text:
\begin{document}

\title{Effects of complex parameters on classical trajectories of\\ Hamiltonian systems}

\author{Asiri Nanayakkara}\email{asiri@ifs.ac.lk}
\author{ Thilagarajah Mathanaranjan$^*$}\email{mathan@jfn.ac.lk}

\affiliation{$^*$Institute of Fundamental Studies Hanthana Road, Kandy, Sri Lanka\\
$^\dagger$ Department of Mathematics, University of Jaffna, Sri Lanka}

\date{\today}

\begin{abstract}
Anderson et al have shown that for complex energies, the classical
trajectories of \textit{real }quartic potentials are closed and periodic
only on a discrete set of eigencurves. Moreover, recently it was revealed
that, when time is complex $t$ $(t=t_{r}e^{i\theta _{\tau }}),$ certain real
hermitian systems possess close periodic trajectories only for a discrete
set of values of $\theta _{\tau }$. On the other hand it is generally true
that even for real energies, classical trajectories of non $\mathcal{PT}$- symmetric
Hamiltonians with complex parameters are mostly non-periodic and open. In
this paper we show that for given real energy, the classical trajectories of
\textit{complex} quartic Hamiltonians $H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is
real, $b$ is complex and $k=1$ $or$ $2$) are closed and periodic only for a
discrete set of parameter curves in the complex $b$-plane. It was further
found that given complex parameter $b$, the classical trajectories are
periodic for a discrete set of real energies (i.e. classical energy get
discretized or quantized by imposing the condition that trajectories are
periodic and closed). Moreover, we show that for real and positive energies
(continuous), the classical trajectories of \textit{complex} Hamiltonian $
H=p^{2}+

\mu

x^{4},$ ($

\mu

=\mu _{r}e^{i\theta })$ are periodic when \ $\theta =4 tan^{-1}[(n/(2m+n))]$
for $\forall $ $n$ and $m\in

\mathbb{Z}

$.

\end{abstract}

\pacs{  03.65.w, 03.65.Sq, 03.65.Ge }

\maketitle

\section{\textbf{Introduction}}\label{sec:1}

In recent years, classical behavior of non-Hermitian Hamiltonian systems\cite{R1,R2,R3,R4,R5} as well as classical motion of Hermitian systems for complex energies\cite{R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16} have attracted much interest. Investigation of classical mechanics in the complex domain is useful for understanding various classical and quantum
mechanical phenomena such as barrier tunneling, dynamical tunneling\cite{R17,R18}, classical and quantum chaos\cite{R7,R19}, quantum correspondence principle,complex forms of uncertainty relations and the semiclassical limit of complex quantum field theories.

Since the earlier work on classical motion of non-Hermitian systems \cite{R6,R7},
there have been several interesting results found on the various aspects of
the subject\cite{R8,R9,R10,R11,R12,R13,R14,R15,R16}. Numerical and analytical investigations have revealed
that when energies are real, classical trajectories of complex $\mathcal{PT}$-symmetric
non Hermitian systems are closed and periodic. However, when energies of
these systems are complex, the periodic trajectories usually become non
periodic and open\cite{R15,R19}. Recently it was shown that even though most of
the trajectories corresponding to complex energies are open and non
periodic, for some systems, there are special discrete sets of curves in the
complex-energy plane for which the trajectories are periodic\cite{R20}. On the
other hand, in non-Hermitian and non $\mathcal{PT}$-symmetric Hamiltonian systems, even
for real energies, almost all trajectories except a few are non-periodic and
open. It was also shown recently that when time is taken as a complex
quantity with a specific fixed phase angle or as a specific complex
function, non periodic trajectories of 1-D Hamiltonian systems become
periodic and closed\cite{R21}.

In this paper we investigate the classical trajectories from a different
point of view. Here we examine clasical behavior of the complex Hamiltonian $
H=p^{2}+ax^{4}+bx^{k}$, (where $k=1$, $2$, and $a$ is real such that $H$ is
not $\mathcal{PT}$-symmetric) for complex parameter $b$ and real energy $E$. Outline of
the paper is as follows. In the section \ref{sec:2}, analytic expressions for complex
trajectories are derived.Expressions for periods of the periodic trajectories as well as time taken by unbounded trajectories to escape to
infinity are found in terms of $b$ and energy $E$. \ We will show that for
given real energy, the classical trajectories of the above quartic
Hamiltonian are open except for a discrete set of parameter values in the
complex $b$-plane. In section \ref{sec:3} we study how trajectories behave when energy
is real and $b$ is a fixed complex parameter. The classical trajectories of
complex Hamiltonian $H=p^{2}+

\mu

x^{4},$ ($

\mu

=re^{i\theta })$ is investigated for real energies in section \ref{sec:4} and
concluding remarks are given in section \ref{sec:5}.

