text:
\section*{1. Introduction}
\vspace*{0.5cm} \hspace*{3ex} Supersymmetric Quantum Mechanics
(SUSY QM) \cite{cooper} is a new framework for analyzing
non-relativistic quantum problems. In particular, it helps to
investigate the spectral properties of different quantum models as
well as to generate new systems with given spectral
characteristics (quantum design).
\par Much less attention has been paid to SUSY QM as a tool to
study individual hidden symmetries of the superpartner
Hamiltonians. This problem is reasonable for considering either
one-dimensional quantum systems with internal degrees of freedom
(with matrix potentials) or systems of higher spatial
dimensionality. Only for these classes of systems may the symmetry
operators, which are in involution with the Hamiltonian (and are
independent of it), exist. Thus, the quantum models in
one-dimensional SUSY QM with matrix potentials and the
higher-dimensional (in particular, two-dimensional) models with
scalar and/or matrix potentials are extremely attractive.
\par Regarding SUSY QM systems with matrix potentials, we refer
to papers \cite{matrix}. The SUSY QM systems with an arbitrary ($d
> 1$) dimensionality of space\footnote{Instead of dimensionality
of the space $d$ can be also interpreted as a number of particles,
for example in Calogero-like models of interacting $N$ particles
on a line \cite{calogero}.} were constructed and investigated in
\cite{2d}, \cite{abei}. Such models include both the scalar and
matrix components of the Superhamiltonian. The latter are
interesting either for a description of interacting
non-relativistic particles with spin \cite{scattering},
\cite{neelov}, or for developing supersymmetric quantum field
theory on a spatial lattice \cite{wipf}. The appearance both of
scalar and matrix Hamiltonians in a unique Superhamiltonian
provides an opportunity to consider (starting from a given scalar
Schr\"odinger operator) SUSY extensions that correspond to the
systems with internal degrees of freedom. Some SUSY extensions of
this type were considered in \cite{neelov1} (Calogero-like models
of $N$ particles on a line) and in \cite{wipf2} (Coulomb potential
in $d$ dimensions).
\par Among multidimensional SUSY QM models,
those most developed are the two-dimensional ones. Namely,
precisely for these systems second-order supercharges were used to
build the higher-order deformation of SUSY algebra \cite{david1},
\cite{david2}, \cite{ioffe}. In the framework of this HSUSY QM
generalization of conventional Witten's
superalgebra\footnote{One-dimensional HSUSY QM was investigated in
detail in \cite{acdi}, \cite{highod}. }
 one can
avoid the appearance of any matrix components of the
Superhamiltonian, so that two {\it scalar} two-dimensional
Schr\"odinger operators are intertwined by second-order
supercharges. As a by-product of this construction, each of the
intertwined Hamiltonians obeys the hidden symmetry: the
differential operators of fourth order in derivatives exist, which
are not reducible to the Hamiltonian and commute with the
Hamiltonian, \cite{david1}, \cite{david2}, \cite{newmet},
\cite{ioffe}. In the two-dimensional context this means the
complete integrability of the system.
\par Another direction in which to investigate SUSY QM models involves
their connections with classical counterparts. Initially, the
arbitrary-space-dimensionality generalization \cite{abei} of SUSY
QM, mentioned above, was obtained by canonical quantization of a
suitably chosen multi-dimensional system of Classical Mechanics.
Then the quasi-classical limit of some supersymmetrical quantum
models investigated afforded new insight into the properties of
the classical models obtained. In the one-dimensional case, this
limit led to a new SWKB quantization rule \cite{swkb}, which
turned out to be more useful than the standard WKB rule. In the
two-dimensional case, the quasi-classical limit provided an
alternative effective method \cite{david4} for the construction of
integrable systems in Classical Mechanics, which essentially
enlarges the list of such models. The SUSY QM approach also
provides new interesting links with the well-known Hamilton-Jacobi
equation in Classical Mechanics \cite{Aai1}.
\par In the present paper we shall combine both known two-dimensional
SUSY QM constructions - the direct generalization of SUSY with
first-order supercharges and HSUSY with second-order supercharges
- in order to investigate the symmetry properties of the models
obtained, both at quantum and classical level. The paper is
organized as follows. In Section 2, known results about
two-dimensional SUSY QM models and their connections with
classical models, necessary for the original part of the paper,
are briefly summarized. In Section 3, the particular case of
two-dimensional models for second-order supercharges with
intermediate twists are studied: the particular models of
generalized Morse and P\"oschl-Teller potentials are presented
within this common framework. In Section 4, the integrability of
these models is extended onto their matrix - both quantum and
classical - superpartners. In Section 5, the links with classical
Hamilton-Jacobi equation are considered. \vspace*{0.2cm}
\section*{2. Two-dimensional SUSY Quantum Mechanics}
\vspace*{0.2cm}
\subsection*{2.1. The representation of SUSY algebra with first-order supercharges}
\vspace*{0.5cm} \hspace*{3ex}
\par In the two-dimensional case $\vec x = (x_1, x_2)$, the direct (i.e. of first order in
derivatives) generalization of SUSY QM satisfies \cite{2d},
\cite{abei} the conventional Witten's \cite{witten} SUSY algebra
\begin{equation} \{ \hat{Q}^+, \hat{Q}^- \} = \hbar\hat{H};\qquad \{
\hat{Q}^+, \hat{Q}^+ \} = \{ \hat{Q}^-, \hat{Q}^- \} = 0;\qquad [
\hat{Q}^\pm, \hat{H} ] = 0.\label{alg}
\end{equation}
by the 4$\times$4 matrix operators:
\begin{equation} \hat H = \left(
\begin{array}{ccc}H^{(0)}(\vec x)&0&0\\ 0& H^{(1)}_{ik}(\vec
x)&0\\0&0&H^{(2)}(\vec x)
\end{array} \right);\quad i,k=1,2;\quad \hat Q^+ = (\hat Q^-)^{\dagger} =
\hbar^{1/2}\left(
\begin{array}{cccc}0&0&0&0\\
q_1^-&0&0&0\\q_2^-&0&0&0\\0&s_1^+&s_2^+&0
\end{array} \right),\label{suham}
\end{equation}
where two scalar Hamiltonians $H^{(0)}, H^{(2)}$ and one
2$\times$2 matrix Hamiltonian $H^{(1)}_{ik}$ of the Schr\"odinger
type take on a quasi-factorized form:
\begin{eqnarray}
H^{(0)}&=&q_l^+q_l^- = -\hbar^2\partial_l^2 + V^{(0)}({\vec x}) =
-\hbar^2\partial_l^2 + \Bigl(\partial_l\chi (\vec x)\Bigr)^2 -
\hbar\partial_l^2\chi (\vec x);\,\, \partial_l^2 \equiv
\partial_1^2 + \partial_2^2; \nonumber\\
H^{(2)}&=&s_l^+s_l^-=-\hbar^2\partial_l^2 + V^{(2)}({\vec x}) =
-\hbar^2\partial_l^2 + \Bigl(\partial_l\chi (\vec x)\Bigr)^2 +
\hbar\partial_l^2\chi (\vec x);\label{hams}\\
H^{(1)}_{ik}&=&q_i^-q_k^+ + s_i^-s_k^+ = -\hbar^2\delta_{ik}
\partial_l^2 + \delta_{ik}\Bigl(\bigl(\partial_l\chi (\vec x)\bigr)^2 -
\hbar\partial_l^2\chi (\vec x)\Bigr) + 2\hbar\partial_i\partial_k
\chi (\vec x),\nonumber
\end{eqnarray}
the components of the supercharges being of first order in
derivatives:
\begin{equation}
q_l^{\pm}\equiv \mp\hbar\partial_l + (\partial_l \chi (\vec
x));\quad s_l^{\pm}\equiv \epsilon_{lk}q_k^{\mp},\label{2p}
\end{equation}
where $\partial_i \equiv \partial/\partial x_i$ and summation over
repeated indices is assumed. The Planck constant was restored in
formulas (\ref{suham}) - (\ref{2p}), and the ({\it normalizable or
unnormalizable}) zero-energy wave functions of the scalar
Hamiltonians $H^{(0),(2)}$ are now written as
$\exp{(\pm\chi/\hbar)}$.
\par The quasi-factorization in (\ref{hams})
ensures that the last equation in (\ref{alg}) holds, and leads to
the following intertwining relations for the components of the
Superhamiltonian (\ref{suham}):
\begin{equation}
H^{(0)}q_i^+=q_k^+H^{(1)}_{ki};\quad H^{(1)}_{ik}q_k^- =
q_i^-H^{(0)}; \quad H^{(1)}_{ik}s_k^-=s_i^-H^{(2)} ; \quad
H^{(2)}s_i^+ = s_k^+H^{(1)}_{ki}.\label{2dintw}
\end{equation}
These relations play the main role in the SUSY QM approach and, in
particular, they lead to the connections between the spectrum of
the matrix Hamiltonian and the spectra of two scalar ones. We
remark that $H^{(0)}$ and $H^{(2)}$ {\it are not} intertwined with
each other and {\it are not} (in general) isospectral since
$q^+_k\cdot s^-_k \equiv 0$. \vspace*{0.2cm}
\subsection*{2.2. Second-order supercharges}
\vspace*{0.5cm} \hspace*{3ex} An alternative opportunity to
include two-dimensional scalar Hamiltonians in the SUSY QM
framework is based on the supercharges of second order in
derivatives \cite{david1}, \cite{david2}, \cite{ioffe}:
\begin{equation}
Q^+ = (Q^-)^\dag = g_{ik}(\vec{x})\hbar^2\partial_i\partial_k +C_i
(\vec{x})\hbar\partial_i + B(\vec{x}) \label{Q}
\end{equation}
where $g_{ik}, C_i, B$ are arbitrary real functions. In this case,
two scalar Hamiltonians $H^{(0)}$, $\widetilde{H}^{(0)}$ are
intertwined directly without any (matrix) intermediary:
\begin{equation}
\widetilde{H}^{(0)}(\vec{x})Q^+ = Q^+H^{(0)}(\vec{x});\qquad
H^{(0)}(\vec{x})Q^- = Q^-\widetilde{H}^{(0)}(\vec{x}).
\label{intw2d}
\end{equation}
Although no method to find the general solution of the
intertwining relations (\ref{intw2d}) has been proposed, a certain
number of models obeying these relations  have been found for the
cases of hyperbolic (Lorentz) and degenerate metric $g_{ik}.$ One
very important specific property of all these models, which
follows from the intertwining relations (\ref{intw2d}), is their
integrability. Indeed, both Hamiltonians possess the symmetry
operators $R^{(0)}, \widetilde{R}^{(0)}$ of fourth order in
derivatives \cite{david1}, \cite{david2}: \begin{equation} [R^{(0)}, H^{(0)}] =
0;\quad [\widetilde{R}^{(0)}, \widetilde{H}^{(0)}] = 0;\qquad
R^{(0)} = Q^-Q^+; \qquad \widetilde{R}^{(0)} = Q^+Q^-,
\label{symm} \end{equation} which are not, in general\footnote{It has been
proved \cite{david2} that only for Laplacian (elliptic) metric
$g_{ik}=\delta_{ik}$ can the symmetry operators be reduced to
second-order operators, and the corresponding Hamiltonians
$H^{(0)}, \widetilde{H}^{(0)}$ allow the separation of
variables.}, polynomials of $H^{(0)}, \widetilde{H}^{(0)}$.
\par For the case  of Lorentz
(hyperbolic) metric $g_{ik} = diag(1,-1)$ \cite{david1} -
\cite{david4}, the intertwining relations (\ref{intw2d}) can be
rewritten in a reduced form:
\begin{equation}
\partial_-(C_-F)  = -
\partial_+(C_+F), \label{2odmain}
\end{equation}
where $x_\pm = (x_1 \pm x_2)/\sqrt{2}$; functions $C_{1,2}$ were
found to satisfy $C_\pm \equiv C_1 \mp C_2 \equiv
C_\pm(\sqrt{2}x_\pm),$ and $F(\vec{x})= F_1(2x_1)+ F_2(2x_2)$.
