\section{Introduction}

  The recent discovery of soft gamma repeaters (SGRs) with weak fields
\citep{2010Sci...330..944R}
has again raised problems related to magnetic field evolution
in isolated neutron stars. The activity of magnetars
with SGRs and
anomalous X-ray pulsars (AXPs) has been
believed to be powered by the decay of ultrastrong magnetic fields
\citep{1995MNRAS.275..255T,1996ApJ...473..322T}.
Their surface dipole field, which is observationally inferred from
the spin period and its time derivative, typically exceeds
electron quantum magnetic field $ B_{Q} =4.4 \times 10^{13} $G.
The field strength of SGR0418+5729 is, however, relatively weak
at $<7.5\times 10^{12} $G\citep{2010Sci...330..944R}.
The critical boundary between magnetars and radio pulsars
has thus become less clear, and magnetars are not sufficiently
characterized by their dipole field strength alone.
Their activity may be explained by hidden magnetic fields such as
poloidal components with higher-order multipoles
or internal toroidal components.
In either case, the field strength should be greater
than $ B=10^{14}$-$10^{15}$G,
since other energy sources are insufficient
(see, for example, the review by \cite{2008A&ARv..15..225M}).

   The importance of Hall drift on the magnetic evolution of neutron stars
was pointed out
before evidence of the existence of magnetars was
available \citep{1988MNRAS.233..875J,1992ApJ...395..250G}.
The effect, which depends on the
field strength, becomes more important in strong regimes where
$ B > 10^{14}$G.
Furthermore, Hall drift may induce instability under certain
conditions \citep{2002PhRvL..88j1103R}.
The effect has been considered
in analytic treatments
\citep{2004ApJ...609..999C,2007A&A...472..233R},
and as plane-parallel slab geometry
\citep{2000PhRvE..61.4422V,2003A&A...412L..33G,2004A&A...420..631R}.
These studies are useful to understand some aspects of the
mechanism, but nonlinear numerical analysis is also required to
examine the behavior in more realistic stars.
Several simulations have also been performed, assuming that
the fields are located in the spherical crust region of a neutron star
\citep{1994MNRAS.266..597N,1997A&A...321..685S,
2002MNRAS.337..216H,2004MNRAS.347.1273H,2007A&A...470..303P}.
In particular, \cite{2002MNRAS.337..216H,2004MNRAS.347.1273H}
extensively studied the effect in a crust with uniform
conductivity and density, and subsequently extended
the discussion to stratified stellar models.
\cite{2007A&A...470..303P} also calculated the evolution
in a realistic stratified model with a thermal history.
These numerical calculations are based on
spectral or quasi-spectral methods, for example expanding
angular functions using spherical harmonics.
Limitations of such an approach are discussed therein.
For example, evolution of a purely toroidal field forms
a steep gradient that cannot be calculated
\citep{2007A&A...470..303P}.
A recent promising approach is using a finite difference scheme to examine
the nonlinear evolution of the Hall instability in a 2D
slab \citep{2010A&A...513L..12P}.
Each numerical scheme has advantages and disadvantages,
so multiple complementary approaches are needed.

  In this paper, we calculate magnetic field evolution using
a finite difference scheme to understand nonlinear
Hall drift dynamics. The model is simplified by assuming
an axisymmetric magnetic field located in a crust with uniform
conductivity and electron number density.
This paper is organized as follows. In section 2,
the model and its assumptions are described.
Magnetic field evolution is governed by an
induction equation with nonlinear Hall drift term.
The relevant boundary conditions and
initial configurations are also discussed.
Sections 3 and 4 give numerical results for two distinct initial
configurations.
One condition is a purely toroidal field in which the poloidal part is always zero
if it is exactly zero at the initial state.
This is in contrast to a purely poloidal initial case,
for which the toroidal part is inevitably induced.
The evolution of a purely toroidal field is furthermore of interest,
due to the similarity to Burgers' equation (e.g., \cite{1974G.B.Whitham}).
\citet{2000PhRvE..61.4422V} study the problem in a stratified
plane-parallel slab, which corresponds to small-scale dynamics
much smaller than the stellar radius.
The results clarify the local mechanism,
but our concern is global aspects.
How do the results change in a spherical shell like the crust?
This problem is numerically studied
in section 3.
Section 4 describes a second study related to the evolution of mixed
fields with poloidal and toroidal components.
Magnetohydrodynamics simulation
\citep{2006A&A...450.1077B,2006A&A...450.1097B,2009MNRAS.397..763B}
shows that dynamically stable configurations are such mixed ones,
in which poloidal and toroidal field strengths are of the same order.
There is little known about the initial configuration of neutron stars
in particular, the location, topology, and field strength.
If both components coexist, their strengths are likely to be similar.
It is therefore important to explore energy transfer by Hall drift
between the components over long-term evolution through Ohmic dissipation.
Section 5 presents our conclusions.

