
Our results are
converged with respect to all numerical solution parameters, which we additionally checked for $\bar{\mbox{H}}(1)$ formation cross sections.

\begin{table}[t!]
{\small
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\backslashbox{}{$E$, a.u.} & 0.27026 & 0.28140 & 0.32017 & 0.36145 & 0.385 & 0.40 & 0.415 & 0.42\\
\hline
$\sigma_{\bar{\mbox{H}}(1)\rightarrow \bar{\mbox{H}}(1)}$ & 0.0353 & 0.0417 & 0.0634 & 0.0836 & 0.0944 & 0.100 & 0.105 & 0.107\\
\cite{mitroy95} & & & 0.0651 & 0.0844 & & 0.100 &  \\
\cite{hu99} & 0.0372 & 0.0429 & 0.0649 & 0.0866 & 0.090 & 0.096 & 0.099 & 0.101\\
\cite{gien99} & & 0.0431 & 0.0650 & 0.0856 & \\
\hline
$\sigma_{\bar{\mbox{H}}(1)\rightarrow Ps(1)}$ & 0.00412 & 0.00430 & 0.00487 & 0.00562 & 0.00565 & 0.00572 & 0.00575 & 0.00574\\
\cite{mitroy95} & & & 0.00490 & 0.00567 & & 0.00581 & \\
\cite{hu99} & 0.00410 & 0.00439 & 0.00487 & 0.00557 & \\
\cite{gien99} & & 0.00422 & 0.00481 & 0.00554 & \\
\hline
$\sigma_{Ps(1)\rightarrow Ps(1)}$ & 3.49 & 7.06 & 9.87 & 8.31 & 7.11 & 6.44 & 5.82 & 5.62 \\
\cite{mitroy95} & & & 9.87 & 8.32 & & 6.45 & \\
\cite{hu99} & 3.500 & 7.060 & 9.866 & 8.312 & 7.09 & 6.44 & 5.83 & 5.63\\
\cite{gien99} & & 6.936 & 9.868 & 8.332 &  \\
\hline
$\sigma_{Ps(1)\rightarrow \bar{\mbox{H}}(1)}$ & 0.0272 & 0.0191 & 0.0111 & 0.0091 & 0.00806 & 0.00763 & 0.00724 & 0.00709 \\
\cite{hu99} & 0.0274 & 0.0195 & 0.0111 & 0.0091 & 0.00815 & 0.00780 & 0.00729 & 0.00715\\
\hline
$\sigma_{\bar{\mbox{H}}(1)\rightarrow \bar{\mbox{H}}(2, s)}$ & & & & & 0.000662 & 0.00137 & 0.00206 & 0.00228\\
\hline
$\sigma_{\bar{\mbox{H}}(1)\rightarrow \bar{\mbox{H}}(2, p)}$ & & & & & 0.000399 & 0.000236 & 0.000421 & 0.000582 \\
\hline
$\sigma_{\bar{\mbox{H}}(2,s)\rightarrow Ps(1, s)}$ & & & & & 1.26 & 0.576 & 0.477 & 0.475 \\
\hline
$\sigma_{\bar{\mbox{H}}(2,s)\rightarrow \bar{\mbox{H}}(1, s)}$ & & & & & 0.0249 & 0.0217 & 0.0212 & 0.0212 \\
\hline
$\sigma_{Ps(1)\rightarrow \bar{\mbox{H}}(2, s)}$ & & & & & 0.0476 & 0.0484 & 0.0581 & 0.0631 \\
\hline
$\sigma_{Ps(1)\rightarrow \bar{\mbox{H}}(2, p)}$ & & & & & 0.0390 & 0.0484 & 0.0512 & 0.0519 \\
\hline
\end{tabular}
}
\caption{Scattering cross sections in $e^-e^+\bar{p}$ system, present results and that of other authors (the energy is measured from the -0.49973 a.u. $\bar{\mbox{H}}$(1) threshold)}
\label{others}
\end{table}

\begin{figure}[t]
\center{\includegraphics[width=0.99\textwidth]{CS_all.png}}
\caption{Cross sections in $e^-e^+\bar{p}$ system. Vertical solid lines denote binary thresholds, vertical dashed lines mark resonances positions.}
\label{CS_H}

\end{figure}

Cross sections for the processes with positron or antiproton colliding with antihydrogen or positronium respectively are presented in Fig.~\ref{CS_H}.
We see that some cross sections associated with excited antihydrogen or positronium states are large, for example the $\bar{\mbox{H}}(2,s)$ - Ps(1) and Ps$(2, s)$ - $\bar{\mbox{H}}(2, s)$ cross sections for rearrangement processes of positronium and antihydrogen formation.
The positronium ground state elastic channel cross section is also large and is almost constant along a wide energy interval.

In the recent work~\cite{lazau18} cross sections for antihydrogen formation in the energy region between $\bar{\mbox{H}}(2)$ and $\bar{\mbox{H}}$(3) thresholds are studied in details.
For the comparison we plot our results on these cross sections in Fig.~\ref{antiH}.
The interest in~\cite{lazau18} was to find to what extent the resonances
can enhance the antihydrogen formation cross sections.
Our energy resolution is better than that in~\cite{lazau18}, so our pictures may be useful for this purpose.
In both Figs.~\ref{CS_H} and~\ref{antiH} the resonances manifest themselves as peaks
in some of the calculated cross sections.
Resonance energies found by different methods~\cite{ho04, varga08, yu12, umair14} are known with good accuracy.
In figures their positions are marked by vertical dashed lines.
All resonances are clearly seen in calculated cross sections, especially those of new processes which became possible at the previous threshold.
The cross sections of processes  $\mbox{Ps}(1,s)\rightarrow\mbox{Ps}(1,s)$ and $\bar{\mbox{H}}(2,s)\rightarrow\bar{\mbox{H}}(2,p)$ (and also $\bar{\mbox{H}}(2,s)\rightarrow\bar{\mbox{H}}(2,s)$ not shown in the figure) have
sharp minima which look like resonances but do not coincide with any of known resonance positions. We agree with associating these minima in \cite{mitroy95} with the Ramsauer-Townsend effect.