Thus, the potentials $V^{(0),(1)}$ and the supercharges $Q^{\pm}$
are expressed in terms of the functions $C_\pm(\sqrt{2}x_\pm)$ and
$F_1(2x_1), F_2(2x_2)$:
\begin{eqnarray}
V^{(0)}, \widetilde{V}^{(0)} & = &
\mp\frac{1}{2}\hbar\Bigl(C'_+(\sqrt{2}x_+) +
C'_-(\sqrt{2}x_-)\Bigr) + \frac{1}{8}\Bigl(C_+^2(\sqrt{2}x_+) +
C_-^2(\sqrt{2}x_-)\Bigr)  +\nonumber\\
&&+\frac{1}{4}\Bigl(F_2(2x_2) - F_1(2x_1)\Bigr); \label{ve2od}\\
Q^+ & = & \hbar^2(\partial_1^2 - \partial_2^2) +
C_1\hbar\partial_1 + C_2\hbar\partial_2 + B;\label{quplus}\\ B & =
& \frac{1}{4}\Bigl(C_+(\sqrt{2}x_+)C_-(\sqrt{2}x_-) + F_1(2x_1) +
F_2(2x_2)\Bigr),\label{be2od}
\end{eqnarray}
where the prime denotes the derivative of function with respect to
its argument. A set of particular solutions of (\ref{2odmain}) was
obtained in \cite{david2}, \cite{david4}, \cite{iv}.
\vspace*{0.2cm}
\section*{3. Two-dimensional models with twisted reducibility of supercharges}
\vspace*{0.5cm} \hspace*{3ex} In the previous Section it was shown
that two different constructions with very different properties
exist in two-dimensional SUSY QM. The first one (Subsection 2.1.)
includes two scalar Hamiltonians $H^{(0)},\,\, H^{(2)}$ (only one
of them has normalizable\footnote{For the case of supersymmetry
not broken spontaneously.} zero-energy ground state wave function
$\Psi_0(\vec x) = \exp (\pm\frac{\chi (\vec x)}{\hbar})$) and
their $2\times2$ matrix partner $H^{(1)}_{ik}.$ The second one
(Subsection 2.2.) contains only the scalar Hamiltonians
$H^{(0)},\,\, \widetilde{H}^{(0)}$ with no information about their
ground-state energy in advance, and both $H^{(0)}$ and
$\widetilde{H}^{(0)}$ {\it a priori} (by construction) obey the
important property of integrability with the symmetry operators
$R^{(0)},\,\widetilde{R}^{(0)}$  of fourth order in derivatives
(see (\ref{symm})). The natural idea is to unite all the above
tempting properties by combining these two constructions, i.e. by
identifying the original Hamiltonian $H^{(0)}$ as the same in both
approaches. More precisely: let $H^{(0)}$ of the form (\ref{hams})
have the superpartners $H^{(1)}_{ik}$ and $H^{(2)}$ in the
first-order scheme, and {\it at the same time} the superpartner
$\widetilde{H}^{(0)}$ in the second-order scheme. It is known
\cite{david2} that the simplest, {\it reducible or
quasi-factorizable}, form of the second order supercharges
$Q^{+}=(Q^-)^{\dagger}=q_i^{+}\tilde q_i^{-}$, which is suitable
for the construction described, leads to the $R-$separation of
variables, and therefore it is not considered here. All other
models (excluding the case of elliptic metric $g_{ik}=\delta_{ik}$
in $Q^{\pm}$) have been proved \cite{david2} {\it not} to be
amenable to separation of variables; they have nontrivial
fourth-order symmetry operators (\ref{symm}). The main idea to
achieve the identification of $H^{(0)}$ in the two approaches is
to consider a class of models with second-order supercharges,
which are quasi-factorizable, {\bf but} with an intermediate {\it
twist} transformation (see also \cite{iv}): \begin{equation}
Q^-=(Q^+)^{\dagger}=q_i^+U_{ik}\tilde q^-_k ,\label{twist} \end{equation}
where $U_{ik}$ is a constant unitary matrix, $q^{\pm}$ were
defined in (\ref{2p}), and $$ \tilde q^{\pm}_k \equiv \mp
\hbar\partial_k + (\partial_k\tilde\chi (\vec x))
$$ with some new superpotential $\tilde\chi .$ Such a generalization
of the notion of reducibility (we shall call it {\it twisted
reducibility}) is somehow reminiscent of the "gluing with shift"
recipe in one-dimensional scalar \cite{acdi} and matrix
\cite{matrix} HSUSY QM. The intertwining relations (\ref{intw2d})
with supercharges (\ref{twist}) and the general expression for
matrix $U_{ik} :$
$$
U = \alpha_0\sigma_0 +
i\overrightarrow{\alpha}\overrightarrow{\sigma}; \quad \alpha_0^2
+ \overrightarrow{\alpha}^2 = 1; \quad \alpha_0, \alpha_i \in
\bf{R},
$$
($\sigma_i$ are the Pauli matrices and $\sigma_0$ is the unit
matrix) give the system of four linear and one nonlinear equations
for two functions $\chi_\pm = \frac{1}{2}(\chi \pm \tilde{\chi}):$
\begin{eqnarray}
\alpha_3\Box\chi_- +2\alpha_1\partial_1\partial_2\chi_- = 0;
\qquad \alpha_1\Box\chi_+ - 2\alpha_3\partial_1\partial_2\chi_+
&= 0;\label{lin1}\\
\alpha_2\Box\chi_+ - 2\alpha_0\partial_1\partial_2\chi_- = 0;
\qquad \alpha_0\Box\chi_- + 2\alpha_2\partial_1\partial_2\chi_+ &=
0; \label{lin2}
\end{eqnarray}
\begin{equation}
(\partial_k \chi_-)(\partial_k\chi_+) = 0, \label{nonlin}
\end{equation}
where $\Box \equiv \partial_1^2 - \partial_2^2$. Precisely the last
equation (\ref{nonlin}) is obviously most difficult to solve. Both
the solutions of linear partial differential equations
(\ref{lin1})-(\ref{lin2}) and the form of (\ref{nonlin}) depend
crucially on the values of $\alpha_i$ chosen. For the most sets of
$\alpha$'s, including the general case with all $\alpha_i\neq 0$ as
well as almost all degenerate cases with some $\alpha_i$ vanishing,
the corresponding potentials allow the separation of variables and
are ignored here.
\par The only exception to the above rule, and therefore the most
interesting quantum models, corresponds to the case\footnote{The
system with $\alpha_0 = \alpha_2 = \alpha_3 = 0;\quad \alpha_1 \ne
0$, i.e. $U = \sigma_1$ leads to analogous results with the
substitution $\chi_{\pm}\leftrightarrow\chi_{\mp}$.} when
$\alpha_0 = \alpha_1 = \alpha_2 = 0;\quad \alpha_3 \ne 0$, i.e. $U
= \sigma_3$. Then, the metric of supercharges $Q^\pm$ is
hyperbolic, i.e. $Q^{\pm}$ belong to the class discussed in
Subsection 2.2. For these models (due to (\ref{lin1}) -
(\ref{lin2}) only), the supercharges are represented in terms of
four arbitrary real functions $\mu_{1,2}$, $\mu_\pm:$ \begin{equation} \chi_- =
\mu_+(x_+) + \mu_-(x_-),\qquad \chi_+  =  \mu_1(x_1) + \mu_2(x_2),
\label{chimu} \end{equation} Hence, the last equation (\ref{nonlin})
rewritten via $\phi \equiv \mu'$ takes the form of the functional
equation: \begin{equation} \phi_1(x_1)\left[ \phi_+(x_+) + \phi_-(x_-) \right]
+ \phi_2(x_2) \left[ \phi_+(x_+) - \phi_-(x_-) \right] = 0.
\label{functional} \end{equation} It is reasonable to formulate here the
important specific property of solution (\ref{chimu}). The
superpotential \begin{equation} \chi (\vec x) =\chi_+ + \chi_- = \mu_1(x_1) +
\mu_2(x_2) + \mu_+(x_+) + \mu_-(x_-); \label{property1} \end{equation} leads
to an expression for the quantum potential $V^{(0)}(\vec x)$ (see
the first of Eq.(\ref{hams})), which also has {\bf the form of the
sum}: \begin{equation} V^{(0)}(\vec x) = \Bigl(\partial_l\chi (\vec x)\Bigr)^2
- \hbar\partial_l^2\chi (\vec x) = v_1(x_1)+v_2(x_2)+v_+(x_+)
+v_-(x_-); \label{4terms1}, \end{equation} with $v_{1,2} = \mu^{\prime\,
2}_{1,2} - \hbar \mu^{\prime\prime}_{1,2}$, $v_\pm = \mu^{\prime\,
2}_{\pm} - \hbar \mu^{\prime\prime}_{\pm}.$ It may be seen that
both terms in quantum potential (\ref{4terms1}) {\it separately}
have the form of the sums as in (\ref{property1}). Therefore, at
the quasi-classical limit $V_{cl}^{(0)}$ (see Sections 4 and 5
below), where only the first term $\Bigl(\partial_l\chi (\vec
x)\Bigr)^2$ survives, the potential is also represented in a form
similar to (\ref{4terms1}) but with truncated $v_{1,2,+,-}$. Both
in the quantum and classical contexts, form (\ref{4terms1}) seems
to be typical for a wide class of integrable two-dimensional
models, considered within very different approaches in the
literature (see \cite{perelomov}, \cite{david4}, as examples).