\section{Model and Formulation}
\subsection{Equations}

  The magnetic field evolution is governed by the induction
equation with the Hall term
\begin{equation}
\frac{\partial }{\partial t}\vec{B}= -\vec{\nabla}\times \left( \frac{c^{2}}{
4\pi \sigma }\vec{\nabla}\times \vec{B}\right) +\vec{\nabla}\times \left[
\frac{c}{4\pi e n_{e}}\vec{B}\times \left( \vec{\nabla}\times \vec{B}\right)
\right] ,
 \label{Evl.eqn}
\end{equation}
where $\sigma $ is the electric conductivity, $n_{e}$ the electron number
density, $e$ the charge density, and $c$ the speed of
light \citep{1992ApJ...395..250G,1994MNRAS.266..597N}.
There are two typical timescales associated with the first and
second terms in eq. (\ref{Evl.eqn}),
the Ohmic decay timescale $ 4\pi \sigma L^{2}/c^{2} $
and the Hall drift timescale $ 4\pi en_{e}L^{2}/(cB_{0})$.
Here, $L$ and $B_{0}$ are typical values
for the spatial length and magnetic field strength.
In general, $\sigma $ and $n_{e}$ depend on
the spatial position and the time through the thermal history. In
this paper, however, for simplicity they are assumed to be constant to
explore the dynamics.
Thus, the behavior of this system is specified by
the ratio of two timescales,
$\mathcal{R}_{m}=\sigma B_{0}/(ecn_{e} )$,
which is called the magnetized parameter
by \cite{2007A&A...470..303P}.
The parameter $\mathcal{R}_{m}$ is given by the initial maximum
value of the magnetic field.
The typical value for a neutron star is
$\mathcal{R}_{m}=$1-10$\times(B_{0}/10^{13}{\rm G}) $
\citep{2007A&A...470..303P},
but this significantly depends on spatial position and
temperature \citep{2004ApJ...609..999C}.
The overall physical timescale is scaled by
$\tau_{d}= 4\pi \sigma r_{s}^{2}/c^{2} $,
where the stellar radius $r_{s}$ is used for normalization.
Note that the characteristic decay timescale is $\tau_{d}$ for node-less
field filled in a sphere, but it becomes smaller for the field
localized in a crust.
A magnetic field with $\mathcal{R}_{m} (> 1)$ is initially
given, and the evolution is numerically followed until
decay at $\sim \tau _{d}$.
In the case where $\mathcal{R}_{m} \gg 1$,
which is relevant to strong fields like those of magnetars,
the second term on the right-hand side of eq. (\ref{Evl.eqn})
dominates. The advection term is nonlinear, and
treating it becomes complicated.