\section{\protect \protect\textbf{Classical trajectories of
}$H=p^{2}+ax^{4}+bx^{k}$}\label{sec:2}

In this section first we study in detail the classical motion of the complex
quartic anharmonic oscillator. We assume that the Hamiltonian has the form
\begin{equation}
H=p^{2}+ax^{4}+bx^{k} \label{eq:1}
\end{equation}
where $a$ is a real positive constant, $b$ is a complex constant and $k=1$
or $2$. First we derive expressions for $x(t)$ and the period for the above
Hamiltonian. When it is needed, value of $k$ is chosen as $1$ or $2$.
Throughout this paper mass of the particle is taken as half $(i.e.2m=1).$
The equation of motion is
\begin{equation}
\frac{dx}{dt}=p=2\sqrt{E-ax^{4}-bx^{k}}\label{eq:2}
\end{equation}
The turning points of this system are taken as $x_{0},x_{1},x_{2}$ and $
x_{3} $ and by integrating equation(\ref{eq:2})we have
\begin{equation}
\int \frac{dx}{\sqrt{\left( x-x_{0}\right) (x-x_{1})(x-x_{2})(x-x_{3})}}=2
\sqrt{a}e^{i\pi /2}t+c  \label{eq:3}
\end{equation}
where $c$ is the constant of integration which depends on initial
conditions. The left-hand side of the above equation is an elliptic integral
of the first kind and hence equation (\ref{eq:3}) becomes
\begin{equation*}
\frac{2}{\sqrt{\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }}{\small
F}\left( \sin ^{-1}\left[ \sqrt{\frac{\left( x-x_{1}\right) (x_{0}-x_{2})}{
\left( x-x_{0}\right) (x_{1}-x_{2})}}\right] ,\frac{\left(
x_{1}-x_{2}\right) (x_{0}-x_{3})}{\left( x_{0}-x_{2}\right) (x_{1}-x_{3})}
\right)
\end{equation*}
\begin{equation}
=2\sqrt{a}e^{i\pi /2}t+c   \label{eq:4}
\end{equation}
where $F$\ is an elliptic function. We invert the above equation in terms of
Jacobian elliptic function `$sn$' as
\begin{equation}
x\left( t\right) =\frac{x_{1}\left( x_{0}-x_{2}\right)
-x_{0}(x_{1}-x_{2})sn^{2}(u)}{\left( x_{0}-x_{2}\right)
-(x_{1}-x_{2})sn^{2}(u)}   \label{eq:5}
\end{equation}
where $u=\sqrt{a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }e^{i\pi
/2}t+\alpha $ and modulus $\kappa =\left[ \frac{\left( x_{1}-x_{2}\right)
(x_{0}-x_{3})}{\left( x_{0}-x_{2}\right) (x_{1}-x_{3})}\right] ^{1/2}$ and $
\alpha $ is an arbitrary constant which is determined by the initial
conditions. Note that $x(t)$ in the above equation is still a solution of
(\ref{eq:3}), when $x_{0},x_{1},x_{2}$ and $x_{3}$ are cyclically changed (e.g. $
x_{3}\rightarrow x_{0}\rightarrow x_{1}\rightarrow x_{2}\rightarrow x_{3}$).
In order to understand how the trajectories behave, we need to recognize the
periodic, bounded and unbounded properties of the function $x\left( t\right)
$. First we find complementary modulus $\kappa ^{\prime }$ and complete
elliptic functions $K$ and $K^{\prime }$. They are defined by
\begin{equation}
\kappa ^{{\prime}^ 2}=1-\kappa ^{2}=\frac{\left( x_{0}-x_{1}\right)
(x_{2}-x_{3})}{\left( x_{0}-x_{2}\right) (x_{1}-x_{3})}    \label{eq:6}
\end{equation}
\begin{equation}
K=\overset{\frac{\pi }{2}}{\underset{0}{\int }}(1-\kappa ^{2}\sin ^{2}\left(
\phi \right) )^{-\frac{1}{2}}d\phi    \label{eq:7}
\end{equation}
\begin{equation}
 K^{\prime }=\overset{1}{\underset{0}{\int }}\left( 1-t^{2}\right) ^{-
\frac{1}{2}}\left( 1-\kappa ^{{\prime}^2}t^{2}\right) ^{-\frac{1}{2}}dt.   \label{eq:8}
\end{equation}
$K$ and $K^{\prime }$ are evaluated directly from the above equations and
they are independent of phase angle $\theta $.