This is why the following statement might be useful (at least, in
the classical framework). Thus, if the general solution for the
superpotential $\chi (\vec x)$ in relation \begin{equation} V_{cl}^{(0)}(\vec
x) = \Bigl(\partial_l\chi (\vec x)\Bigr)^2 =
v_1(x_1)+v_2(x_2)+v_+(x_+) +v_-(x_-) \label{relation} \end{equation} is of
the form of (\ref{property1}), precisely the functional equation
(\ref{functional}) must be fulfilled. This equation ensures the
mutual cancellation of crossed terms in (\ref{relation}) and is
therefore very important for this class of model. The general
solution of (\ref{functional}) was found by
D.Nishnianidze\footnote{Private communication.} (see \cite{iv}):
\begin{equation} {\phi}'^2_{1,2}=a\phi_{1,2}^4 + b\phi_{1,2}^2 + c;\qquad x =
\pm\int\frac{d\phi_1}{\sqrt{a\phi_1^4 + b\phi_1^2 + c}};\quad
a,b,c=const , \label{ell} \end{equation} and explicit expressions for
$\phi_{\pm}$ can be obtained from (\ref{functional}). Among the
functions that satisfy conditions (\ref{ell}) there exists a set
of solutions of Eq.(\ref{functional}) possessing the periodicity
property. For example,
\begin{equation}
\phi_+(x) = A\frac{\text{sn}(ax|k)\,
\text{cn}(ax|k)}{\text{dn}(ax|k)};\quad \phi_-(x) =
A\frac{\text{dn}(ax|k)}{k^2\text{sn}(ax|k)\,
\text{cn}(ax|k)};\label{per}
\end{equation}
\begin{equation*}
\phi_1(x) = \phi_2(x) = B\,\,\text{dn}(\sqrt{2}ax|k),
\end{equation*}
where $A, B, a$ are real constants and $\text{sn}(\cdot|k)$,
$\text{cn}(\cdot|k)$ and $\text{dn}(\cdot|k)$ are Jacobi elliptic
functions \cite{bather} with modulus $k$. They are doubly periodic
on the complex plane of argument $x$, but in the case $k=1$ the real
period becomes infinite and the elliptic functions turn into the
hyperbolic functions $\sinh$ and $\cosh$. Restricting ourselves in
(\ref{ell}) to non-periodic functions on a whole plane $(x_1, x_2)$
satisfying (\ref{ell}), which do not happens in systems with
separation of variables, two families of models exist (see
\cite{iv}). One of them is represented by the two-dimensional {\bf
Morse} potential, with $\phi_1=\phi_2 = Be^{-\alpha x}; \phi_+ = 2A;
\phi_-=2A\coth{(\alpha x/\sqrt{2})}$:
\begin{eqnarray}
V^{(0)} =& (B^2e^{-2\alpha x_1} + \hbar B\alpha e^{-\alpha x_1}) +
(B^2e^{-2\alpha x_2} + \hbar B\alpha e^{-\alpha x_2})\nonumber\\ &
+ 2A(2A +
\hbar\frac{\alpha}{\sqrt{2}})\left[\sinh{(\frac{\alpha(x_1-x_2)}{2})}\right]^{-2}
+ 8A^2. \label{morse}
\end{eqnarray}
and the second by the two-dimensional {\bf P\"oschl-Teller}
potential with $\phi_1=-\phi_2 = A\left[ \sinh{(\sqrt{2}\alpha x)}
\right]^{-1}; \phi_+ = \phi_- =B\tanh{(\alpha x)}$:
\begin{eqnarray}
V^{(0)} &= \left( B^2 - \frac{B(B +
\hbar\alpha)}{\cosh^2{(\frac{\alpha}{\sqrt{2}}(x_1 + x_2))}}
\right) + \left( B^2 - \frac{B(B +
\hbar\alpha)}{\cosh^2{(\frac{\alpha}{\sqrt{2}}
(x_1 - x_2))}} \right) +\nonumber\\
&+A\left[ \frac{A - \hbar\sqrt{2}\alpha\cosh{(\sqrt{2}\alpha
x_1)}}{\sinh^2{(\sqrt{2}\alpha x_1)}} + \frac{A +
\hbar\sqrt{2}\alpha \cosh{(\sqrt{2}\alpha
x_2)}}{\sinh^2{(\sqrt{2}\alpha x_2)}}\right]. \label{pot1}
\end{eqnarray}
Other members of these families can be obtained by using two
discrete symmetries of solutions of Eq.(\ref{functional}): \begin{eqnarray}
S_1:\  \biggl\{ \phi_1(x_1), \phi_2(x_2), \phi_+(x_+), \phi_-(x_-)
\biggr\} &\longrightarrow &\biggl\{\phi_+(x_1), \phi_-(x_2),
\phi_1(x_+),
\phi_2(x_-)\biggr\};\nonumber\\
S_2:\  \biggl\{\phi_1(x_1), \phi_2(x_2), \phi_+(x_+),
\phi_-(x_-)\biggr\} &\longrightarrow & \biggl\{\phi_1(x_1),
-\phi_2(x_2), \phi_+^{-1}(x_+), \phi_-^{-1}(x_-)\biggr\}
\label{S1S2} \end{eqnarray} and different combinations thereof.
\section*{4. Supersymmetric extensions of scalar Hamiltonians and their integrability}
\vspace*{0.3cm} \hspace*{3ex} In the previous Section we presented
the explicit forms (\ref{morse}), (\ref{pot1}) of the terms in
Eq.(\ref{4terms1}) for a certain class of quantum integrable
Hamiltonians. In this Section we shall build their classical and
quantum SUSY extensions and we shall also demonstrate their
integrability properties. \vspace*{0.2cm}
\subsection*{4.1. The classical limit for $H^{(0)}$}
\hspace*{3ex} First, we shall consider for $H^{(0)}$ its classical
limit $H^{(0)}_{cl},$ for which the integral of motion $R_{cl}$
exists: $\{H^{(0)}_{cl}, R_{cl}^{(0)} \}_P =0$ ($\{\cdot ,
\cdot\}_P$ denotes standard Poisson brackets). This can be done by
the simple limit procedure $\hbar \to 0$ in Eq.(\ref{hams}). The
practical recipe is as follows. One has to replace all {\it
operators} $-i\hbar\partial_i$ by momenta $p_i$ and skip all {\it
derivatives of functions}, which include $\hbar$ as a multiplier.
One thus obtains\footnote {Here and below we will omit the
arguments of $\phi_{1,2}$ and $\phi_{\pm}$, which implies that
$\phi_{1,2} \equiv \phi_{1,2}(x_{1,2})$ and $\phi_\pm  \equiv
\phi_\pm(x_\pm).$}:
$$
 H^{(0)}_{cl} = p_jp_j + \phi_1^2 + \phi_2^2 +
\phi_+^2 + \phi_-^2; $$ $$ Q_{cl}^ \pm = p_1^2 - p_2^2 \pm
i\sqrt{2}(\phi_+ + \phi_-)p_1 \mp i\sqrt{2}(\phi_+ - \phi_-)p_2 +
\phi_1^2  -\phi_2^2 - 2\phi_+\phi_-. $$ The integral of motion has
the form $R^{(0)}_{cl}=Q_{cl}^+Q^-_{cl}.$ Its involution with the
Hamiltonian can be checked either by direct calculation or by a
simpler two-step procedure, proposed in general form in \cite{pomi}.
It is instructive to perform it in the context of the models
considered here. First, one has to prove the intermediate relations:
\begin{equation}
\{ H^{(0)}_{cl}, Q^\pm_{cl} \}_P = \pm 2i(\phi_+' +
\phi_-')Q^\pm_{cl}. \label{interm}
\end{equation}
From the definition of Poisson brackets
\begin{eqnarray}
\{H^{(0)}_{cl}, Q^\pm_{cl} \}_P  &=&  \pm2i(\phi_+' +
\phi_-')(p_1^2 - p_2^2 \pm i\sqrt{2}(\phi_+ + \phi_-)p_1 \mp
i\sqrt{2}(\phi_+ -
\phi_-)p_2 - 2\phi_+\phi_-) \mp \nonumber\\
&\mp& 2i\sqrt{2}\left(\phi_+(\phi_1\phi_1' - \phi_2\phi_2') +
\phi_-(\phi_1\phi_1' + \phi_2\phi_2')\right).\label{interm2}
\end{eqnarray}
The last term can be transformed with the use of the functional
equation (\ref{functional}):
\begin{eqnarray}
&&\phi_+(\phi_1\phi_1' - \phi_2\phi_2') + \phi_-(\phi_1\phi_1' +
\phi_2\phi_2') = \frac{1}{\sqrt{2}}\phi_+\partial_-\left(\phi_1^2
+ \phi_2^2\right) +
\frac{1}{\sqrt{2}}\phi_-\partial_+\left(\phi_1^2 +
\phi_2^2\right) = \nonumber\\
&&=\frac{1}{\sqrt{2}}\left[ \partial_-(\phi_+(\phi_1 + \phi_2)^2)+
\partial_+(\phi_-(\phi_1 - \phi_2)^2) - 2\phi_+\partial_-
(\phi_1\phi_2) + 2\phi_-\partial_+(\phi_1\phi_2) \right] =
\nonumber\\ &&=\frac{1}{\sqrt{2}}\left[
-\partial_-(\phi_-(\phi_1^2 - \phi_2^2)) -
\partial_+(\phi_+(\phi_1^2 - \phi_2^2)) - 2\phi_+\partial_-
(\phi_1\phi_2) + 2\phi_-\partial_+(\phi_1\phi_2) \right]  =
\nonumber\\
&&=-\frac{1}{\sqrt{2}}(\phi_+' + \phi_-')(\phi_1^2 -
\phi_2^2).\nonumber
\end{eqnarray}
Substituting this into (\ref{interm2}), we prove (\ref{interm}).
Then, (\ref{interm}) leads to the involution of $R^{(0)}_{cl}$ with
$H^{(0)}_{cl}$:
$$\{ H^{(0)}_{cl}, Q^+_{cl}Q^-_{cl} \}_P = \{ H^{(0)}_{cl},
Q^+_{cl} \}_PQ^-_{cl} + Q^+_{cl}\{ H^{(0)}_{cl}, Q^-_{cl} \}_P =
0.$$ It is easy to check that the same classical limit procedure for
the second scalar Hamiltonian $H^{(2)}$ and for the components of
the matrix $H_{ik}^{(1)}$ in (\ref{hams}) leads to the simple
results: $H^{(2)}_{cl} = H^{(0)}_{cl}$ and $H^{(1)}_{ik,\, cl}=
\delta_{ik} H^{(0)}_{cl}.$ Naturally, the corresponding integrals of
motion coincide too: $R^{(2)}_{cl} = R^{(0)}_{cl}$ and
$R^{(1)}_{ik,\, cl}= \delta_{ik} R^{(0)}_{cl}.$ In the next
Subsection we shall construct another classical limit of $\hat{H}$
that will also include Grassmanian dynamical variables in addition
to $p_j$ and $x_j,$ and this can be interpreted as {\it a
SUSY-extension} of $H^{(0)}_{cl}$. \vspace*{0.2cm}
\subsection*{4.2. The SUSY extension of classical scalar Hamiltonians}
\hspace*{3ex} It is well-known \cite{abei}, \cite{wipf} that the 2D
representation of SUSY algebra, reviewed in Section 2, can be
obtained by the canonical quantization from the classical system
with the Hamiltonian:
\begin{equation}
H_{cl} = p_lp_l + (\partial_l \chi (\vec x))(\partial_l \chi (\vec
x)) + 2i\partial_i\partial_l \chi (\vec x)\psi^1_i\psi^2_l,
\label{superH}
\end{equation}
where $\psi^1_i$ and $\psi^2_j$ are Grassmanian anticommuting
variables: $\{\psi^\alpha_i, \psi^\beta_j\} = 0$ ($i,j = 1,...,d$;
$\alpha, \beta=1,2$). One can define the Poisson bracket on the
phase space of the system with classical bosonic and fermionic
variables \cite{bei} as follows:
$$\{F, G\}_P = \frac{\partial F}{\partial p_j}\frac{\partial
G}{\partial x^j} - \frac{\partial F}{\partial x^j}\frac{\partial
G}{\partial p_j} + iF\frac{\overleftarrow{\partial}}{\partial
\psi^\alpha_i}\frac{\overrightarrow{\partial}}{\partial
\psi^\alpha_i}G,$$ such that the canonical brackets are $\{p_i,
x_j\}_P=\delta_{ij};\,\,\{\psi^{\alpha}_i,
\psi^{\beta}_j\}_P=i\delta_{ij}.$ Thus, the Hamiltonian is involved
in the SUSY algebra:
$$\{ Q_\alpha, Q_\beta \}_P = i\delta_{\alpha \beta} H_{cl}; \qquad
\{ H_{cl}, Q_\alpha \}_P = 0$$ with classical supercharges
$$Q_1 = p_j\psi^1_j - (\partial_j\chi)\psi^2_j; \qquad Q_2 =
p_j\psi^2_j + (\partial_j\chi)\psi^1_j.$$ To quantize this model,
one has to introduce the bosonic operators $\hat{p}_i$ and
$\hat{x}_j$ with canonical commutation relations, and fermionic ones
$\hat{\psi}^\alpha_j$ obeying $\{ \hat{\psi}_i^\alpha,
\hat{\psi}_j^\beta\} = \hbar\delta_{ij}\delta^{\alpha \beta}$. At
this stage, it is convenient to introduce the fermionic operators
$\hat{\psi}^\pm_j = (\sqrt{2})^{-1}(\hat{\psi}^1_j \mp
i\hat{\psi}^2_j)$ with anticommutation relations $\{\hat{\psi}_i^+,
\hat{\psi}_j^-\} = \hbar\delta_{ij}$ and $\{\hat{\psi}_i^-,
\hat{\psi}_j^-\} = \{\hat{\psi}_i^+, \hat{\psi}_j^+\} = 0$. These
can be treated as creation/annihilation operators in the system of
$d$ spin $1/2$ fermions. Thus, the quantum counterpart of the
Hamiltonian (\ref{superH}) - the Superhamiltonian - in terms of
these operators takes the form:
\begin{equation}
\hat{H} = -\hbar^2\partial_j\partial_j + \left[
(\partial_j\chi)(\partial_j\chi) - \hbar(\partial_j\partial_j\chi)
\right] + 2(\partial_i\partial_j\chi)\hat{\psi}^+_i\hat{\psi}^-_j.