Special attention should be paid to just above the $\bar{\mbox{H}}(2)$ threshold oscillations of the cross sections in Fig.~\ref{antiH}.
We give more detailed plots of $\mbox{Ps}(1)\to\bar{\mbox{H}}(1)$ and $\mbox{Ps}(1)\to\bar{\mbox{H}}(2)$ cross section in the energy region above this $\bar{\mbox{H}}(2)$ threshold in Fig.~\ref{oscill}. Prominent oscillations of both cross sections and their character suggest to associate these oscillations with phenomenon predicted in \cite{gaili63, gaili63b}. According to \cite{gaili63, gaili63b} the energy position $E_n$ of the $n$th maximum of the oscillations must follow the rule
\begin{equation}
\log(E_n-E_{\mbox{th}})=A n + B,
\label{Grule}
\end{equation}
where $A$ and $B$ are constants and $E_{\mbox{th}}$ is the  threshold energy.  We plot the respective quantities for $\mbox{Ps}(1)\to \bar{\mbox{H}}(1)$ and $\mbox{Ps}(1)\to \bar{\mbox{H}}(2)$ near threshold oscillations in Fig.~\ref{oscill}.
As one can see the linear behaviour of $\log(E_n-E_{\mbox{th}})$ is near perfect in both cases of rearrangement cross sections except for last points. The latter can indicate the range of validity of approximations made in
\cite{gaili63, gaili63b} leading to (\ref{Grule}).
As for the behaviour of the $\mbox{Ps}(2)\to \bar{\mbox{H}}(n\le2)$ cross section on the right panel of Fig.~\ref{antiH}, we obviously cannot make such quantitative analysis of above the $\mbox{Ps}(2)$ threshold oscillations. Nevertheless we
can agree with \cite{lazau18} that there is an oscillation with the energy position close to $-0.06194\ a.u.$, which was also found earlier in \cite{Hu2002}.

\begin{figure}[t]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{CS_1SPs-1S.png}}\\
Ps(1)--$\bar{\mbox{H}}(1)$
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{CS_1SPs-2S+P.png}}
Ps(1)--$\bar{\mbox{H}}(2)$
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{CS_2S+PPs-1S+2S+P.png}}\\
Ps(2)--$\bar{\mbox{H}}(n\le2)$
\end{minipage}
\caption{Antihydrogen formation cross sections. Black triangles mark points given in~\cite{lazau18}.}
\label{antiH}
\end{figure}

\begin{figure}[t]
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[width=0.99\textwidth]{CS_1SPs-1S_enlarged.png}}\\
(a)
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[width=0.99\textwidth]{CS_1SPs-2S+P_enlarged.png}}
(b)
\end{minipage}
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{Gailitis_1SPs-1S.png}}\\
(c) Ps(1)--$\bar{\mbox{H}}$(1)
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{Gailitis_1SPs-2S.png}}\\
(d) Ps(1)--$\bar{\mbox{H}}$(2,s)
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=0.99\textwidth]{Gailitis_1SPs-2P.png}}\\
(e) Ps(1)--$\bar{\mbox{H}}$(2,p)
\end{minipage}
\caption{(a), (b) --- detailed plots of $\mbox{Ps}(1)-\bar{\mbox{H}}(1)$ and $\mbox{Ps}(1)-\bar{\mbox{H}}(2)$ cross sections in the energy region above the $\bar{\mbox{H}}(2)$ threshold. For these cross sections, the logarithm of the relative energy positions $\log(E_n-E_{th})$ of oscillations maxima with respect to their numbers $n$ are depicted in figures (c)--(e).}
\label{oscill}
\end{figure}

\subsection{$e^+e^-\mbox{He}^{++}$ scattering}
Positron-positive helium ion is an example of positron--atomic target scattering
in which asymptotic Coulomb interaction is present in one of the configurations.
There are a number of calculations in a wide energy region, among them are close coupling calculations using two center basis functions expansion~\cite{bransd01, kadyr18} and EFS-CDW method~\cite{jiao15}.
But to the best of our knowledge, there is lack of published results of calculation for the low-energy region.
In this work we have calculated K-matrices of all possible scattering processes in $e^+e^-\mbox{He}^{++}$ system in the total energy range from $-1.9997\ a.u.$ to $-0.12496\ a.u.$ with the energy step of calculation $0.0007\ a.u.$
In this interval elastic, excitations and rearrangement processes leading to He$^+$($n=1,2,3$) and Ps($n=1$)
atom states are possible.
The maximum linear size of K-matrix equals 7.
We proceeded as in the case of $e^+e^-\bar{p}$ system, using different sets of discretisation and cut-off function parameters for subsequent energy intervals between thresholds.
These intervals and corresponding parameters are given in Table~\ref{Hep_discr}.
We hold the accuracy of calculation  with errors not exceeding  1$\%$.