Magnetic fields with axial symmetry ($\partial/\partial \phi=0 $)
are described by two functions, a flux function $G$
describing the poloidal magnetic field and a stream
function $S$ describing poloidal current flow:
\begin{equation}
\vec{B}=\frac{1}{R}(\vec{\nabla}G\times \vec{e}_{\phi})
+\frac{S}{R}\vec{e}_{\phi},
   \label{DefB.eqn}
\end{equation}%
where $R=r\sin \theta $ is the cylindrical radius in spherical coordinates
$(r, \theta, \phi)$. Ampere's equation gives the current density as
\begin{equation}
\frac{4\pi}{c} \vec{j}= \vec{\nabla }\times \vec{B}
 = \frac{1}{R}(\vec{\nabla}S\times \vec{e}_{\phi})
-\frac{1}{R}\mathcal{D}(G)\vec{e}_{\phi},
  \label{DefJ.eqn}
\end{equation}
where ${\mathcal{D}}(G)$ is given by
\begin{equation}
{\mathcal{D}}(G) = \left( \frac{\partial ^{2}}{\partial r^{2}}
+\frac{\sin\theta }{r^{2}}\frac{\partial } {\partial \theta }
\frac{1}{\sin \theta }\frac{\partial}{\partial \theta } \right)G,
\end{equation}%
or in cylindrical coordinates $(R, Z, \phi)$ by
\begin{equation}
{\mathcal{D}}(G) = \left( R\frac{\partial}{\partial R} \frac{1}{R}
\frac{\partial}{\partial R}
+\frac{\partial ^{2}}{\partial Z^{2} }\right)G.
\end{equation}
The magnetic field evolution (\ref{Evl.eqn})
is written in terms of $G$ and $S$ (e.g., \cite{2007A&A...472..233R}) as
\begin{equation}
\frac{\partial G}{\partial t} = \frac{1}{ \tau_{d} }\mathcal{D}(G)
+\frac{\mathcal{R}_{m}}{\tau_{d} R}
 (\vec{\nabla}G\times \vec{\nabla}S)\cdot \vec{e}_{\phi},
  \label{EvlG.eqn}
\end{equation}
\begin{equation}
\frac{\partial S}{\partial t} =\frac{1}{ \tau_{d} }\mathcal{D}(S)
+\frac{\mathcal{R}_{m}R}{\tau_{d}}
\left[ \vec{\nabla} \times \left\{ \frac{1}{R^{2}}(\mathcal{D}(G)
\vec{\nabla}G +S\vec{\nabla}S)\right\} \right]\cdot \vec{e}_{\phi},
\label{EvlS.eqn}
\end{equation}
where $G$, $S$, and the spatial length are appropriately normalized.
As has been pointed out (e.g., \cite{2000PhRvE..61.4422V,2007A&A...470..303P}),
the evolutionary equation of a purely toroidal
magnetic field is very similar to Burgers' equation, which is a simple
example of nonlinear propagation with diffusion in one spatial
dimension \citep{1974G.B.Whitham}.
Equation (\ref{EvlS.eqn}) for $G=0$ is thus reduced to
\begin{equation}
\frac{\partial S}{\partial t} = \frac{1}{ \tau_{d} }
\left( R\frac{\partial}{\partial R} \frac{1}{R} \frac{\partial}{\partial R}
+\frac{\partial ^{2}}{\partial Z^{2} }\right)S
+ \frac{2\mathcal{R}_{m} S}{\tau_{d} R^{2}}\frac{\partial S}{\partial Z}.
  \label{PureS.eqn}
\end{equation}
If the function $S$ depends only on $Z$, eq. (\ref{PureS.eqn}) is exactly
Burgers' equation.

The functions $G$ and $S$ in eq. (\ref{DefB.eqn}) can be expressed by a sum
of Legendre polynomials $P_{l}(\theta)$:
\begin{equation}
G=-\sum_{l=1}g_{l}(r,t)\sin \theta
\frac{\partial P_{l}(\theta )}{\partial\theta },
   \label{Gexpd.eqn}
\end{equation}
\begin{equation}
S=-\sum_{l=1}s_{l}(r,t)\sin \theta
\frac{\partial P_{l}(\theta )}{\partial\theta }.
   \label{Sexpd.eqn}
\end{equation}
The functions $g_{l}$ and $s_{l}$ independently evolve in the absence
of the Hall term ($\mathcal{R}_{m} =0$). However, nonlinear coupling
among $g_{l}$ and $s_{l}$ with different indices $l$ becomes important
with the increase of $\mathcal{R}_{m} $.
Most numerical calculations of magnetic decay with the Hall
effect have been performed by such an expansion
\citep{1994MNRAS.266..597N,
2002MNRAS.337..216H,2004MNRAS.347.1273H,2007A&A...470..303P},
with an exception\citep{1997A&A...321..685S}.
Angular part of the field is expanded by spherical harmonics, but
finite difference is used for radial direction in
quasi-spectral method\citep{1994MNRAS.266..597N,2007A&A...470..303P}.
Tchebycheff polynomials are used for radial direction in
spectral method\citep{2002MNRAS.337..216H,2004MNRAS.347.1273H}.
The limitation of spectral or quasi-spectral methods is
also discussed, for example by \cite{2007A&A...470..303P}.
A numerical Gibbs oscillation appears when the function
evolves to form a steep gradient caused by advection.
This is likely to occur in the large magnetized parameter $\mathcal{R}_{m} $.
The finite difference method is used in this paper as an alternative
approach. The numerical scheme used is a simple stable one, first-order
forward time differencing and second-order centered space (FTCS)
differencing (see, for example, \cite{1992nrfa.book.....P}).
There are more sophisticated schemes, but it is not easy to apply
them to the nonlinear Hall drift term, which is most important here. The
grid is staggered one with equal spacing.
Typical number of grid points is $100 \times 120$.
Our numerical results are verified using previous results based on the spectral
method in \cite{2002MNRAS.337..216H}, and the method works well for most parameters.