The trajectory $x(t)$ is given by $\ x\left( t\right) =\frac{x_{1}\left(
x_{0}-x_{2}\right) -x_{0}(x_{1}-x_{2})sn^{2}(u)}{\left( x_{0}-x_{2}\right)
-(x_{1}-x_{2})sn^{2}(u)}.$ The condition of trajectory become unbounded and
the particle escapes to infinity is
\begin{equation}
\left( x_{0}-x_{2}\right) -(x_{1}-x_{2})sn^{2}(u)=0 \bigskip   \label{eq:9}
\end{equation}
where $u=\sqrt{a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right)
}e^{i\pi /2}t+\alpha $ satisfied for some real positive $t$ and the time
taken for the particle to escape to $\infty $ is give by

\begin{equation}
T_{\infty }=\frac{\left( sn^{-1}(z_{0})-\alpha \right) {\small e}^{-i\pi /2}
}{\sqrt{a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }}   \label{eq:10}
\end{equation}
where $z_{0}=\sqrt{\frac{(x_{0}-x_{2})}{(x_{1}-x_{2})}}.$
`$sn$' is doubly periodic with period $4mK+2niK^{\prime }$ where $m$ and $n$
are integers. Therefore the condition for the trajectory to become periodic
and particle does not escape to infinity is
\begin{equation}
\sqrt{a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }e^{i\pi
/2}t=4mK+2niK^{\prime }\ ;\ \ \ \ m,\ n\ \epsilon \

\mathbb{Z}

.   \label{eq:11}
\end{equation}
and $t<T_{\infty }.$
Then the trajectory is periodic with the period.
\begin{equation}
T_{p}=\frac{\left( 4mK+2niK^{\prime }\right) {\small e}^{-i\pi /2}}{\sqrt{
a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }}.    \label{eq:12}
\end{equation}
Note that if $T_{p}>T_{\infty }$ the trajectory is still nonperiodic. By
imposing the condition that Im$\left( T_{p}\right) =0$, we have
\begin{equation}
r\equiv \frac{n}{m}=\frac{Im[2iK/z]}{Im\left[ K^{\prime }/z
\right] };\ \ \ \ m,\ n\ \epsilon \

\mathbb{Z}

  \label{eq:13}
\end{equation}
where $z=\sqrt{a\left( x_{0}-x_{2}\right) \left( x_{1}-x_{3}\right) }.$
Since $n$ and $m$ are integers and the energy $E$ is fixed, $r$ is rational
and the equation (13) provides a discrete set of parameter values in the
complex $b$ plane for which classical trajectories are periodic. Let $
b=b_{r}e^{i\theta }.$ Figures \ref{fig1}a and \ref{fig1}b show how the ratio $r$ varies with
discrete values of $\theta $ for the cases $k=1$ and $k=2$ respectively.
Without loss of generality, the energy $E$ is taken as unity as it is real.
The results can be generalized for any real energy $E$ by simple rescaling
of $x$ and $t$.

\section{Discretization of classical energy}\label{sec:3}

Next we consider the case when parameter $b$ is a fixed complex number and $
E $ is a variable (Assume $a=1$ and $b=1+i.$). As a result, the equation
(\ref{eq:13}) allows only a discrete set of values of $E$ for which trajectories are
periodic. It was found that these discrete values of $E$ can be either real
or complex satisfying the condition (\ref{eq:13}). Table \ref{tab:Table_1} and Table \ref{tab:Table_2} show some real
discrete values of $E$ which make trajectories periodic when $k=1$ and $k=2$
respectively. Figures \ref{P_1} and \ref{P_2} show the periodic trajectories of systems $k=1$
and $k=2$ for two values of real energies.

\begin{figure}[h]
\centering
\includegraphics[width=7.5cm,height=6.0cm]{G_1.png}
\includegraphics[width=7.5cm,height=6.0cm]{G_2.png}
\caption{\small{By applying the condition that classical trajectories are
periodic, we obtain discrete values of $\theta $ for a fixed value of $b_{r}$
. \textbf{(a)} shows how the ratio $r$ \ varies with complex phase angle $
\theta $ of the potential $V(x)=x^{4}+b_{r}e^{i\theta }x\ $. In order to
have periodic trajectories, $r$ ($r\equiv \frac{n}{m}$) has to be rational
and hence only discrete values of $\theta $ satisfy the condition (13). Each
point in the graph represent such a value. \textbf{(b)} same as (a) but for
the potential $V(x)=x^{4}+b_{r}e^{i\theta }x^{2}$}}
\label{fig1}
\end{figure}