\label{SHquant}
\end{equation}
Together with the quantum supercharges $\hat{Q}^\pm = \pm i
(\sqrt{2})^{-1}(\hat{Q}^1 \mp i\hat{Q}^2) = (\pm i\hat{p}_j +
(\partial_j\chi))\hat{\psi}^\pm_j$ this generates the algebra
(\ref{alg}). To reproduce the matrix form (\ref{suham}) -
(\ref{hams}) one should choose the matrix representation for the
creation and annihilation operators $\hat{\psi}_j^\pm$. For $d=2$,
these operators are $4\times4$ matrices, and a possible choice is as
follows \cite{abei}: $\hat{\psi}_1^+ = \hbar^{1/2}(E_{2,1} -
E_{4,3})$, $\hat{\psi}_2^+ = \hbar^{1/2}(E_{3,1} + E_{4,2})$ and
$\hat{\psi}^-_j = (\hat{\psi}^+_j)^\dag$ (here matrices $E_{m,n}$
are defined as $(E_{m,n})_{ik}\equiv \delta_{mi}\delta_{nj}$). Thus,
we have obtained a matrix realization of two-dimensional SUSY QM
(\ref{alg}) - (\ref{hams}) by means of the canonical quantization of
a certain classical model (\ref{superH}).
\par One can see that in this representation $\hat{H}$ has
a block-diagonal structure. The origin of this feature of the model
is the conservation of the fermion number $[\hat{H}, \hat{N}] = 0$,
with fermion number operator $\hat{N} =
\hat{\psi}^+_j\hat{\psi}^-_j$. Therefore, each component of the
Superhamiltonian acts in a space of states with a fixed fermion
number.
\par In our case $d=2$, this structure is rather simple.
Let us define a basis in the state space: the vacuum $|00\!\!>$,
which is annihilated by $\hat{\psi}^-_i$, and the excited states
$|10\!\!> = \hbar^{-1/2}\hat{\psi}^+_1|00\!\!>$; $|01\!\!> =
\hbar^{-1/2}\hat{\psi}^+_2|00\!\!>$; $|11\!\!> =
\hbar^{-1}\hat{\psi}^+_2\hat{\psi}^+_1|00\!\!>$ (the
$\hbar$-dependent multipliers provide the proper normalization of
the state vectors: $<\!\!mn|mn\!\!> = 1$, $\forall m,n=0,1$). Thus
$H^{(0)}$ acts in one-dimensional space with a fermion number of $0$
and a single basis element $|00\!\!>$; for $H^{(2)}$, the fermion
number is $2$ and the basis element is $|11\!\!>$. The Hamiltonian
$H^{(1)}_{ik}$ acts in two-dimensional state space $\{ |01\!\!>,
|10\!\!> \},$ where the fermion number is $1$.
\par The conclusion to be drawn from this derivation is as
follows. Having the classical system with $H^{(0)}_{cl} =p_jp_j +
(\partial_j\chi)(\partial_j\chi)$, one can construct its SUSY
extension of the form (\ref{superH}). This classical SUSY extension
can be quantized canonically to obtain the quantum Superhamiltonian
(\ref{SHquant}), the original $H^{(0)}_{cl}$ being the classical
limit of $H^{(0)}$ - the first scalar component of the quantum
Superhamiltonian. In our case, $H^{(0)}_{cl}$ was integrable (see
previous Subsection), and we shall explicitly find the integral of
motion $R_{cl}$ for its quantum SUSY-extension (Superhamiltonian).
An analogous problem was investigated by alternative methods in
\cite{Aai}, but for a much more narrow class of models (amenable to
separation of variables). \vspace*{0.2cm}
\subsection*{4.3. Integrals of motion for the quantum and classical SUSY extensions}
\hspace*{3ex} We start from construction of the quantum integral of
motion $\hat{R}$ for the Superhamiltonian (\ref{SHquant}):
$[\hat{H}, \hat{R}] = 0,$ i.e. of conserved operators $R^{(i)}$ for
each component of the Superhamiltonian:
\begin{equation}
[H^{(0)}, R^{(0)}] = 0;\quad [H^{(2)}, R^{(2)}] = 0; \label{hr02}
\end{equation}
\begin{equation}
[H^{(1)}, R^{(1)}] = 0;\label{hr1}
\end{equation}
(note that the last commutator is the matrix one). The explicit
expression for $R^{(0)}$ can be obtained from (\ref{symm}) and
(\ref{twist}):
\begin{equation}
R^{(0)} =
q^+_iU_{ik}\tilde{q}^-_k\tilde{q}^+_mU_{ml}q^-_l.\label{R0}
\end{equation}
One can obtain $R^{(2)}$ from $R^{(0)}$ by the substitutions
$q^\pm_j \to -q^\mp_j$ and $\tilde{q}^\pm_j \to -\tilde{q}^\mp_j$
(since $H^{(0)}$ turns to $H^{(2)}$ after the substitution
$\chi(\vec{x}) \to -\chi(\vec{x})$):
\begin{equation}
R^{(2)} = q^-_iU_{ik}\tilde{q}^+_k\tilde{q}^-_mU_{ml}q^+_l.
\label{R2}
\end{equation}
With respect to $R^{(1)}$, the form of intertwining relations
(\ref{2dintw}) tells us how to build this symmetry operator
explicitly. One can check that the following matrix operator {\it
of sixth order in derivatives}
\begin{equation}
R^{(1)}_{ik} = q^-_iR^{(0)}q^+_k, \label{R1}
\end{equation}
satisfies Eq.(\ref{hr1}).
\par It is clear from the material of the previous Subsection
that knowledge of $R^{(i)},\,\, (i=0,1,2)$ provides a symmetry
operator $\hat R$ for the whole block-diagonal Superhamiltonian
$\hat H.$ In order to construct an all-sector expression for $\hat
R:$
\begin{equation}
\hat{R} = R^{(0)}P^{(0)} + R^{(2)}P^{(2)} +
R^{(1)}_{ik}P^{(1)}_{ik}, \label{decomp}
\end{equation}
we introduce the corresponding projection operators
$P^{(i)}\,\,(i=0,1,2)$ onto subspaces with definite fermion numbers.
The scalar projectors $P^{(0)}$ and $P^{(2)}$ give unity when acting
on $|00\!\!>$ and $|11\!\!>,$ respectively, and give zero otherwise.
The components $P^{(1)}_{ik} \,(i,k=1,2)$ of the $2\times 2$ matrix
projector operators $P^{(1)}$ transform the $k-$th component of the
state vector into its $i-$th component\footnote{By definition, "the
first component" is $|10\!\!>,$ and "the second component" is
$|01\!\!>$.} and are zero on other components of the state vector.
One can check directly that an explicit form of these operators $P$
leads to:
\begin{eqnarray}
\hat{R} &=&
-R^{(0)}\hbar^{-2}\hat{\psi}_1^-\hat{\psi}_2^-\hat{\psi}_1^+\hat{\psi}_2^+
-
R^{(2)}\hbar^{-2}\hat{\psi}_1^+\hat{\psi}_2^+\hat{\psi}_1^-\hat{\psi}_2^-
-
R^{(1)}_{11}\hbar^{-2}\hat{\psi}_1^+\hat{\psi}_2^-\hat{\psi}_1^-\hat{\psi}_2^+
- \nonumber \\ & & -
R^{(1)}_{22}\hbar^{-2}\hat{\psi}_1^-\hat{\psi}_2^+\hat{\psi}_1^+\hat{\psi}_2^-
+ R^{(1)}_{12}\hbar^{-1}\hat{\psi}_1^+\hat{\psi}_2^- +
R^{(1)}_{21}\hbar^{-1}\hat{\psi}_2^+\hat{\psi}_1^-, \label{hatr}
\end{eqnarray}
where $R^{(i)}$ are given by (\ref{R0}) - (\ref{R1}). Naturally,
(\ref{hatr}) can be simplified by employing anticommutation
relations for $\hat{\psi}^\pm_i$. One should not be confused by the
presence of the negative powers of the Planck constant in
Eq.(\ref{hatr}) since they disappear in all matrix elements for
$\hat{R}$.
\par Let us prove straightforwardly that the $\hat{R}$ constructed commutes
with $\hat{H}$. The Superhamiltonian can be presented similarly to
(\ref{decomp}):
$$\hat{H} = H^{(0)}P^{(0)} + H^{(2)}P^{(2)} +
H^{(1)}_{ik}P^{(1)}_{ik},$$ where $H^{(i)}$ are given by
Eqs.(\ref{hams}). By definition $[P^{(0)}, P^{(2)}] = [P^{(1)}_{ik},
P^{(2)}] = [P^{(1)}_{ik}, P^{(0)}] = 0$, and therefore, due to
(\ref{hr02}), we have:
$$
[\hat{H}, \hat{R}] = [H^{(1)}_{ij}P^{(1)}_{ij},
R^{(1)}_{kl}P^{(1)}_{kl}].
$$
Employing the explicit form (\ref{hatr}) of $P^{(i)}$ and
Eq.(\ref{hr1})), one can see that the commutator in the rhs
vanishes, completing our proof.
\par From expression (\ref{hatr}), its classical limit $R_{cl}$
can be constructed by means of the substitution
$\hbar^{-1/2}\hat{\psi}^\alpha_j \to \psi^\alpha_j$ for fermionic
operators. Finally, the total classical integral of motion reads:
\begin{eqnarray}
R_{cl} & = & R^{(0)}_{cl} + \frac{i}{2}(R^{(1)}_{11\,cl} -
R^{(1)}_{22\,cl})(\psi^1_1\psi^2_1 - \psi^1_2\psi_2^2) +
\frac{1}{2}(R^{(1)}_{21\,cl} - R^{(1)}_{12\,cl})(\psi_1^1\psi_2^1
+ \psi_1^2\psi_2^2) +\nonumber \\ && +
\frac{i}{2}(R^{(1)}_{12\,cl} + R^{(1)}_{21\,cl})(\psi_1^2\psi_2^1
+ \psi_1^1\psi_2^2) +
(R_{11\,cl}^{(1)}+R_{22\,cl}^{(1)}-2R^{(0)}_{cl})\psi_1^1\psi_1^2\psi_2^1\psi^2_2,
\label{Rfinal}
\end{eqnarray}
where its components are:
\begin{eqnarray}
R^{(0)}_{cl} & = & R^{(2)}_{cl} = \left( p_1^2 - p_2^2 + \phi_1^2
- \phi_2^2 - 2\phi_+\phi_- \right)^2 + 2\left( (\phi_+ +
\phi_-)p_1 - (\phi_+
- \phi_-)p_2 \right)^2; \nonumber\\
R^{(1)}_{11\,cl} & = & \left(p_1^2 + \phi_1^2 +
\frac{1}{2}(\phi_+^2 +
\phi_-^2) + 2\sqrt{2}\phi_+\phi_- \right)R^{(0)}_{cl}; \nonumber\\
R^{(1)}_{22\,cl} & = & \left(p_2^2 + \phi_2^2 +
\frac{1}{2}(\phi_+^2 -
\phi_-^2) - 2\sqrt{2}\phi_+\phi_- \right)R^{(0)}_{cl}; \label{iint}\\
R^{(1)}_{12\,cl} & = & \left( ip_1 + \phi_1 +
\frac{1}{\sqrt{2}}(\phi_+ + \phi_-) \right) \left( -ip_2 + \phi_2
+ \frac{1}{\sqrt{2}}(\phi_+ - \phi_-) \right)R^{(0)}_{cl}; \nonumber\\
R^{(1)}_{21\,cl} & = & \left( ip_2 + \phi_2 +
\frac{1}{\sqrt{2}}(\phi_+ - \phi_-) \right) \left( -ip_1 + \phi_1
+ \frac{1}{\sqrt{2}}(\phi_+ + \phi_-) \right)R^{(0)}_{cl}.