\begin{table}[t!]
\centering
\begin{tabular}{c|c|c|c}
\hline
 & He$^+$(1) -- He$^+$(2) & He$^+$(2) -- Ps(1) & Ps(1) -- He$^+$(4)\\
\hline
First component\\
\hline
$x_{\max}$ & 10.6 & 21.2 & 35.4 \\
$y_{\max}$ & 35.4  & 99 & 120 \\
$n_x$ & 60 & 60 & 90 \\
$n_y$ & 270 & 810 & 960 \\
$n_z$ & 15 & 27 & 21 \\
$x_0$ & 2.1 & 2.8 & 5.8 \\
\hline
Second component\\
\hline
$x_{\max}$ & 15 & 50 & 60 \\
$y_{\max}$ & 7.5  & 20 & 35 \\
$n_x$ & 60 & 90 & 105 \\
$n_y$ & 60 & 120 & 195 \\
$n_z$ & 15 & 18 & 42 \\
$x_0$ & 1 & 1 & 3.5 \\
\end{tabular}
\caption{Discretisation and cut-off function parameters used in calculations of $e^+e^-\mbox{He}^{++}$ system cross sections. The column headings denote the energy intervals associated with atoms excitations thresholds.}
\label{Hep_discr}
\end{table}

The calculated values of cross sections are tabulated in Table~\ref{Hep_CS}.
Cross sections for the processes with positron or helium core colliding helium ion or positronium are presented in Fig.~\ref{CS_Hep}.
The largest cross sections are, as in the case of $e^+e^-\bar{H}$ system, the elastic ground state positronium cross section, and also that one which is associated with excited helium ion states.

\begin{table}[t!]
{\small
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\backslashbox{}{$E$, a.u.} & 1.55 & 1.60 & 1.65 & 1.70 & 1.77 & 1.80 & 1.83 & 1.86 \\
\hline
$\sigma_{\mbox{He}^+(1)\rightarrow \mbox{He}^+(1)}$ & 0.000855 & 0.00101 & 0.00116 & 0.00133 & 0.00158 & 0.00168 & 0.00178 & 0.00188 \\
\hline
$\sigma_{\mbox{He}^+(1)\rightarrow \mbox{He}^+(2,s)}$ & $\sim$e-09 & $\sim$e-08 & 2e-07 & 6e-07 & 2.6e-06 & 4.4e-06 & 6.9e-06 & 1.1e-05\\
\hline
$\sigma_{\mbox{He}^+(1)\rightarrow \mbox{He}^+(2,p)}$ & $\sim$ e-10 & $\sim$e-08 & 3e-07 & 2.5e-06 & 1.1e-05 & 1.8e-05 & 2.6e-05 & 3.6e-05\\
\hline
$\sigma_{\mbox{He}^+(1)\rightarrow Ps(1)}$ & & & & & 1e-07 & 1e-07 & 2e-07 & 3e-07\\
\hline
$\sigma_{Ps(1)\rightarrow Ps(1)}$ & & & & & 20.6 & 19.6 & 8.82 & 3.00\\
\hline
$\sigma_{Ps(1)\rightarrow \mbox{He}^+(2,s)}$ & & & & & 0.366 & 0.102 & 0.0433 & 0.0199\\
\hline
$\sigma_{Ps(1)\rightarrow \mbox{He}^+(2,p)}$ & & & & & 0.0944 & 0.0214 & 0.00876 & 0.00584\\
\hline
$\sigma_{\mbox{He}^+(2,s)\rightarrow \mbox{He}^+(2,s)}$ & 1.12 & 3.35 & 6.64 & 6.63 & 5.11 & 4.59 & 4.10 & 3.66 \\
\hline
$\sigma_{\mbox{He}^+(2,p)\rightarrow \mbox{He}^+(2,s)}$ & 5.34 & 4.57 & 2.76 & 1.35 & 0.866 & 0.832 & 0.820 & 0.815\\
\hline
$\sigma_{\mbox{He}^+(3,s)\rightarrow \mbox{He}^+(3,s)}$ & & & & & & 9.87 & 18.4 & 11.7\\
\hline
$\sigma_{\mbox{He}^+(3,s)\rightarrow \mbox{He}^+(3,p)}$ & & & & & & 15.7 & 1.62 & 1.21\\
\hline
\end{tabular}
}
\caption{Scattering cross sections in $e^-e^+\bar{p}$ system (the energy is measured from the -1.9997 a.u. He$^+$(1) threshold)}
\label{Hep_CS}
\end{table}

\begin{figure}[t]
\center{\includegraphics[width=0.99\textwidth]{CS_all_Hep.png}}
\caption{Cross sections in $e^-e^+\mbox{He}^+$ system. Vertical solid lines denote binary thresholds, vertical dashed lines mark resonance positions.}
\label{CS_Hep}

\end{figure}

All scattering cross sections associated with non excited atom states follow the well known laws of threshold behaviour~\cite{baz69}, which are presented briefly in Table~\ref{thresh} where $p$ means the relative momentum between
target and projectile.
The only exception is the cross section for $\mbox{He}^+(1)\to \mbox{Ps}(1)$ process.
At the $-0.25\ a.u.$  threshold, according to Table~\ref{thresh}, it should tend to zero linearly as $p\to 0$, but instead it grows up to some constant value.
This anomaly can be regarded as a sign of near threshold resonance, see the discussion below.
The near $\mbox{He}^+(2)$ threshold oscillations of cross sections for $\mbox{He}^+$  excitation to $\mbox{He}^+(2, s)$ and $\mbox{He}^+(2, p)$ reactions are evidently due to their very small values being disturbed by numerical errors.
The threshold behaviour of scattering cross sections associated with excited atom states is more complicated due to the degeneracy of energy levels and requires special treatment~
\cite{gaili63,gaili82,kvits89}.

\begin{table}[t!]
\begin{tabular}{c|c|c}
Process: & not charged & charged \\
\hline
elastic & const & $1/p^2$ \\
slow$\to$fast rearrangement & $1/p$ & $1/p^2$ \\
fast$\to$slow rearrangement & $p$ & const
\end{tabular}
\caption{Threshold behaviour of cross sections in the system of two compound particles $a$ and $X$, for elastic scattering and rearrangement to $b$ and $Y$. $p$ is the relative momentum of ingoing or outgoing slow particles, either oppositely charged or not charged, the angular momentum of the system equals zero.\cite{baz69}}
\label{thresh}
\end{table}