Moreover, it was found that if energy $E$ is corresponding to the periodic
trajectories of $p^{2}+ax^{4}+bx$ then $-E$ will be the energy which makes
trajectories of $p^{2}-ax^{4}+i\overline{b}x$ ( $\overline{b}$ is the
complex conjugate of $b$) periodic. Further $E$ and $-E$ are solutions
corresponding to the same $n$ and $m$ in the periodic condition (\ref{eq:13}) for
these two Hamiltonians respectively. In other words if $S_{E}$ is the
discrete set of energies for which classical trajectories of $
p^{2}+ax^{4}+bx $ are periodic then $S_{-E}$ is the set of energies for
which trajectories of $p^{2}-ax^{4}+i\overline{b}x$ are periodic. Figures \ref{P_3}a and \ref{P_3}b show two periodic trajectories illustrating the above claim.

\begin{table}[h]
\centering
\begin{tabular}{ccc}
\hline
\textbf{m} &\hspace{1in}\ \textbf{n}\hspace{1in} & \textbf{E} \\
 \hline
1 & 1 & 0.27499 \\
1 & 2 & 0.71624 \\
1 & 3 & 0.78605 \\
2 & 3 & 0.60480 \\
2 & 5 & 0.74280 \\
2 & 1 & -0.28103 \\
3 & 1 & -0.53968 \\
3 & 2 & -0.07449 \\
5 & 2 & -0.42562 \\
\hline
\end{tabular}
\caption{ Classical energy spectrum corresponding to periodic trajectories of
$V(x)=x^{4}+(1+i)x$ for various $(m,n).$}
\label{tab:Table_1}
\end{table}

\begin{table}[h]
\centering
\begin{tabular}{ccc}
\hline
\textbf{m} &\hspace{1in}\ \textbf{n}\hspace{1in} & \textbf{E} \\
 \hline
1 & 1 & -0.02143 \\
1 & 2 & -0.16951 \\
1 & 3 & -0.32417 \\
2 & 3 & -0.08940 \\
2 & 5 & -0.24827 \\
2 & 1 & 1.45802 \\
3 & 1 & 2.99725 \\
3 & 2 & 0.81963 \\
5 & 2 & 2.17849 \\
\hline
\end{tabular}
\caption{ Classical energy spectrum corresponding to periodic trajectories of
$V(x)=x^{4}+(1+i)x^{2}$ for various $(m,n)$}
\label{tab:Table_2}
\end{table}

\section{\textbf{Periodic Classical trajectories of }$H=p^{2}+\protect\mu
_{r}e^{i\protect\theta }x^{4}$}\label{sec:4}

Next we assume that $a=\mu $ and $b=0$ in the Hamiltonian (\ref{eq:1}). Then new
Hamiltonian has the form
\begin{equation}
H=p^{2}+\mu x^{4}  \label{eq:14}
\end{equation}
\ where $\mu $ is complex and $\mu =\mu _{r}e^{i\theta }$.
The equation of motion is
\begin{equation}
\frac{dx}{dt}=2\sqrt{E-\mu x^{4}}   \label{eq:15}
\end{equation}
where $E$ is the total energy. Following the same procedure as in section \ref{sec:2},
we obtain required equations. By integrating (\ref{eq:15}) we have
\begin{equation}
\int \frac{dx}{\sqrt{E-\mu x^{4}}}=2t+c  \label{eq:16}
\end{equation}
where $c$ is the constant of integration which depends on initial
conditions. The left-hand side of the above equation is an elliptic integral
of the first kind and hence equation (\ref{eq:16}) becomes
\begin{equation}
{\small F}\left( \sin ^{-1}\left[ \left( \frac{\mu }{E}\right) ^{1/4}x(t)
\right] ,-1\right) {\small =2}\left( \mu E\right) ^{1/4}{\small t+\alpha }
  \label{eq:17}
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=7.5cm,height=6.0cm]{P_1.png}
\caption{\small{A periodic trajectory corresponding to $(m,n)=(1,2)$ for the
quartic potential $V(x)=x^{4}+(1+i)x$ with real energy $E=0.71624$}}
\label{P_1}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=7.5cm,height=6.0cm]{P_2.png}
\caption{\small{A periodic trajectory for the quartic potential $
V(x)=x^{4}+(1+i)x^{2}$ corresponding to $(m,n)=(3,2)$ with real energy $
E=0.81963$}}
\label{P_2}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=7.5cm,height=6.0cm]{P_3a.png}
\includegraphics[width=7.5cm,height=6.0cm]{P_3b.png}
\caption{\small{A typical periodic classical trajectories of the potentials (a) $
V(x)=x^{4}+(2+3i)x$ and (b) $V(x)=-x^{4}+(3+2i)x$\ . Energies of
trajectories\ corresponding to figures (a) and (b) are $E=2.42227$ and $
E=-2.42227$ respectively. The four turning points are marked as dots.}}
\label{P_3}
\end{figure}