\nonumber
\end{eqnarray}
\vspace*{0.2cm}
\section*{5. The flipped potentials and the classical Hamilton-Jacobi equation}
\vspace*{0.5cm} \hspace*{3ex} In this Section we shall establish
links between the Hamilton-Jacobi equations of Classical Mechanics
and the equation for the superpotential. Starting from
Eq.(\ref{relation}), one can see that the condition for the
classical Hamiltonian
$$H_{cl} = p_l^2 + V(\vec{x})
$$
to be supersymmetric with superpotential $\chi(\vec{x})$ takes the
form:
\begin{equation}
V(\vec{x}) = \frac{\partial\chi(\vec{x})}{\partial
x_l}\frac{\partial\chi(\vec{x})}{\partial x_l}. \label{ham}
\end{equation}
On the other hand, for the Hamiltonian $h_{cl}=p_l^2+U(\vec x)$ with
potential $U(\vec x)$ and the classical action functional $S$, the
well known Hamilton-Jacobi equation \cite{hamilton} reads:
\begin{equation}
 \frac{\partial S}
{\partial t} + h_{cl}(\frac{\partial S} {\partial
x_1},\frac{\partial S}{\partial x_2}, \cdots \frac{\partial S}
{\partial x_d}; x_1,x_2,\cdots ,x_d)=0 \label{HJ}.
\end{equation}
There being no explicit dependence on time in $h_{cl}$, one looks
for its solutions of the form $S(t,x_1,x_2,\cdots
,x_d)=W(x_1,x_2,\cdots ,x_d)-Et$, and the time-independent
Hamilton-Jacobi equation becomes:
\begin{equation}
E=(\partial_lW)^2+U(\vec{x}), \label{eq:thj}
\end{equation}
where $W(x_1,x_2,\cdots ,x_d)$ is usually referred to as the
Hamilton characteristic function. Solutions of (\ref{eq:thj}) in the
case $E=0$ are obviously connected with those of (\ref{ham}):
\begin{equation}
\chi(\vec{x}) = \pm iW(\vec{x}). \label{conn}
\end{equation}
Eq.(\ref{eq:thj}), with zero energy, can alternatively be thought of
as a condition for the ''flipped'' classical potential $V = -U$ to
be supersymmetric, i.e. $V$ should satisfy (\ref{ham}), with $\chi$
and $W$ related by (\ref{conn}). Thus, we find
\[
W(x_1,x_2)=\mp
i\left[\mu_1(x_1)+\mu_2(x_2)+\mu_+(x_+)+\mu_-(x_-)\right]
\]
as the Hamilton characteristic function of the system. The system of
equations of motion for $E=0$ \begin{eqnarray} \dot{x}_1&=&\frac{\partial
W}{\partial x_1}=\mp
i\left[\phi_1+{1\over\sqrt{2}}\phi_++{1\over\sqrt{2}}\phi_-\right]\nonumber
\\ \dot{x}_2&=&\frac{\partial W}{\partial x_2}=\mp
i\left[\phi_2+{1\over\sqrt{2}}\phi_+-{1\over\sqrt{2}}\phi_-\right]\label{mot}
\end{eqnarray} is not amenable to separation of variables and in general has
(non-physical) complex solutions. One may see that system
(\ref{mot}) becomes real, and bona fide solutions exist for the
specific complexification of the P\"oschl-Teller model (\ref{pot1}):
namely, with purely imaginary $\alpha$. In contrast to the
complexification of two-dimensional Morse potential in
\cite{pseudo}, this one is PT-invariant.
\par Analogous classical systems with "flipped" potentials
were investigated in \cite{Aai} for the case of $d=2$ integrable
models of the Liouville type. For these systems the Hamilton-Jacobi
equations were separable in elliptic, polar, parabolic and Cartesian
coordinates. The structure of related supersymmetric models (also
with separation of variables) {\bf in the quantum domain} has been
investigated in \cite{Aai1} via canonical quantization. In
particular, it was shown that there are two essentially different
supersymmetric extensions (two different superpotentials) for a
given separable classical solution of the Hamilton-Jacobi equation.
In the present paper our strategy is just the opposite. Namely,
starting from scalar quantum $d=2$ systems which {\bf do not allow}
for separation of variables, but do have non-trivial symmetry
operators, we construct their quantum SUSY-extension. Then, we
describe corresponding classical SUSY-extended systems and their
integrals of motion. Finally, the link between this kind of
classical system and the Hamilton-Jacobi approach for related
systems with "flipped" potential provides the integrability of these
"flipped" systems too. The necessary explicit expressions for
integrals of motion can be obtained from (\ref{Rfinal}),
(\ref{iint}) by the substitution $\phi \rightarrow \pm i\phi$, which
is equivalent to (\ref{conn}). It should be remarked that
(analogously to \cite{Aai1}) besides an arbitrary common sign in
(\ref{conn}) there is the additional non-uniqueness of the
superpotential for this class of model. Indeed, equation (\ref{ham})
has {\it two independent} solutions for fixed original classical
potential: $\chi (\vec x)$ and $\tilde\chi (\vec x).$ To prove this
statement, one has to check that $(\partial_l\chi)^2 =
(\partial_l\tilde\chi)^2$
 for $\chi = \mu_1+\mu_2+\mu_++\mu_-$
and $\tilde\chi = \mu_1+\mu_2-\mu_+-\mu_-$ (see (\ref{chimu})) due
to the nonlinear equation (\ref{nonlin}). \vspace*{0.2cm}
\section*{Acknowledgements}
\vspace*{0.5cm} \hspace*{3ex} The work was partially supported by
the Spanish MEC grants SAB2004-0143 (M.V.I.), MTM2005-09183
(M.V.I.), BFM2003-00936 (J.M.G.), by the project VA013C05 of the
Junta de Castilla y Le\'on (J.M.G. and M.V.I.) and by Russian
grants (M.V.I. and P.A.V.) RFFI 06-01-00186-a and RNP 2.1.1.1112.
P.A.V. is indebted to the International Centre of Fundamental
Physics in Moscow, the non-profit foundation "Dynasty" and the
Grant Centre of Natural Sciences (grant M05-2.4D-56) for financial
support. M.V.I. is grateful to the University of Salamanca and the
University of Valladolid for kind hospitality and to J.Negro and
L.M.Nieto for helpful discussions.

\vspace{.2cm}

redpajama_set_name: RedPajamaArXiv


From farmland to final resting place, Roseland Park Cemetery in many ways is intricately entwined with the history of Berkley. The land on which the cemetery now sits was originally farmed by Native American tribes for corn and squash. In the mid-1800s, John Benjamin and his son John Jr. purchased the land and erected a two-story structure in which they produced the Benjamin Muley Grain Cradle. This device, combining the sickle with a cradle to separate wheat from the chaff, enabled farmers to quadruple their yield.
In 1906, a group of forward-thinking businessmen who anticipated population expansion northward from Detroit founded Roseland Park Cemetery. A stock purchase converted the north section of the Cromie Dairy Farm on Twelve Mile Road from cattle pasture to graveyard.
Although John Tuttle was the first person buried here in 1908, Roseland Park was officially dedicated in June of 1910. More than 1,000 people attended, including the mayors of Detroit and Pontiac. Poet Edgar Guest read a poem he had written for the ceremony.
The Rose Chapel was designed by world-renowned Detroit architect, Louis Kamper and was officially dedicated in 1914. Many mausoleums in North America were modeled after the famed Rose Chapel. Certified as a Michigan Historical Site, the Chapel is currently undergoing renovations to restore its timeless beauty. This century-old mausoleum has stood the test of time. Louis Kamper, the designer, is interred in the Rose Chapel.
From its modest one-acre beginning, the cemetery gradually expanded to its current 135 acres. During the Depression, though, the cemetery sold a portion of its land to a developer who created the St. Johns Woods subdivision.
Note: Contrary to legend, Arthur Godfrey is NOT interred in Roseland Park. Although he was a salesman for the cemetery before becoming a famous entertainer, he is actually buried in Florida.


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Cryptocurrencies such as Bitcoin, LiteCoin, Ether, YOCoin, and many others have been designed as a non-fiat currency. In other words, its backers claim that there's actual worth, even through there is absolutely no physical representation of that worth. The worth increases due to computing power, that is, is the lone way to create new coins distributed by allocating CPU electricity via computer programs called miners. Miners create a block after a time frame which is worth an ever diminishing amount of currency or some type of wages to be able to ensure the shortage. Each coin contains many smaller components. For Bitcoin, each component is called a satoshi. Operations that take place during mining are just to authenticate other transactions, such that both creates and authenticates itself, a simple and elegant alternative, which will be among the appealing aspects of the coin. Once created, each Bitcoin (or 100 million satoshis) exists as a cipher, that is part of the block that gave rise to it. The blockchain is where the public record of all trades lives. Most all cryptocurrencies function as Bitcoin does.
The fact that there's little evidence of any increase in using virtual money as a currency may be the reason there are minimal efforts to control it. The reason behind this could be merely that the marketplace is too small for cryptocurrencies to justify any regulatory attempt. It is also possible the regulators just don't comprehend the technology and its consequences, awaiting any developments to act.
Only a fraction of bitcoins issued so far are available on the exchange markets. Bitcoin markets are competitive, meaning the price a bitcoin will rise or fall depending on supply and demand. Many people hoard them for long term savings and investment. This limits the variety of bitcoins that are truly circulating in the exchanges. Moreover, new bitcoins will continue to be issued for decades to come. Thus, even the most diligent buyer could not purchase all present bitcoins. This scenario isn't to suggest that markets usually are not vulnerable to price manipulation, yet there exists no requirement for big amounts of cash to transfer market prices up or down. The merest events on the planet market can change the price of Bitcoin, This can make Bitcoin and any other cryptocurrency explosive.
Bitcoin is the principal cryptocurrency of the net: a digital money standard by which all other coins are compared to. Cryptocurrencies are distributed, worldwide, and decentralized. Unlike conventional fiat currencies, there's no governments, banks, or every other regulatory agencies. Therefore, it's more resistant to wild inflation and tainted banks. The advantages of using cryptocurrencies as your method of transacting money online outweigh the protection and privacy risks. Security and seclusion can easily be achieved by simply being smart, and following some basic guidelines. You'dn't place your whole bank ledger online for the word to see, but my nature, your cryptocurrency ledger is publicized. This can be fastened by removing any identity of possession in the wallets and thereby keeping you anonymous.
Since among the oldest forms of making money is in money lending, it truly is a fact that you can do this with cryptocurrency. Most of the giving sites currently focus on Bitcoin, some of those sites you are demanded fill in a captcha after a particular period of time and are rewarded with a bit of coins for seeing them. It is possible to see the www.cryptofunds.co site to locate some lists of of these sites to tap into the money of your choice. Unlike forex, stocks and options, etc., altcoin marketplaces have quite different dynamics. New ones are always popping up which means they don't have lots of market data and historical outlook for you to backtest against. Most altcoins have fairly poor liquidity as well and it is hard to come up with a fair investment strategy.