For $e^+e^-\mbox{He}^{++}$ system, resonance energies are worse known, there exist a number of discrepancies of results~\cite{igara97,igara04,han12} (and references therein).
Most authors agree that there are two broad resonances at $-0.371\ a.u.$ and $-0.188\ a.u.$~\cite{igara04} and one narrow resonance slightly below the positronium ground state formation threshold at $-0.250\ a.u.$~\cite{igara04,han12}.
These resonances positions are marked in Fig.~\ref{CS_Hep} by dashed vertical lines (dashed vertical line at
$-0.250\ a.u.$ almost coincides with vertical line denoting the positronium ground state threshold and is not visible).
We do not see the usual singular behaviour in the cross sections in the vicinity of the narrow resonance.
However, the discussed above anomalous threshold behaviour of the $\mbox{He}^+(1)\to \mbox{Ps}(1)$ process cross section at $\mbox{Ps}(1)$ threshold can indicate the presence of a resonance.
Broad resonances are not seen in the cross sections expectedly.

To check the existence of broad resonances we have used another approach based on the complex rotation method applied to the Schr\"odinger equation~\cite{yar98}.
We have found these broad resonances, their positions and widths are given in Table~\ref{res} and compared with results of~\cite{igara04}.

\begin{table}[t!]
\begin{tabular}{c|c|c}
Present work & (-0.3704, 0.1297) & (-0.1857, 0.0395) \\
\hline
\cite{igara04} & (-0.3705, 0.1294) & (-0.1856, 0.0393) \\
\end{tabular}
\caption{Broad resonance in the $e^+e^-\mbox{He}^{++}$ system energies ($E_r$, $\Gamma$) (in a.u.)}
\label{res}
\end{table}

The sharp local minimum is again seen in the $\mbox{Ps}(1)\to \mbox{Ps}(1)$ cross section for the direct process with neutral target. As above we assign this  minimum with the Ramsauer-Townsend effect.

\section{Conclusion}

In this paper the detailed calculations of low-energy reactive scattering in $e^-e^+{\bar p}$ and $e^+e^-\mbox{He}^{++}$ systems
for the zero total angular momentum have been performed with the use of the FM equations in total angular momentum representation. The total angular momentum representation for FM equations assumes all partial waves in subsystems to be included in the formalism, thus in this respect  the results obtained should be considered as complete.

The calculated cross sections in $e^-e^+\bar{p}$ system reproduce all known resonant peaks.
The sharp minima in elastic cross sections
${\bar p}-\mbox{Ps}(n=1)$ and $\mbox{He}^{++}-\mbox{Ps}(n=1)$ at low relative energies  display the  Ramsauer-Townsend effect. The Gailitis Damburg oscillations of the $\mbox{Ps}(n=1) \to \bar{\mbox{H}}(n=1)$ and $\mbox{Ps}(n=1) \to \bar{\mbox{H}}(n=2)$ cross sections just above the $\bar{\mbox H}(n=2)$ threshold are discovered
and the proper spacing with respect to the threshold of oscillation maxima are verified.

The two known  broad resonances \cite{igara04}  in $e^+e^-\mbox{He}^{++}$ system do not contribute into the cross section profile. The anomalous threshold behavior of the $\mbox{He}^+(n=1)\to \mbox{Ps}(n=1)$ cross section we suggest to explain by the existence of the narrow resonance found in~\cite{igara04,han12}.

\ack
This work was supported by Russian Foundation for Basic Research grant No. 18-02-00492. The calculations were performed on resources of the Computational Center of St Petersburg State University.