where $\alpha =\left( \mu E\right) ^{1/4}c$ and $F$\ is an elliptic
function. We invert the above equation in terms of Jacobian elliptic
function `$sn$' as
\begin{equation}
x\left( t\right) =\left( \frac{E}{\mu }\right) ^{1/4}sn\left( 2\left( \mu
E\right) ^{1/4}t+\alpha \ ;\ -1\right) .    \label{eq:18}
\end{equation}
Note that modulus $\kappa ^{2}=-1$ for the above problem. $K$ and $
K^{\prime }$ are defined in (\ref{eq:7}) and (\ref{eq:8}) and
\begin{equation}
\kappa^{\prime^{2}}=1-\kappa ^{2}=2.    \label{eq:19}
\end{equation}
Then $K$ and $K^{\prime }$ are obtained as
\begin{equation}
K=\frac{\sqrt{\pi }\Gamma \left( 1/4\right) }{4\Gamma \left( 3/4\right) }
  \label{eq:20}
\end{equation}
\begin{equation}
K^{\prime }=\frac{\sqrt{\pi }\Gamma \left( 1/4\right) }{4\Gamma \left(
3/4\right) }\left( 1-i\right)    \label{eq:21}
\end{equation}
As in the previous sections the condition of trajectory become
periodic and particle does not escape to infinity is
\begin{equation}
2\left( \mu E\right) ^{1/4}t=4mK+2niK^{\prime }\ ;\ \ \ m,\ n\ \epsilon \

\mathbb{Z}

.   \label{eq:22}
\end{equation}
Then the trajectory is periodic with the period
\begin{equation}
T_{p}(\mu _{r})=\frac{2mK+niK^{\prime }}{\left( \mu E\right) ^{1/4}}.
  \label{eq:23}
\end{equation}
Since $\mu =\mu _{r}e^{i\theta }$
\begin{equation}
T_{p}(\mu _{r})=\frac{K}{\left( \mu _{r}E\right) ^{1/4}}\left[ \left(
2m+n\right) +in\right] \left( \cos \left( \theta /4\right) -i\sin \left(
\theta /4\right) \right) .   \label{eq:24}
\end{equation}
\begin{equation}
T_{p}(\mu _{r})=\frac{K}{\left( \mu _{r}E\right) ^{1/4}}\left[ \left( \left(
2m+n\right) \cos \left( \theta /4\right) +n\sin \left( \theta /4\right)
\right) +i\left( n\cos \left( \theta /4\right) -\left( 2m+n\right) \sin
\left( \theta /4\right) \right) \right] .    \label{eq:25}
\end{equation}
Since $K$ is \ real and $E$ is real and positive, by imposing the condition
that Im$\left( T_{p}\right) =0$, we have

\begin{equation}
\frac{m}{n}=\frac{\cot \left( \theta /4\right) -1}{2};\ \ \ \ m,\ n\
\epsilon \

\mathbb{Z}

.  \label{eq:26}
\end{equation}
or
\begin{equation}
\theta =4\tan ^{-1}\left[ \frac{n}{2m+n}\right] ;\ \ \ \ m,\ n\ \epsilon \