Anyone can become a Bitcoin miner running applications with specialized hardware. Mining applications listen for transmission transactions on the peer-to-peer network and perform the appropriate tasks to process and confirm these transactions. Bitcoin miners do this because they can get transaction fees paid by users for faster transaction processing, and new bitcoins in existence are under denominated formulas.
Cryptocurrency is freeing people to transact money and do business on their terms. Each user can send and receive payments in an identical way, but they also take part in more complex smart contracts. Multiple signatures allow a trade to be supported by the network, but where a particular number of a defined group of folks agree to sign the deal, blockchain technology makes this possible. This permits advanced dispute arbitration services to be developed in the future. These services could allow a third party to approve or reject a trade in the event of disagreement between the other parties without checking their money. Unlike cash and other payment procedures, the blockchain always leaves public evidence that a transaction occurred. This can be possibly used within an appeal against companies with deceptive practices.
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You've probably heard this many times where you generally spread the great word about crypto. It's not risky? What happens when the value accidents? sofar, several POS devices provides free conversion of fiat, alleviating some concern, but before volatility cryptocurrencies is resolved, most people will undoubtedly be reluctant to keep any. We have to discover a way to struggle the volatility that is inherent in cryptocurrencies.
Many people prefer to use a currency deflation, particularly those that desire to save. Despite the criticism and disbelief, a cryptocurrency coin may be better suited for some applications than others. Fiscal solitude, for example, is great for political activists, but more problematic when it comes to political campaign financing. We need a steady cryptocurrency for use in trade; if you're living pay check to pay check, it would take place within your riches, with the rest earmarked for other currencies.
For most users of cryptocurrencies it's not crucial to understand how the procedure operates in and of itself, but it's fundamentally vital that you understand that there is a process of mining to create virtual money. Unlike monies as we know them now where Governments and banks can just select to print unlimited amounts (I am not saying they are doing so, just one point), cryptocurrencies to be operated by users using a mining application, which solves the complex algorithms to release blocks of monies that can enter into circulation.
Ethereum is an incredible cryptocurrency platform, nevertheless, if growth is too fast, there may be some issues. If the platform is adopted immediately, Ethereum requests could increase dramatically, and at a rate that surpasses the rate with which the miners can create new coins. Under a situation like this, the entire platform of Ethereum could become destabilized because of the raising costs of running distributed applications. In turn, this could dampen interest Ethereum platform and ether. Uncertainty of demand for ether can lead to an adverse change in the economic parameters of an Ethereum based business that could result in business being unable to continue to manage or to discontinue operation.
The physical Internet backbone that carries data between the different nodes of the network has become the work of a number of firms called Internet service providers (ISPs), including firms offering long distance pipelines, occasionally at the international level, regional local conduit, which ultimately connects in families and businesses. The physical connection to the Internet can only happen through any of these ISPs, players like amount 3, Cogent, and IBM AT&T. Each ISP runs its own network. Internet service providers Exchange IXPs, owned or private businesses, and occasionally by Governments, make for each of these networks to be interconnected or to move messages across the network. Many ISPs have arrangements with providers of physical Internet backbone providers to offer Internet service over their networks for last mile-consumers and businesses who need to get Internet connectivity. Internet protocols, followed by everyone in the network makes it possible for the information to flow without interruption, in the appropriate area at the perfect time.
While none of these organizations owns the Internet collectively these businesses determine how it functions, and recognized rules and standards that everyone remains. Contracts and legal framework that underlies all that's occurring to ascertain how things work and what happens if something goes wrong. To get a domain name, for example, one needs consent from a Registrar, which has a contract with ICANN. To connect to the Internet, your ISP must be physical contracts with providers of Internet backbone services, and suppliers have contracts with IXPs from the Internet backbone to attach to and with her. Concern over security problems? A working group is formed to work on the problem and the solution developed and deployed is in the interest of most parties. If the Internet is down, you might have someone to phone to get it repaired. If the issue is from your ISP, they in turn have contracts in position and service level agreements, which govern the manner in which these problems are worked out.
The benefit of cryptocurrency is that it uses blockchain technology. The network of nodes the make up the blockchain isn't governed by any centralized business. No one can tell the miners to upgrade, speed up, slow down, stop or do anything. And that's something that as a dedicated promoter badge of honour, and is identical to the way the Internet works. But as you understand now, public Internet governance, normalities and rules that govern how it works current built-in problems to the user. Blockchain technology has none of that.
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Entrepreneurs in the cryptocurrency movement may be wise to investigate possibilities for making gigantic ammonts of money with various forms of online marketing.There could be a rich reward for anyone daring enough to brave the cryptocurrency marketplaces.Bitcoin architecture provides an instructive example of how one might make lots of money in the cryptocurrency marketplaces. Bitcoin is an incredible intellectual and technical achievement, and it's generated an avalanche of editorial coverage and venture capital investment opportunities. But very few people understand that and pass up on very profitable business models made available due to the growing use of blockchain technology.
It is certainly possible, but it must be able to recognize opportunities regardless of market conduct. The market moves in relation to cost BTC … So even if it's in a BTC tendency down can make money by purchasing the altcoins which are altcoin oversold trading ratios-BTC. Sure, your purchasing power in DOLLARS may be lower, but as long as your purchasing power in BTC is still growing you'll be okay.
The creation of websites has changed many lives, but there is always a concern in regards to the security of websites. There are other individuals with ill intentions who'll see what you are doing online. They could monitor your trends over time. Some of the things they can check online include seeing your online photographs, what you post online and even monitor your financial transitions over time with an intent of stealing from you. Even if there are many options which have been executed, there is always danger due to third parties. For example, when buying online using a credit card, you may be giving away lots of your personal info to the third party. Additionally, there are transaction fees which make online payment expensive.
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You've probably seen this often where you often distribute the nice word about crypto. It is not risky? What happens if the value failures? So far, many POS devices delivers free conversion of fiat, relieving some issue, but until the volatility cryptocurrencies is resolved, many people is likely to be resistant to put on any. We have to find a method to struggle the volatility that is inherent in cryptocurrencies.


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How to Make Your Car Last Longer
Statistics show that the average driver will keep his or her car for roughly six years before buying a new one. Buying a new car isn't cheap, but there are ways to prolong the life of your existing vehicle, which is something that we're going to discuss in today's blog post.
If there's one type of automotive maintenance that you don't want to overlook, it's changing the oil. Oil is responsible for lubricating the engine and its respective components while also wicking heat away from the cylinders. Over time, however, dirt, debris and "sludge" will accumulate inside the oil, making it less effective at performing these tasks. Refer to your vehicle's owner manual for specific guidance on when and how to change the oil; however, a good rule of thumb is to change it at least once every 5,000 miles.
Watch the Engine Temperature
Allowing your engine to overheat is just asking for trouble, which is why it's a good idea to monitor coolant levels. When the cooling system fails, it may cause the engine to melt down, literally. Check your car's coolant levels, and when necessary, add extra coolant to protect against overheating. Keeping the engine well-lubed will also help to protect against overheating.
Related Reading: Common Causes of Engine Overheating
Drive Smart
Of course, the way in which you drive will also play a role in the longevity of your car. If you have a habit of slamming the gas pedal when the light turns green, only to stop just 100 yards down the road at the next red light, you'll increase wear and tear of your car's engine and components. To prolong the life of your vehicle, lighten up on the gas pedal by driving more cautiously.
Wash and Wax it
We can't talk about ways to prolong the life of a car without mentioning washing and waxing. When dirt settles on the surface of a car, it will slowly rub away at the protective coating of paint. This problem is compounded by the fact that cars are exposed to ultraviolet (UV) light from the sun. You can protect your car from this destructive and damaging phenomenon, however, by washing and waxing it on a regular basis.
Tire Alignment
When was the last time that you had your tires align? Whenever you drive over a pothole, speed bump, or any uneven surface/terrain, it will gradually knock your tires out of alignment. When this occurs, it will pull your car to either the left or right side, causing additional tread wear while subsequently forcing you to replace the tires sooner rather than later. If you notice your car pulling to one side, take it to an auto shop for a diagnosis.
Related Reading: DIY Wheel Alignment Guide

redpajama_set_name: RedPajamaCommonCrawl


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About OCRA
The Cancer Registrar is a respected and valued member of the hospital's cancer program. The program's main purpose is to oversee state-of-the art cancer care to all patients. The registrar works closely with physicians, hospital administration, and other allied health professionals in maintaining a database on patient care, which is evaluated and studied periodically.
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/ Performance Zone
Internet License Plates? We Already Have 'em!
Web security is tracked quite heavily, and recently a Homeland Security official suggested "internet license plates." Read more about this Orwellian concept.
Christopher Lamb
Jan. 25, 16 · Performance Zone ·
A few days ago, Erik Barnett, a US Department of Homeland Security official, penned a piece suggesting that the US Government require "license plates" for those of us who cruise around the internet. This, of course, was met with the usual disdain from expected corners of the interwebs. It initial suggestion certainly exudes the usual "trust us, we won't track you if you don't have anything to hide" kind of thinking we've come to expect from law enforcement today. Look, I'm not claiming that Erik's interested in turning the US into a police state, or that he honestly believes that internet access and use should be regulated like it is in slightly more oppressive regimes like, say, Iran or North Korea. I don't think he believes this. I choose to believe in the examples he outlines in his article as motivation for his thinking. Which are pretty horrible. But then, whenever people bring these kinds of things up, these are always the kinds of examples used to justify these kinds of ideas, so perhaps I should be a bit more skeptical of his motivations than I am.
Anyway, it's a bit too late for this debate anyway. Erik doesn't know this, but we already have internet "license plates". With browser fingerprinting, just about everybody out there already has a unique fingerprint associated with their browsing (and please, don't bring up Tor, it was broken late last year). The flexibility of modern browsers and operating systems gives each browser on each computer a potentially unique profile that can be tracked and associated with internet use. We can look at the browser resolution, we can examine accepted and submitted HTTP Headers, we can look at fonts and language preferences, and other things, too. With all of this in hand we can identify who's looking at what, when, and from where, especially when tied to IP addresses. Not only can the somebody already identify who you are, they can identify where you go. They can see your wifi preferences and locations - do you use your laptop with wifi at a friends house? the analysts could see that. Internet cafes? that too. Libraries? you bet. Navigate over to https://panopticlick.eff.org/ to see this kind of fingerprinting in action.
Advertisers have been doing this kind of thing for a year or two now. It's not a leap to think that someone else has had the same kind of ideas.
How to Set Up Your Own Personal iOS Network Monitor
Canvas Fingerprinting: Cookies on Steroids
When It Comes to Web Performance, We Are All Winners
Introduction to Docker Monitoring
monitoring ,web ,privacy
Performance Partner Resources


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The Oath of Dr. Issa Oakmont
by Joseph Patrick 12 months ago in bitcoin
Photo by Yang Xia on Unsplash
Tyler walked into my office carrying a cup of coffee and a bagel. "Hey Mitch, what are you on today," he asked, as he sat down in a chair on the other side of my desk. He put his feet up on the corner, his brown wingtips knocking over a framed drawing my six-year-old gave me. "Best Dad Ever". It was a crayon drawing of me in a superman suit, slaying a monster.
"Dammit, Tyler! Get your feet down," I yelled at him, as I picked up the drawing and checked for damages. The glass was still intact.
"Jim has me investigating unusual trading activity in a micro-cap biotech stock," I said. "Looks like naked shorting again. I've got a meeting with a hedge fund manager at 1:00. Wish me luck."
"God, how do you always get the best assignments?" Tyler sighed, "I'm writing a really boring retrospective about deflation in the post cold war polyester market."
"Oof. Well, it's sad that you think mine is the best assignment," I replied. "When you started studying journalism, didn't you think you'd be covering troops on the front lines, interviewing Fidel Castro, maybe doing an expose on a drug cartel kingpin?" I asked. "I mean, don't get me wrong, The Wall Street Journal is a great gig, I'm not complaining. I just thought things would be different."