\section*{References}

redpajama_set_name: RedPajamaArXiv


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I start this song with the notion that when I see injustice, I will do as I can to expose it. The basis of this song surrounds subjects of gender inequality, racial tension, political strangulation, and my frustration.
Our selfish, egotistical lifestyle which continually disrupts the planet. Held down by our own greed and desire for materials. Losing touch with who and what we are to the Earth and how eventually Mother Nature will strike us down.
I wrote this song when I was 18 just after Lost In Transit. My innate desire for the road. The burning sensation that always kept me urging for more and all things different. Bound to a lifestyle of seeking out this world. Embracing it.
A song about my undeniable hatred toward politics, Donald Trump and the facade of democracy through capitalistic greed. Reaching through history to show how the political game has been corrupted for centuries. How the United States continually embarks in wars for profit.
Thoughts that go beyond the scope of the breadth of our existence. Seeing our infinitely small world and how we are all stardust; moving parts throughout a cosmic realm we have no possibility to yet imagine. Coming to terms with that, finding beauty and meaning in life regardless of the fact that our lives are so minimal to space and time.
A song about coming into our own. My contemplation of whether or not this world (society) will actually allow humanity to progress through the stagnant mindsets that have belittled and held down love of all kind. How knowledge is a pretext of false opinions that have constantly been upheld by ancestors who wronged us with their prudish morality.
Clearly a song written for Tyema. It is meant for her and many would not understand the dynamics of our social life which ultimately led to this. Friends and family alike have tried to pull our love down in many ways, but in the end, we always remain strong and true to each other. This is an affirmation to that.
A rap parody of sorts that makes light of the "ice" lyrics heard in many hip hop songs over the last decade. Really, this song is strictly about skiing. Check out the music video on YouTube : On Ice Music Video
Written about my constant and inexhaustible addiction for travel, adventure, excitement and worldly knowledge. The world is my home. It always has been. Also an understanding that in this way of life I am a Phantom to others. You may never see me again, and if you do, surely I will be a new man, for I am always growing and changing.
My distaste toward religion and all those who cower under the veil of fallacy. How religion has destroyed more than it has ever aided. A vicious plot to pull resources from the weak, to gain trust for malice and disengage worldwide unity.
Meant to be a comical play on hip hop topics today, but with a spin. This song is about cooking, loving, and traveling in our bus.
Another love song for Tyema. Simply, the world is at our feet and we have so much to live, see and do. A brief meeting that turned into a lifelong adventure together.
This is a new song about finding the strength to seek out truth of self, and in that realization an overwhelming sense of clarity pours through the mind and body. It also communicates how I felt that I had to banish the thoughts, feelings, and actions of the past that no longer serve my life for any good in order to reach this truth. No longer holding onto the past to determine my present state of mind.
Simply a fun summer song about love and travel. The need to explore, to live and let be. To remain in constant ecstasy through love. To not hold onto pain, to let it all go. Be free.
Understanding that I spent many years wallowing in darkness, forgetting that I was actually alive. Taking the time to cope with it, understand it, and then forget the past. Move forward in love, light and happiness.
Exposing those who may listen to the horrid advancements in technology that have removed people from each other. Hiding behind cellphone lights. Clinging to social media. Brain dead to the world around them. Lost in cyberspace. Drifting endlessly in a stagnant state of delirium with a removed sense of focus toward anything real.
Incorporates many ideas and motifs associated with a previous story "The Tunnels Left Within" I have been writing for quite some time. The dark days of young and how I have turned those feelings into inspiration for a renewed sense of life.
All to do with our drug infested cities, towns and parks. The disgust and disease. How we are turning into zombies who have no regard to ourselves, our fellow people or our planet.
* All of these songs below were written and recorded between the ages of 17-21 (2009-2013) *
This song was the iconic character that changed my perception of how I should live my life. It was written during and just after my first trek with the bus across the states when I was 18 years old. This is when I realized the bus was my home, and life on the road needed to be a priority.
Is the anticipation of another journey, the determination to fulfill the void that was ever-present. It is the simple understanding that without exploration and adventure, I am lost in the world.
Was a special moment in my life when I was backpacking through the Virgin Islands and British Virgin Islands. It was a moment of pure bliss and a noticeable detriment of miscalculated reasoning. It was the beauty and shambles of a whimsical adventure.
This song is a straight-forward attack at the irony of the American democratic system. It exemplifies the similarities of kingship and our current state of affair; our transgression into what we originally steered away from in the name of a free country. It explains the repeated lies and scandals that drew us into international battles, and the wit of the presidency under the Elastic Clauses.
Autonomy is the essence of political corruption through stagnant greed. It is the basis that our government is owned, and even our elected representatives have not the power to control the motives of the real players.

redpajama_set_name: RedPajamaCommonCrawl


Our regular monthly meeting to discuss how to make Toluca Lake a safe and great place to live is usually held in a stuffy conference room. But, our President, Richard Bogy, treated to meeting at Lakeside Country Club which was a really nice change of pace.
Steve Hampar (V.P.). Not pictured: Reyn Blight, Susanne Hayes, Ryan Kugler.
Toluca Lake Homeowners Association presents $7000 check to East Valley PALS!
6th Annual Toluca Lake Turkey Trot raises $7000 for PALS!


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What motivates you through life and why do we do the things we do? Do you only do things for the rewards (or to avoid negative consequences)? Is there anything you do just for the fun of it anymore? In this episode, we talk about the difference between intrinsic and extrinsic motivation and what role each plays in our lives – from childhood to adulthood. Touching on some of our earlier motivations – playing piano, dusting stairs, and working at a job that wasn't entirely rewarding, we dive into this topic with the 2016 INTRINSIC Cabernet Sauvignon from Washington.