\mathbb{Z}

.   \label{eq:27}
\end{equation}
When $n=0$ and $m\neq 0$ , $H=p^{2}+\mu _{r}x^{4}$ and it is Hermitian. Then
$H$ possesses periodic trajectories and the period $T_{p}(\mu _{r})$ becomes
 $T_{p}(\mu _{r})=\frac{2mK}{\left( \mu _{r}E\right) ^{1/4}}$ but
the period is corresponding to the minimum non zero $m\ $and the resulting
period is
\begin{equation}
T_{p+}(\mu _{r})=\frac{\sqrt{\pi }\Gamma \left( 1/4\right) }{2\left( \mu
_{r}E\right) ^{1/4}\Gamma \left( 3/4\right) }   \label{eq:28}
\end{equation}
On the other hand when $n\neq 0$ and $m=0$ ,$H=p^{2}-\mu _{r}x^{4}$ and it
is the non-Hermitian `wrong sign' potential which also possesses periodic
trajectories. The period is

\begin{equation}
T_{p-}(\mu _{r})=\frac{\sqrt{\pi }\Gamma \left( 1/4\right) }{2\sqrt{2}\left(
\mu _{r}E\right) ^{1/4}\Gamma \left( 3/4\right) }=\frac{\sqrt{\pi }\Gamma
\left( 1/4\right) }{2\left( 4\mu _{r}E\right) ^{1/4}\Gamma \left( 3/4\right)
}    \label{eq:29}
\end{equation}
It is evident from equations (29) and (30) that the Hamiltonians $p^{2}+4\mu
_{r}x^{4}\ $\ and $p^{2}-\mu _{r}x^{4}$ have the same classical period (i.e.
$T_{p+}(4\mu _{r})=T_{p-}(\mu _{r})$). Note that these two Hamiltonians are
the classical limit of the quantum mechanical isospectral Hamiltonians as
shown in \cite{R22, R23, R24, R25, R26}.

\bigskip

\section{Concluding Remarks}\label{sec:5}

In this paper we have presented three main results. The first is that, for
given real energy, the classical trajectories of quartic Hamiltonians $
H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is real, $b$ is complex, and $k=1$ $or$ $
2 $) are closed and periodic only for a discrete set of parameter curves in
the complex $b$-plane.

The second result is that given complex parameter $b$, the classical
trajectories are found to be periodic only for a discrete set of real
energies. As a result, real classical energies get discretized or quantized
by the condition that trajectories are periodic and closed. This result is
analogous to what was obtained by Anderson et al in \cite{R20} for real potential
parameters with complex $E$ (Here it is for complex potential parameters
with real energies). Further we showed that if $S(E)$ is the discrete set of
energies for which classical trajectories of $p^{2}+ax^{4}+bx$ are periodic
then $S(-E)$ is the set of energies for which trajectories of $p^{2}-ax^{4}+i
\overline{b}x$ are periodic. We presented our results with illustrations. It
is important to note that when $b$ is complex and not pure imaginary, the
entire quantum eigen spectrum corresponding to the Hamiltonian $H$ is
complex and eigenenergies do not come as complex conjugate pairs. Therefore $
H$ cannot be pseudo Hermitian and cannot have any antilinear symmetry.

As the third result, we showed that for real energies , the classical
trajectories of complex Hamiltonian $H=p^{2}+

\mu

x^{4},$ ($

\mu

=re^{i\theta })$ are periodic only for discrete values of $\theta $
satisfying the condition \ $\theta =4tan^{-1}[(n/(2m+n))]$ for $n$ and $m\in

\mathbb{Z}

$. Further it was found that Hamiltonians $p^{2}+4\mu _{r}x^{4}$ $\ $\ and $
p^{2}-\mu _{r}x^{4}$ which are the classical limit of the quantum mechanical
isospectral Hamiltonians introduced in \cite{R22,R23,R24,R25,R26}, have the same classical
period.

\bigskip

config: arxiv


TITLE: Two Ito processes : are they a 2-dim Brownian motion?
QUESTION [2 upvotes]: I am stuck with the following problem :
I have a Brownian motion $B_t$ and an Ito process
$$X_t:=\int_0^t sgn(B_s)\ d B_s,$$
where $sgn(x)=1$ when $x \geq 0$ and $sgn(x)=-1$ when $x<0$.
I have to say whether the couple $(B_t, X_t)$ is a 2-dimensional Brownian motion or not.
I have a hint saying "you may show that the components of the vector are not independent".
Therefore, I guess the answer is NO.
My idea to prove that $X_t$ and $B_t$ are not independent is to show that the expectation of the product is different from the product of the expectations, which is zero.
To compute $\mathbb{E} [X_t B_t]$, I would compute the distribution of $X_t B_t$ via Ito's lemma (integration by parts, actually) and then consider the expectation.
Is this feasible ? Once I have the distribution of $X_t B_t$ I am not so sure about how to procede. Could you help me ?
Thank you very much for any help !!!