Tyler nodded his head in agreement and then took a sip of his coffee and a bite of his bagel. "Shouldn't have minored in Finance, I guess," he chuckled with his mouth full. At that time, Chuy Martinez from the mailroom came in with the day's deliveries.
"Thanks Chuy," I said, as he handed me a stack of manila envelopes. Chuy was from Brookline, MA, and was wearing a Boston Celtics jersey. He got this job in the mailroom at the Journal part-time, while finishing his bachelor's degree at NYU. It was game night, and Boston was playing the Knicks at Madison Square garden at 7pm.
"Good lord, Chuy. You've got some stones wearing that in here. You guys are gonna get crushed tonight," I razzed him.
"Oh yeah, we'll see about that Mr. Goldberg," Chuy pushed back. "You guys ain't been nothin' since Ewing was traded to the Supersonics."
"Okay, okay. True, but I'm not a front runner. I'm a Knickerbocker through and through. I bleeeed blue," I joked and smiled. "And it's not Mr. Goldberg, it's Mitch," I said, pointing at him. "Look, I'll see you at the game tonight -- but do yourself a favor and put on a different shirt!"
"Alright, Mr. Goldberg," Chuy said as he walked out of the office.
I started flipping through the mail as Tyler picked up a copy of the New York Times and kept drinking his coffee. There were some opinion articles, research requests, and a rejected expense report from accounting. Forgot to attach a receipt. Then I got to a large, beaten up manila envelope. It was mailed from Chicago and post-marked March 17, 2007. There was no return address. Bizarre. It was December 2010, and this thing took almost four years to get to me.
"Hey Ty, check this out," I said. Tyler set the New York Times down and took the envelope with both hands. "God, that's odd. Look," he said pointing at the stamps. "They are upside down."
"A sign of distress," I said.
"Yeah," Tyler said. "Open it."
"With my luck, it's going to be a bomb or anthrax. Been nice working with you, Ty," I joked as I reached for the scissors. The envelope was partially sealed and had been taped over by the post office. I used the scissors to cut the tape. No bomb, no white powder, just a hand-typed manuscript.
I pulled it out and showed Tyler.
The Oath of Dr. Issa Oakmont: a Manifesto
Tyler grabbed it from me and read the title out loud, "The Oath of Dr. Issa Oakmont...a Manifesto. Who actually calls their own manuscript a Manifesto?" he asked.
"I...don't...know, the Unabomber maybe," I replied. "Well, this is going to be more interesting than post cold war polyester. Should we show Jim?"
"No Way! He'd just give it to Phillips," Tyler replied. "Let's check it out."
There was a letter attached to the manifesto, and it was written directly to me:
Mr. Mitchel Goldberg:
For reasons that I shall make clear in the pages that follow, I have set out to revolutionize the way in which the world considers currency. The fiat currencies, for which we pay dearly, will soon be a relic of a bygone era.
No more shall we submit to the ruthless tactics of world banks. No more shall we pay interest and fees to use our own money.
No more shall we submit fees to middle-man operators to transact between two willing parties.
I submit this Oath to you, Mr. Goldberg. I solemnly swear, I shall create this currency and unleash it on the world. Because of the disruption to the current world order, I shall be hated. I shall be an outlaw. I shall be a martyr for my cause. This is the cross that I must bear.
My only request is that you publish this manifesto in its unedited form so that the world may know my true intentions and pursue the reform that I seek.
Dr. Issa Oakmont
I set the manifesto back down on my desk and looked at Tyler. "This sounds like Bitcoin, Ty," I said.
"Yeah, it does, but there's other crypto currencies now. Besides, the guy who created Bitcoin wasn't named Oakmont. It was a Japanese guy...I can't recall his name just now," said Tyler.
"Satoshi Nakamoto," I said. "But he's supposedly not real. There's a belief that it's actually a group of people." I sat back in my chair, which made a creaking sound as I leaned back and held my hands in front of me, thinking about the manifesto.
"Maybe we should google Dr. Issa Oakmont, see what comes up," Tyler suggested. He took a sip of his coffee and stood up. He reached over and threw the rest of his bagel in my trash can.
"Seriously, Ty?" I asked. "Use the kitchen trash can. This is just going to attract bugs, and I don't know if the cleaning staff is coming tonight. You're kind of a mess."
"Sorry, Mitch. I am what I am," said Tyler.
"Okay, look. I'm curious about this Oakmont guy, but I don't have time to research right now. I need to get ready for my 1:00. Why don't you have Lisa Dabrowski in Research do some digging, see what she finds," I suggested.
"Okay, will do Mitch," Tyler responded. "By the way, I'm not going to the game tonight. Sarah's parents are coming over for dinner. So, I'll catch up with you in the morning."
Tyler turned to walk away. "Hey, Ty. Take the trash dude," I said. Tyler turned and smiled, stopped at the trash can and removed the bag before leaving my office.
The meeting with the hedge fund manager went pretty much the way I was expecting. Naked shorting is illegal. Proving it is almost impossible, and these guys are never going to admit to it. I still needed to turn in an article, but I hit a dead end here.
It was 4:00 pm, and I was beat. I thought I'd get a beer before heading off to the game. The hedge fund was three blocks from McSorley's on 7th street. I walked down there and ponied up to the bar.
McSorley's is one of the oldest bars in the city. There's sawdust on the floors and they never dust behind the bar. There's a hundred years worth of cobwebs, ghosts, and memories. My kind of place.
I sipped on a dark pint and pulled out the Oakmont manifesto. It was filled with pages of drawings and descriptions of how the alternative currency would work. Oakmont went on for several pages about monetary policy and how the Federal Reserve is running a racket on the middle class. A great deal of it I understood, some of it was too abstract. Some of it was true, most of it strained my credulity.
I understood how in some way our system was like a house of cards. The Federal Reserve was responsible for maintaining the amount of money in circulation. There was supposed to be actual currency for every dollar represented in a savings account, checking account, mortgage, car loan. Pension. But was there?
I thought about how I paid for my pint. I used my debit card. My bank transferred the money from my account to McSorley's account. I thought about the fees they would charge for it -- how stupid it would be if we used the same bank. I thought about It's a Wonderful Life, and how George Bailey explained that he could not give everybody their money when they wanted to close their accounts. Their money was loaned out. One missing deposit nearly ruined him.
I understood the attraction of the world Oakmont sought to create. But I did not trust his vision. How would it be any different? Just like Bitcoin, it would only be an illusion -- something that has value because we have decided it has value. And how is that any different than our paper money? Those thoughts chilled me to the bone. I finished my pint. It was 5:30 PM, and I decided it was time to head to the Garden for the game. I tightened my overcoat, tied my scarf, and stepped outside. I walked to 3rd Avenue and hailed a cab.
The Garden was packed to the gills, even though it was a Wednesday. Chuy was right. The Knicks had not been very exciting for years, but the season was off to a great start. Heading into this game with Boston, we were 16-9 and on an 8-game winning streak.
I loved the Knicks. I came out because my wife was in upstate New York with the kids, visiting her sister. I had too many deadlines at the paper to join them.
I settled into the Wall Street Journal luxury suite. This was one of the benefits of working for the paper. I grabbed a beer and saw Chuy sitting at the edge of the suite. I couldn't believe it; he was still wearing that Celtics jersey. I walked over to him.
"Chuy, Chuy, Chuy. You'll be lucky to get out of here in one piece. I told you to change that thing. Knicks fans are like wild beasts, son," I said to him.
"I know, Mr. Goldberg, but I don't change for anybody or anything. I do what's right for me," Chuy said with a smile.
I sat down next to him and opened a beer. "Well, I gotta respect that, my man," I said. "Cheers!" I lifted my bottle of beer and took a sip. The Rolling Rock was not nearly as good as the pint I had at McSorley's, but it would do for now.
The basketball was great. Stoudemire, Chandler, and Felton were a well-oiled machine, and they weren't making it easy for the Celtics, but Boston's Pierce was a force. Chuy and I spent the first three quarters chatting, snacking, drinking beer, and cheering and jeering at opposite times.
"You ready?" I asked Chuy, holding up a Rolling Rock.
"Yeah, beer me, Mr. Goldberg," he said.
"Come on, seriously, Chuy, call me Mitch."
"Right on, Mitch," he replied.
We opened our beers and looked out at the game. The fourth quarter was underway, and the Knicks were leading.
"Hey Chuy," I said. "What was it like growing up in Brookline?"
"Oh, Brookline is a great place. I miss it," he replied. "But I didn't grow up there. My parents moved after I graduated high school."
"Oh, I didn't know that," I said. "What do you miss about it?"
"Oh, it's just so steeped in history, you know? Beautiful buildings and parks. I'm a bit of a history buff. You know, it was the birthplace of JFK," he said.
"Really? I don't know that much about JFK," I said. "You got a girl back home?"
"Nah, no girl. Not in Brookline," he said. "I was going with a girl in high school, but she went off to Chicago for art school, and I moved to Brookline with my folks. I visited her once, but that was a long time ago."
"Got it. Well, lots of fish out there, my friend. Whoa!! Look at that! Hahaha! You guys are going down now," I cheered. Stoudemire had just dunked to widen the lead by 4. The score was 111-107, and the Knicks looked like they may take this one.
"Hey, plenty of time on the clock, Mitch," Chuy said.
He was right. There was still almost 3 minutes left.
"Yeah, we'll see Chuy. I think we got you guys this time," I said. "So, tell me more about your hometown-- where you went to High School."
"Yeah, Westminster. It's a tiny little place in Worcester County. There's not a lot going on there," Chuy said.
"How did your family end up there?" I asked.
Chuy said his dad was the Director of IT for Aubuchon Hardware's distribution center there, and that was how he got interested in computers.
"So, you're a computer guy. You know anything about Blockchain, Chuy," I asked.
Chuy nodded. "I'm interested in it. If a guy could figure out how to quickly mine a lot of coins, he could make a lot of money," Chuy said with enthusiasm.
"Got it," I said. "So, how'd you get the name Chuy, anyway?"
"Oh, my parents are HUUUGE Star Wars fans," Chuy said, and then laughed. "Nah man. My real name is Jesus. In Spanish, it's common to call people with my name, Chuy. It's like Richard and Dick, or Charles and Chuck."
"Huh. Learn something new...", I started to say.
"Whoa!!! Hahahahaha! Mitch, you guys are going down now!" Chuy yelled, jumping from his seat and spilling his beer.
Boston's Pierce had just nailed a 2 point jumper with only 0.4 seconds left on the clock, giving Boston a 2 point lead. The score was now 118-116, and the Knicks had just one possession left...and a prayer.
After a brief timeout, Landry inbounded the ball to Stoudemire just below mid-court, high in the 3-point range. He didn't hesitate and put the ball up as the buzzer went off. The high arc of the ball was insane. Swish, it went right through the hoop. It was deafening as The Garden exploded in cheers. Strangers were hugging each other. People were crying. The whole Knicks bench came rushing onto the court, jumping on each other and screaming. I was elated. Chuy was pissed.
Suddenly, there was a lot of commotion. The officials were questioning the shot. Then, we heard the dreaded announcement. The shot was after the buzzer and did not count. The Celtics had beaten our 8 game winning streak. We were heart-broken.
More than heart-broken, I was also in a bit of trouble, having bet money on this game. I owed $50 large to a bookie from Staten Island: Kenny the Chiropractor -- paying him would keep me from needing one.
Chuy and I finished our beers and said our goodnights. I told him to be careful on the way out wearing that jersey. Fans were angry. I walked out in the cold December air and hailed a cab. It was a long ride back to my empty brownstone on the Upper East Side.
I was back in my office the next morning, drinking a cup of coffee and reading the Oakmont manifesto again when the phone rang. I answered.
"Goldberg," I said.