Is Objectivism Compatible with Nationalism?
The Nationalist Community of today is comprised of many newcomers, many of whom (myself included), came from a Libertarian background. The Trump phenomenon of 2016, propelled by a newfound awareness of the severity of the migrant crisis, caused many of us to reconsider our lofty and often impractical Libertarian ideals due to their incompatibility with a non-White supermajority. Although my views have shifted significantly because of this, I still believe there is a great deal of good that can be gained from the Objectivist philosophy of people like Ayn Rand, Leonard Peikoff and Nathaniel Branden.
An unfortunate reality of philosophy is that very few people have the ability to "take the good and leave the bad". To most people, everything is either black or white, good or bad, when life is much more complex than that. Many people read Rand, fall in love with her and accept everything she says as gospel. This is obviously destructive and unphilosophical. Others read her (or more often hear something about her) and totally reject every word she ever wrote. This is equally as bad.
Rand was very clear on emphasizing that "pursuing one's own happiness" does not mean doing harm to others as a means of achieving those ends nor does it mean participating in hedonistic and trivial pursuits. As someone who lived through the horrors of the Soviet Union, her philosophy focused on the morality of value and a disdain for the concept of self-sacrifice. She saw with her own eyes that the proselytizers of this dogma of self-sacrifice were the most wicked and selfish people on Earth. Much confusion and outright anger has erupted over her condemnation of altruism and self-sacrifice. As a moral philosopher and epistemologist, her definitions are precise and any attempt to understand Objectivism must be made with her definitions in mind. Many would call her selfish for her views and she would accept this as a badge of honor, not because she accepted their definition of selfishness, but because she accepted her own.
I believe we can apply her philosophy to the immigration crisis in a way that is compatible with Nationalist views.
Western nations are sacrificing our safety and prosperity for the sake of third-world migrants who are not only ungrateful, but are openly hostile towards us. By subsidizing third-worlders with billions in foreign aid and by encouraging parasitic behavior in the form of welfare and taxpayer funded social entitlements we are hurting everyone of all races! The rest of the world is dependent upon White nations and by sacrificing ourselves on the altar of political correctness, pathological altruism and neurotic self-loathing, we are allowing the destruction of the entire species! These parasitic migrants might receive some short-term gain in the form of free "gimmies", but even that hurts them long-term because it disincentivizes hard work and productivity, which is overall detrimental to their self-esteem and self-efficacy because it makes them dependent upon another race for their very survival.
This madness cannot go on forever, eventually it will reach a point where we have nothing left to sacrifice. Ayn Rand's novel Atlas Shrugged is a prophetic portrayal of what happens when the producers, builders and inventors in society are forced by the lazy, weak and wicked to sacrifice everything for the "collective good"; civilization collapses and the world in thrust into chaos and upheaval!
Ayn Rand has received a tremendous amount of criticism from both the Far Left and the Far Right because of her criticisms of Collectivism. While I don't fully agree with her conclusions, I tend to think that the "Individualism vs Collectivism" debate is a false dichotomy that can be easily resolved when reframed in the context of "Microcosm vs Macrocosm". The old "secure your own oxygen mask before securing the masks of those around you" airplane analogy is my go-to when simplifying this unnecessarily complex philosophical misunderstanding. It is not evil to put on your own mask before helping those who are unable. You do not wish for them to suffocate, you simply understand that if you, as an able-bodied adult want to save both yourself as well as those who are unable to save themselves, you must take care of your own immediate interests first!
Now apply this to the question of immigration and foreign aid. I do not wish for the third world to perish. I do not laugh at their misfortune. I do not wish to see these people suffer. I want them to get their act together! They are suffering because we made them dependent on our help! Whites cured most deadly diseases that afflicted the third world and globally we give them hundreds of billions of dollars every year! This has caused their population to skyrocket to unsustainable levels. Because of the shortage of resources, skills and infrastructure needed to maintain their exploding population, their nations break out in crime, war, famine and political instability. Then, the media shows pictures of poor starving Africans and it makes Whites feel guilty for having prosperity. These Whites then go on to donate their money and enact policy that gives aide to these countries which continues the vicious cycle of misguided altruism and self-sacrifice. We are giving these people the rope to hang themselves with, and in the process, we are bankrupting our own nations which inevitably turns our own homelands into third world nightmares!
The results of generations of these destructive policies are proof of why they don't work and why we need to build two walls, stop all immigration and cut all foreign aid to these unsustainable nations. I have written about this extensively elsewhere, but suffice to say, we have our own problems and we need to take care of those first before we even consider helping anyone else. In the meantime, we need to deeply contemplate what "helping" really means and what it should look like in practice.
Sometimes the right thing to do isn't always the "nice" thing to do. For Rand, Objectivism was all about doing away with the niceties, social stigmas and philosophical fallacies of "the crowd" and getting to the root of what is and is not ethical. Her pioneering, abrasive and argumentative Aquarian personality didn't make her many friends but did make her plenty of enemies on all sides of the political spectrum. She was often characterized as an "enemy of the worker" and "pro-cronyism" but ironically, her books were most popular with blue-collar workers and least popular with stuffy, establishment, white-collar academics.
A specific gripe I have with Rand was her love for Capitalism. As someone who grew up in the Soviet Union where the motto of the worker was: "we pretend to work, they pretend to pay us", I can see how she was attracted to Capitalism when she moved to New York. . Obviously, we know that Capitalism has many problems, she was not unaware of this. Her novel Atlas Shrugged is full of examples of corrupt Capitalists working behind closed doors with government to stifle competition through monopolization and legislative regulation. To be fair, I believe as a moral philosopher, her love was with the concept of Capitalism as fair exchange of value for value. It is, however, important to highlight the fact that it doesn't work that way in practice. It is, in fact, many of these wealthy Capitalists who went on to become the biggest supporters and financiers of Communism, as she depicted in Atlas Shrugged with the character James Taggart.
Where I fault Rand the most was for her inability to take these lofty principles of moral philosophy and apply them to the individual. As much of an "Individualist" as she was, her biggest failure in my opinion was in taking these concepts that she portrayed in a fictional macrocosm or an intangible philosophical non-fiction and not making them practical and implementable for the individual in the microcosm.