REPLY [3 votes]: One way of disproof is to check the quadratic variation: If $B_t, X_t$ are jointly Brownian, then the quadratic variation should be
$[B,X]_t = 0$
But by the condition, we actually have
$[B,X]_t = \int_0^t sign(B_s) ds$ , which is not even deterministic,


TITLE: Proofs of "Table of Integrals, Series, and Products" formulas
QUESTION [2 upvotes]: http://www.amazon.com/Integrals-Series-Products-Seventh-Edition/dp/0123736374
This book is amazing. But there are no proofs in it. Is there book or site which contains all of the proofs of this book's formulas?

REPLY [1 votes]: From the Wikipedia article for Gradshteyn and Ryzhik:

Victor Hugo Moll with George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references. In the foreword of the book Irresistible Integrals (2004), they wrote:

It took a short time to realize that this task was monumental.

Nevertheless, the efforts have meanwhile resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles of which papers 1 to 28 have been published in issues 14 to 26 of Scientia, Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015 and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015) already.

The two volume work can be purchased on amazon: volume 1 and volume 2. I've been tempted to get it myself, the proofs are probably more valuable than the formulae themselves to those interested in preserving mathematical knowledge, and they're probably more useful to anyone able to dedicate the time studying them.


TITLE: Rational approximation of an integer combination of two irrationals
QUESTION [6 upvotes]: Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$,
$$d(nx,\mathbb{Z})>C n^{-\beta}.$$
It is known that for a.e. irrational number $x$, the irrationality index is $1$.
It is even known that some numbers satisf the above  for $\beta=1$ (for instance, $x=\sqrt{2}$).
By arguments from measure theory, I have been able to prove that if $a_n$ satisfies $$\sum_{n=1}^{\infty}na_n<\infty,$$ almost every couple $(x,y)$ of $\mathbb{R}^2$ satisfies for some $C>0$ $$d(nx+my,\mathbb{Z})>Ca_{|n|+|m|},n,m\in \mathbb{Z}.$$
Ideally, I would like, as for a single number $x$, find irrational numbers $(x,y)$ such that this holds in the limiting case $a_n=n^{-2}$.
Has anyone an idea? Or has anyone a useful reference for such things?

REPLY [9 votes]: Yes, such $(x,y)$ exist; for example,
$x = \root 3 \of 2$ and $y = x^2$.
For $l,m,n \in \bf Z$, define
$$
N(l,m,n) := l^3 + 2m^3 + 4n^3 - 6lmn \in {\bf Z};
$$
this is the algebraic norm
$$
 (l + mx + nx^2)
 (l + m\rho x + n \bar\rho x^2)
 (l + m\bar\rho x + n \rho x^2)
$$
of $l + mx + nx^2$,
where $\rho$ is the cube root of unity $e^{2\pi i/3} = (-1 + \sqrt{-3})/2$.
But if $(m,n) \neq (0,0)$ then
$l + mx + nx^2 \neq 0$, so $\left|N(l,m,n)\right| \geq 1$ and
$\left|l + mx + nx^2\right| \gg (|l|+|m|+|n|)^{-2}$.
Taking for $l$ the integer nearest to $-(mx+nx^2)$ we deduce that
$d(mx+nx^2,{\bf Z}) \gg (|m|+|n|)^{-2}$, as claimed.
The same argument (which generalizes the familiar one for $\sqrt 2$)
shows that in general if $x$ is an algebraic number of degree $D$ then
$$
d\Bigl(\sum_{j=1}^{D-1} n_j x^j, {\bf Z}\Bigr) \gg
 \left( \sum_{j=1}^{D-1} \left| n_j \right| \right)^{1-D}
$$
for $n_1,\ldots,n_{D-1} \in \bf Z$ not all zero.
This is best possible up to the value of the implicit constant, because
Dirichlet's celebrated "pigeonhole" argument shows that conversely
for any $N$ one can find integers $n_1,\ldots,n_{D-1} \in [-N,N]$, not all zero,
such that $d(\sum_{j=1}^{D-1} n_j x^j, {\bf Z}) \ll N^{1-D}$.


TITLE: Nuclear fusion reactors and neutrons
QUESTION [11 upvotes]: The majority of energy produced by nuclear fusion is harnessed by neutrons or protons that split out from the product.
Given the dominant fusion method today is Deuterium + Tritium which produces He and a neutron (a neutron that has most of the energy from the fusion), what do current experimental fusion reactors do to harness the energy from said neutrons?
This question is asked using the context that neutrons cannot be controlled using electromagnetic forces. Hence, the energy contained in neutrons would (in my mind, at least) be difficult to capture without resorting to some sort of fission technique.