"Hey, Mitch, it's Lisa Dabrowski, from Research," said the voice on the other end.
Lisa was the highest educated person working at the Journal. She had a PhD in Linguistics and Masters in World History.
"Oh, Hi Lisa, how are you doing?" I asked.
"Oh, I'm good, Mitch. How are you?" she asked.
"Mmm, been better," I said as I took a sip of coffee. "Wish the Knicks could have pulled it off last night, but...what are you going to do. So what you got for me? Any luck with Issa Oakmont?" I asked.
"Yeah, no, not really. I can't find a trace of a Dr. Issa Oakmont. There are a lot of doctors with the last name Issa, but it isn't a common first name. I cannot find any record of Oakmont being a last name. That said..." she continued as a mailroom worker came into my office.
"Hold on Lisa," I said, and covered the phone with my hand.
"Hey, where's Chuy today?" I asked the mailroom worker.
"Didn't show up," the worker said.
"Huh. Okay, thanks," I replied.
"Hey Lisa, can I give you a call back? I need to check in on a friend," I said.
"Sure, Mitch. No worries," she said and hung up the phone.
I was worried that Chuy may have been jumped wearing that Celtics jersey last night. I did not have his phone number, so I set off to HR. They were on the next floor up, and I decided to take the stairs.
When I found myself at HR, I knew it was going to be a battle. Marsha Collins was working the front desk, and she was a formidable gate keeper. A real stickler for policy, she was not particularly fond of me. I poked fun at her monthly Safety meetings five years ago at the company holiday party, and she had never forgotten it.
"Hey Marsha, how are you doing today?" I asked.
"Mitchel. What can I do for you?" she replied, deadpan. No smile.
"Well, Chuy Martinez, in the mailroom-- We were at the game together last night, and he didn't show up for work today. I was hoping I could get his phone number so I could check in on him," I explained.
"Oh, hmmm, sorry Mitchel. We have policy about not sharing personal information. I can't give you his number," Marsha said with a glad look in her eye. She sat back in her chair and folded her arms across her chest.
"Come on Marsha. Look, I know you're still upset..." I said.
"I have no idea what you are talking about," she said, glibly. "It's P-O-L-I-C-Y, policy."
"Oh come on. Look, this kid could be in trouble. He was wearing a Celtics jersey, and..."
"Mitchel, I don't care if he was wearing a Kimono, you're not getting his phone number. Now please, I have work I need to do," she said and waved me away.
I left the office and called Tyler. I explained the situation. Tyler understood Knicks fans, and he knew that Chuy could be in trouble. He agreed to help me get Chuy's file out of HR. There was remodeling going on in the office next to Marsha's. I had a plan.
Tyler burst into the HR office.
"Marsha!" he said with intensity.
Marsha leapt to her feet.
"Tyler, what is it?" she questioned.
Tyler put both of his hands on Marsha's desk and leaned forward. In a loud whisper he said, "I was walking by the office next door, where they are painting, and I saw one of the maintenance workers on the highest rung of the ladder, straining to reach a corner. It doesn't look right."
"Well! That will not stand," she exclaimed. She grabbed a clipboard and a red Sharpie and marched out of the office with Tyler in tow.
I had been hiding behind a wall in the hallway. I glanced around the corner. Tyler looked over his shoulder and gave me a thumbs up. I quickly darted into Marsha's office and began scrambling for the key to the filing cabinets. I found it in the pull out drawer under her computer. I was running out of time.
Through the walls, I could hear Marsha berating the maintenance worker.
"It says in Paragraph 3 of page 32 of the Employee Handbook...that you must obey all safety policies..."
Ty told me later that the poor man just looked on in shock, occasionally fighting to get a word in edgewise. He ended up getting written up for violating company safety policy.
I found the cabinet that had Chuy's file and ran out of the office. I was already in the stairway on my way back down to my floor when I realized I left the key in the cabinet. It was too late. I would have to deal with Marsha soon.
I was back in my office. I knew it would be mere minutes before Marsha came down here to confront me about the file. I found Chuy's cell phone number from his application and called him. His phone was either turned off, or his battery was dead, because it clicked over to voicemail right away. The mailbox was full.
I called Lisa back.
"Hey, Lisa. Sorry about that. There's was something I had to do," I explained.
I was looking at Chuy's file when Lisa continued what she was trying to explain earlier.
"What I was starting to say, Mitch," she continued. "is that Oakmont is not a last name, but it is a very common geographical name. You'll find streets, buildings, golf courses with this name. It's quite ubiquitous."
Just then, I found Chuy's resume.
"Yeah, would you look at that," I said. "Chuy graduated from Oakmont Regional High School. What a coincidence."
"Chuy? Who is Chuy, Mitch?" Lisa asked.
"A friend from the mailroom," I replied.
"Mitch, Chuy is a nickname for Jesus, and.." Lisa started to say before I cut her off.
Marsha had entered our offices and was looking for me. Thankfully, Tyler caught her before she could make it to my side of the room.
"Lisa. Shit. Lisa-- meet me in the lobby right now. Hurry!" I cried and slammed the phone. I grabbed Chuy's file and the manifesto and ran out of my office, trying to avoid Marsha. It was impossible. I was halfway to the elevator when I heard her scream.
"Stop! Mitchel Goldberg. Stop!"
But I didn't stop. I made it to the elevator and pressed the button for the lobby. The doors were closing as Marsha came running towards me, her eyebrows slanted and nostrils flaring. I was on the fifth floor, and I knew I would barely beat her to the lobby.
The doors opened, and I saw Lisa standing by a potted plant near the drinking fountain. I ran toward her telling her we didn't have much time. We headed for the front doors.
As we approached the front doors, I saw Kenny the Chiropractor enter the rotating door. He had come to collect.
"Shit. Dammit. Lisa we can't go this way," I exclaimed.
We turned around and headed for the back door, when Marsha came busting through the stairway doors.
"Mr. Goldberg. You need to STOP!" she yelled.
Building security looked over at us. Lisa grabbed my hand and said, "Come on, I know another way."
We ran through a side corridor to an emergency exit. The alarm went off as we pushed through and spilled onto W 47th street. We picked ourselves up and grabbed a Yellow Cab.
Out of breath and dazed, we felt safe in the cab.
Lisa said, "I was trying to tell you that Chuy is a nickname for Jesus."
"Yeah, I know. Chuy told me that last night. It's very common, I guess," I said.
"No, you don't understand," Lisa said emphatically. "Issa is a name in Arabic that means Jesus. Jesus Martinez, graduating from Oakmont Regional High School...?"
I was shocked. "He's also the one who delivered the manifesto to me," I said. "Do you think he wrote it?"
"The odds of another explanation are extremely slim," she explained.
The cab stopped in front of Chuy's apartment building on Avenue B and 10th Street. We got out and rang the bell for his place. His roommate answered.
"Who is it?" he asked.
"Hey, it's Mitchel Goldberg," I said. "I work with Chuy."
"Oh, yeah. Chuy's not here," the roommate said.
"Where is he? He didn't come to work this morning," I said. "I'm worried about him."
"He's fine. He said not to worry. But he doesn't live here anymore," the roommate said. "He left a package for you. I'll meet you down there. Hold on."
The roommate handed me a box and a card from Chuy. The card read:
Mitch. I'm sorry I had to skip town without saying goodbye. I'm not really a student at NYU, and I will not be found again. The truth is, I am the creator of Bitcoin. The manifesto, which I'm guessing you already know, was written by me. I created Bitcoin after my father lost his job in Westminster, and the bank foreclosed on our family home. I needed a way to survive in this world without having to submit to the banks and the US Dollar. I took the job at the Journal so that I could get close to you. I trust you to handle the manifesto. I did write that in 2007, but I never mailed it. I had my girlfriend in Chicago send me some canceled stamps, and I used them on the old envelope, hoping to throw you off the trail.
I leave you with this parting gift. In the box, you will find a keycode protected hard drive containing nearly 2 million bitcoins. I will not tell you the passcode, but you have all you need to figure it out. It's 15 characters. Good luck- your friend, Chuy.
P.S. - I am sorry about the game last night. Knicks do look good this year.
I looked at Lisa dumbfounded. "Wow, 2 million Bitcoins. It's trading at 23 cents per coin right now," I said.
"That's nearly $500k dollars!" Lisa cried.
"Help me crack the code, Lisa. Do that, and I'll give you half the coins," I said. She nodded in agreement.
We walked across the street to a corner computer cafe to start brainstorming ideas for the codes. We rented a computer and plugged in the hard drive.
"Mitch, in the letter, Chuy said 'you have all of the information you need to figure it out'," Lisa explained. "Do you think there's a message in the card?"
"Yeah, maybe," I said. "Look. He underlined Chicago and Bitcoins. Those represent two things he loved, and together they are 15 characters."
Lisa agreed that it was a good possibility and worth a try.
I typed in the passcode...it failed. Worse yet, the computer told me I had only 3 more attempts and then the hard drive would encrypt itself.
"Wait, Mitch. BITCOINSCHICAGO," she said, "That could be it?"
"Yeah, let's try it."
Failed. We had 2 more attempts until the $5o0k was lost forever, and I would have to take out a second mortgage to pay off Kenny the Chiropractor.
"What about Oakmont Manifesto?" I suggested.
"No, that's 16 characters," she said in a disappointed tone. "Hmmm. Maybe it is something from the manifesto, though. Can I see it?
I handed her the manifesto and she started to pore over the letter that "Dr. Issa" had written to me.
She said, "Mitch. The words Oath, Issa, and Oakmont are exactly 15 characters when put together. I think that is the passcode, and likely in that order."
"Okay, but we only have 2 attempts left, Lisa. What if you're wrong?" I said.
"Well, we aren't going to get anywhere by not trying it. It makes sense," she said.
I tried it. My hands were shaking as I slowly pecked each letter, making sure I my fat fingers wouldn't cost us $500k. I looked at Lisa and took a deep breath. She said a prayer. I hit [ENTER].
Fail. 1 attempt left.
I was deflated. It seemed we were never going to unlock the hard drive.
"God dammit, Lisa," I cried and sighed heavily. "This is really fucked up. I can't believe a 17 year old kid from a tiny town in Massachusetts is Satoshi Nakamoto."
"Who?" Lisa asked.
"Satoshi Nakamoto. He's the creator of Bitcoin. Nobody thought he was a real pers..."
"Hold on Mitch. Hand me that paper and pencil," Lisa said, while motioning to a pad of paper next to me.
I did as she asked, and Lisa went to work.
She had written 'Satoshi Nakamoto' on the paper and stared at it for minutes.
"What is it, Lisa?" I asked.
"Shhh. Hold on. I'm thinking," she said.
I waited patiently, staring at the computer and hearing the sounds of the East Village. It was nearing 11am, and traffic was buzzing. Horns were honking. I didn't understand how Lisa could concentrate under these conditions.
Suddenly, she looked up, and showed me the note pad. Alongside 'Satoshi Nakamoto', she had written 'OathIssaOakmont'. They were both 15 characters.
She smiled brightly and had a glimmer in her eye. "This is it, Mitch," she said, sounding victorious.
"What do you mean?" I asked. "What have you figured out there?"
"Satoshi Nakamoto is an anagram of Oath Issa Oakmont," she explained. "When I rearrange the letters, it spells Satoshi Nakamoto, the founder of Bitcoin." She laughed. She was giddy.
"So, Satoshi Nakamoto is the passcode!" I cried.
Lisa nodded, and I typed it into the computer. I hit enter and looked over at Lisa as tears came streaming down her cheeks, and she smiled from ear to ear.
I picked up my cell phone and called Kenny the Chiropractor.
"Kenny. I've got your money," I said. "Meet me at Grand Central tomorrow morning at 9am."
Joseph Patrick
former rocker, father, husband, day trader, metal detectorist, Bordeaux collector
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