This is where the work of Nathaniel Branden comes in. Branden, a psychologist, former student, colleague and lover of Ayn Rand, highlights the epidemic lack of self-esteem that plagues mankind. Those without self-esteem will be inclined to seek power over others, to steal from others, to disrespect others and to force their neurotic will upon others. The modern Left is an extreme example of a lack of self-esteem. Rather than holding themselves accountable for their lack of skills, intelligence, work ethic, fortitude, ambition and ethics, they blame everyone else (particularly a certain orange-skinned father-figure) for all of their failures. Rather than shrugging off name-calling and teasing, they demand "safe spaces", "acceptance" and immunity from criticism. Rather than working hard to earn a living like the rest of us, they demand wealth redistribution, Universal Basic Income and social entitlements that keep them in a state of complete dependency on others. This dependency decreases their sense of self-esteem and proportionally increases their sense of self-loathing, which fuels their anger, vitriol and "social activism" that enslaves us in the clown-world that they are creating in their own wretched self-image.
In Atlas Shrugged, Dagni's brother James is the epitome of a man who is deficient in self-esteem. He is incompetent, insecure and corrupt. He feels undeserving of his title of CEO of Taggart Transcontinental and overcompensates for his ineptitude by turning his business (upon which millions of people rely) into a vessel for his self-destructive, virtue signaling social agenda. He takes credit for all of his sister's triumphs and blames her for all of his failures. It is this kind of mentality that has so viciously afflicted mankind today. Rather than working hard to become our very best and develop the self-esteem that is so vital to our psychological well-being, people will make excuses for why they can't do this or that. They will point fingers over who is to blame for their plight and while these may be valid in some cases, it is a slippery slope that results in the disempowerment of the individual. When the individual makes a habit of abdicating personal responsibility, they fail to learn the lessons from the personal shortcomings that are within their control. It is crucial that you take responsibility for your failures and develop the emotional honesty to hold yourself accountable when you make mistakes. It is, however, much easier to place blame on others or depend on others to rescue you from the prison that you caged yourself in! The danger of this mentality on the macrocosmic level comes when an entire society believes itself always just one law, one election, one purge away from the absolute resolution of all their problems. This is extremely dangerous for everyone as the consensus trance of the crowd serves as judge, jury and executioner for the innocent, as we saw in the case of James Fields and many other pro-White activists. The "court of public opinion" and the political pressure for prosecution it creates is the product of the perpetual uninformed outrage of the masses who are seeking villains to blame for all the problems of the world. Again, many of these problems are external in nature, we are all surely aware and vaguely in agreement as to who or what creates these problems. I am not absolving these bad actors of their misdeeds, nor do I think they deserve to get away with their crimes. I am simply saying that we must hold ourselves equally as responsible for our actions as they are for theirs. You can't criticize the corrupt media and then continue to watch CNN every day. They have power because we give them power. I'm not saying "don't place blame" I'm saying "place blame where blame is due". The problem we have now is no one is being held accountable for their actions. The evil-doers in government, finance, media and business are never held accountable for their actions and the individual never holds themselves accountable for the plethora of ways that they empower these evil-doers.
Another important disagreement I have with both Branden and Rand was their rejection of race. I believe Whites are suffering because we have developed racial self-loathing due to being indoctrinated with anti-White propaganda. Self-esteem is important, I think we would all agree on that, but a part of self-esteem is being proud of your race. Ultimately, you are who you are because of your race, not solely because of it, but it is a major factor that contributes to why you are who you are. I think their failure to mention race was due to their emphasis on the individual's ethics rather than the collective and from the perspective of a moral philosopher this makes sense. A major problem we have within the Nationalist Community today is the belief that race is the sole factor that determines your greatness. You are the "Ubermensch" because you are White and everyone who isn't White is automatically an "Untermensch". The Ubermensch is something you become, not something you magically are. Granted, genetics helps, but it won't take you 100% of the way. That kind of magical thinking disincentivizes individual achievement and moral responsibility and places the emphasis on the achievement and morality of the race as a given, rather than the product, of hard work and dedication. The greatness of the race is a secondary outcome of individual achievement, not the primary. Mozart didn't create such beautiful music because he wanted to show how great the White Race is. He did it because he was passionate about it and dedicated his life to making music that would stand the test of time. We are at a place now where we are lacking in modern examples of White achievement and greatness. We tend to look at White greatness as an anachronism and because of this, we become stuck in a desire to go back to a caricature of the 1950's. We have idealized fantasies of a time that we never knew. It is impossible to base your philosophy on returning to a foggy memory. You must play the hand you were dealt, move forward and build something better. At some point we forgot that and we stopped building. Because the individuals stopped building, so did the race. Microcosm, Macrocosm.
We must bear in mind that most people will never live the heroic, creative and productive existence of a Dagni Taggart or a John Galt. The Untermensch of all races are too traumatized and neurotic to escape their own "mind-forged manacles", it is these people who are waiting on a "savior". Politically, we can provide them with such a "savior" as well as a nationalistic culture of morals, tradition and hierarchy to preserve the stability of society and prevent the collapse into absolute chaos and mayhem (which will surely ensue if we continue down our current trajectory). However, we must also allow enough freedom and space for the Ubermensch to reach their intellectual and productive heights without being weighed down by the envious and self-destructive "looters". For our nations to flourish we must have an intellectual, moral and productive aristocracy. A hierarchy of merit and virtue, not of theft, corruption and deception. We need to get back to what made the White Race great: Innovation, productivity, courage, honor and justice. It is because we abandoned these principles in favor of whatever the meek, poor and downtrodden hordes of foreigners demanded of us that we lost control of our destiny. We need to return to a rule of the strong over the weak, not the weak over the strong as we have today. Strength is not about hurting others and it certainly is not about sacrificing your people's interests for those of another people. Strength is about commanding respect, not demanding it. We forfeited our personal responsibility in hopes that a new leader, a new law or a new movement would come along and save us. We adopted a passive approach to life rather than an active one. Because of this, foreign interests, unethical individuals and groups were more than happy to take advantage of our weaknesses and exploit our misguided virtues for their own self-interest.
As much as we have been wronged by others, as reprehensible as these actions were, we must never forget that responsibility is a two-way street. We cannot blame others for what they've done to us if we refuse to accept our complicity and passivity that enabled them to hurt us for so long. Only when man frees himself from the chains he willfully places upon himself can we ever remove the chains placed upon us by others. With freedom comes responsibility. Objectivism teaches you to free yourself so that you may be empowered to fight for the freedom of all.