REPLY [8 votes]: In fusion power plants the energy from neutrons is captured in the blanket, which is immediately behind the first wall that surrounds the plasma. One candidate for the blanket is FLiBe molten salt, which is also being studied in the context of fission and concentrated solar power, and another candidate is dual-coolant Lead-Lithium/Helium. An important aspect of the blanket is also to ensure very little neutron energy is absorbed by the superconducting magnets and support structures. The neutron energy is carried away as thermal energy in the liquid blanket and then used to power a steam turbine.
The neutron energy is also used to breed Tritium in the liquid blanket with the reaction:
$^6\text{Li} + \text{n} \rightarrow \alpha  + \text{T}  + 4.78 \text{ MeV}$
This Tritium can then be used as fusion fuel, while the Tritium decay product Helium-3 is rare on earth and important for neutron radiation detectors and magnetic resonance imaging of the lung. While Tritium is currently bred inside fission reactors (e.g. by replacing some control rods with rods containing Lithium), a major research goal is for fusion power plants to generate their own Tritium.


TITLE: Polynomial bound for the log function
QUESTION [0 upvotes]: For $x>0$, is $\log{x}$ bounded by $x^{-\epsilon}$ for any $\epsilon>0$ when x is small?

REPLY [1 votes]: Let's try to show that $\log x  = o(\varepsilon)$ for suitable $\varepsilon$.
Let us first prove the probably more common fact, any growing exponential function grows faster than a monomial.
For any $\alpha>0$, $\beta > 1$, we can choose an integer $c\gt \alpha$. Applying L'Hôpital's rule $c$ times we get
$$0\le\lim_{x\to\infty}\frac{x^\alpha}{\beta^x}\le\lim_{x\to\infty}\frac{x^c}{\beta^x} 0=\lim_{x\to\infty}\frac{cx^{c-1}}{(\log\beta)\beta^x}\\
=\lim_{x\to\infty}\frac{cx^{c-1}}{(\log\beta)\beta^x}=\lim_{x\to\infty}\frac{c(c-1)x^{c-2}}{(\log\beta)^2\beta^x}\\
=\cdots=\lim_{x\to\infty}\frac{c(c-1)\cdots2\cdot1}{(\log\beta)^c\beta^x}=0$$
where the last equality holds since $\beta^x = (1+(1-\beta))^x\gt1+x(1-\beta)\to\infty$ as $x\to\infty$.
So, $x^\alpha = o(\beta^x)$ for any $\alpha>0$, $\beta > 1$.

Let $\alpha = 1, \beta=e^\varepsilon$. By a change of variable $x=\log n$, we can see that
$$\log n = o(e^{\varepsilon\log n})=o(n^\varepsilon)$$
Similarly, we have
$$\log\log n = o((\log n)^\varepsilon) \quad\text{ for any }\varepsilon\gt 0 $$
$$e^n = o(\lambda^{e^n}) \quad\text{ for any }\lambda\gt 1 $$


TITLE: Why is the constant velocity model used in a projectile motion derivation?
QUESTION [1 upvotes]: I was re-studying university physics last week, I'm now in the chapter about kinematics in 2 dimensions and specifically the one treating projectile motion. In page 86 of his book (Serway - Physics for scientists and engineers) he derives the equation of the range of the projectile motion to be: $$R=\frac{{v_i}^2\sin2\theta_i}{g}$$
But I don't know why he used one of his assumptions

$\color{red}{\bf Question1:}$ Why $v_{xi}=x_{x\rlap\bigcirc B}$? Where $\rlap\bigcirc {\,\sf B}$ is the time when the projectile stops.
$\color{darkorange}{\bf Question2:}$ Why did he use the particle under constant velocity model to derive that formula, whereas here we deal with a projectile under constant acceleration?
Any responses are welcome, I'm disappointed a lot about those matters!

REPLY [0 votes]: If we assume there is no wind and air, the only force acting on a projectile in the air is the force of gravity. The force of gravity acts in vertical direction, thereby affecting the vertical component or Y component of velocity only.
Newton's 1st law says that a particle will continue to be in its state of rest or motion if no external force is impressed on it. Can you think of any force acting on the projectile in horizontal direction? There is none if we assume no wind/ air resistance etc. Thus there is no reason for velocity of the projectile to change in X direction. Hope this will make the above derivation simpler for you to understand.
You can also watch this video I made, in case it is still not clear
Analysing Projectile Motion in X and Y direction