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*1: Guideline operating hours before the panel brightness is reduced to half when the panel is used to display motion pictures in the Standard mode.
Afterimages (burned-in images) and malfunctions are not taken into consideration.
*2: An optional 3D IR Transmitter and 3D Eyewear are required for viewing 3D images.
*3: Based on IEC 62087 Ed.2 measurement method.


Over the nearly 40 years that Solar-X of Sarasota has been in the window film business, there have been tremendous changes in the industry. We've gone from six or seven films to hundreds. Manufacturers have improved the quality, and extended the life of the products. Window film has "come of age," and is being taken seriously by building mangers and homeowners alike as an important part of their overall energy savings strategy.
Have window film dealers grown in sophistication along with the increased technology of their products? Well…some have. But many today can't tell you with any degree of competence about window film and its compatibility with today's more complicated glazing systems. They can't explain the actual testing that window films undergo, and they can't explain why those tests are important in meeting codes. They might be able to tell you how much heat their films stop, but they can't tell you how the film works in order to do it.
As this area's exclusive Panorama® Window Film dealer, all members of our sales and installation staff are required to participate in continuing education – from the President of the company to the newest installer – if we wish to maintain our active status. That means we are going to be up-to-date on the latest products, technologies, installation techniques, and glazing systems. We'll know which films are the best for any given application, and be able to explain why. Yet Panorama® is the only window film brand that requires this stringent training. Why is that?
As our clients, we feel you deserve nothing less than the best. Our product recommendations are based on knowledge and training – not hype. And the quality of our workmanship is unmatched. That's why discerning homeowners and business professionals in the Bradenton, Sarasota, and Venice area have been putting their energy-savings and comfort needs in the hands of the Solar-X professionals since 1973. Contact Solar-X today!


The South Carolina Corps of Cadets' first parade of 2004 is at 3:45 p.m., Friday, Jan. 16 on Summerall Field. Click on the headline for a complete schedule of spring semester parades.
Who was Johnson Hagood and why is The Citadel's football stadium named after him? Click on the headline to learn the answer. For more about the future of the stadium.Click here.
Registration for the spring semester for The Citadel's College of Graduate and Professional Studies is now underway and will end with late registration on January 5th, the day that classes begin for the spring term for CGPS students. Current students may register online anytime.
Sen. Lindsey O. Graham, R-S.C., will deliver a Greater Issues address to the South Carolina Corps of Cadets, faculty and staff at 11:15 a.m., Tuesday, Dec. 9, 2003 in McAlister Field House.
The South Carolina Corps of Cadets will conclude its annual Christmas Candlelight service for the Lowcountry at 7:30 p.m. today in Summerall Chapel.
The college's dining services provider, celebrating the 35th anniversary of its partnership with The Citadel, honored employees with 25 years or more service on Friday, Dec. 5.
Cadets, faculty and staff will remember Cadet Gregory F. Pope, a sophomore in Mike Company, who died in a single-car accident Nov. 23. The memorial service begins at 3:15 p.m. Friday in Summerall Chapel.
Reports that there has been an outbreak of flu at The Citadel have been overstated, college officials announced Monday.


Friday 30 November is #BlueBeanieDay. Show your support for #webstandards & #a11y. Put blue on your bean!
Web standards hold the key to accessible, cost-effective web design and development, but you wouldn't know it from surveying most big commercial sites.
It is worth the reading and then watching the video to see the pinball machine made entirely from Legos in action.


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Bourbon barrel shortage to last two years, distillers warn
6th May, 2014 by Becky Paskin
The current shortage of American oak blighting the Bourbon industry will be "short lived", but is expected to last up to two years, distillers have warned.
The American oak shortage may continue to affect the Bourbon and Tennessee whiskey industry for two years
A shortfall in the amount of lumber delivered to cooperages over the past six months has heavily impacted the number of new American oak barrels available to distilleries.
While some larger distilleries claim to be able to secure enough barrels to maintain supply, the pressure is being felt across the entire industry, with some distillers predicting the plight to continue for between 12-24 months.
"There is definitely a shortage of barrels on the marketplace," said Paul Hletko, founder of Few Spirits in Evanston, Illinois. "I have friends that have had to spend US$100,000 on new stainless steel tanks to put their new make spirit into while they wait for new barrels. People have been talking about it for a few months, but the full pain is just starting to be felt."
An especially harsh, wet winter in the States has caused difficulties for the logging industry, which in turn, has impacted the number of barrels produced by the country's cooperages. The situation has exasperated an existing shortage caused by swift and strong global demand for Bourbon and Tennessee whiskey.
Harsh winter was "bad timing"
While several cooperages contacted by The Spirits Business refused to comment on the issue, many distillers predicted an end to the crisis in the short to medium term.
"It will change this year now we're coming out of the winter," said Jeff Arnett, master distiller of Jack Daniel's in Tennessee, which despite owning its own cooperages, has also felt the impact of the wood shortage.
"There's a lot of active logging going on around us here, and while we've seen our inventory of wood at our stave mills reduced down to a minimum pile, we're beginning to regain inventory there. Even though people aren't getting their orders satisfied right now, it's a very short lived issue."
Meanwhile Darek Bell, founder of Corsair Distillery in Nashville, Tennessee, believes the deficiency will continue for up to two years, "so long as there's not another harsh winter again".
"The barrel shortage has ramped up over the last 4-5 months; it just happened to be bad timing, as we had an especially harsh winter. It will clear itself up. It will take a while to sort out but it will. Loggers will now over-harvest the wood as prices have shot up, then next year there will be too much and prices will tank."
The shortage has prompted discussions over the use of rejuvenated casks in the Tennessee whiskey production process, which Diageo claims would help new craft distillers enter the market.
For a more in-depth review of the American oak shortage, and its effect on the Tennessee whiskey industry, see the May issue of The Spirits Business.
The Rock names Mast-Jägermeister Teremana Tequila distributor
Tito's overtakes Smirnoff as top-selling spirit in US
Constellation Brands' wine and spirits sales drop 9%
Powers Irish whiskey unveils new bottle design
Barbados rum GI proposal puts 'no restrictions' on production
Diageo begins restoring famous Edinburgh clock
Conor McGregor's Proper No. Twelve whiskey joins US delivery platform
South Korea 'most attractive' drinks market in Asia
Monkey Shoulder creates '100% accurate' jigger
Baileys Red Velvet Cupcake launches in UK
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