\subsection{Initial configuration and boundary conditions}

The magnetic field is assumed to be located outside the
superconducting core. The crust ranges from $r_{1} $ to the
surface $r_{s}$. A typical size in neutron star models
is $(r_{s}-r_{1})/r_{s} \sim 0.1$, and depends on the
equation of state and the stellar mass. A slightly thicker crust model is
chosen here, $r_{1}/r_{s} =0.75 $, because this allows an easier demonstration of the numerical results.
In actual implementation it is necessary to use more realistic models
that include stratified conductivity, number density, and so on,
but this simple model is useful for understanding the fundamental dynamics.

The magnetic field cannot penetrate into the core ($r < r_{1}$).
The condition for the toroidal field is simply $S=0$ at $r_{1}$.
The condition for the poloidal field means that $G$ is a constant at $r_{1}$,
which is chosen as $G=0$.
As discussed in \cite{2002MNRAS.337..216H,2007A&A...470..303P},
the tangential components of the electric field,
$E_{\theta }$ and $E_{\phi }$, should vanish.
They are explicitly given by
\begin{equation}
E_{(\theta , \phi)} = \frac{1}{\sigma} j_{(\theta , \phi)} +\frac{1}{e n_{e}}
( \vec{j} \times \vec{B} )_{(\theta , \phi)}.   \label{Etan.eqn}
\end{equation}
The second term on the right-hand side vanishes because
$\vec{B} = 0$ at $r_{1}$. The first term may be negligible
for a large conductivity $\sigma $. Otherwise, some conditions should
be imposed to cause the tangential current to vanish.
The condition $j_{\theta}=0$ is satisfied if
the function $S$ rapidly approaches $0$ as $r$ goes to $r_{1}$.
The condition $j_{\phi}=0$ should be imposed on the function $G$,
more precisely on $\mathcal{D }(G)$, which is not easily treated.
In the numerical simulation, the conditions $G=S=0$ are used
at $r_{1}$, assuming a large $\sigma $.
The current distributions should be set up at the
initial time for the conditions to be satisfied.

  The exterior of the star is assumed to be vacuum. The toroidal field should
vanish, so that the boundary condition is $S=0$ at $r_{s}$. The vacuum
solution of the poloidal field is expressed by a sum of multipole fields,
\begin{equation}
G=-\sum_{l=1} \frac{ a_{l} }{r^{l}} \sin \theta
\frac{\partial P_{l}(\theta)}{\partial \theta },
  \label{Vac.eqn}
\end{equation}
where the coefficients $a_{l}$ represent the multipole moments.
For example, the dipole moment is $ a_{1}=\mu$.
The radial component $B_{r}$ should be continuous to
the exterior vacuum solution across the surface
by the condition ${\vec \nabla } \cdot {\vec B} =0$,
while $B_{\theta}$ may be discontinuous if a surface current is
allowed. In this paper, neglecting the surface currents,
both components are assumed to be continuous,
and the coefficients $a_{l}$ are calculated by the
interior numerical function $G$ at the surface $r=r_{s}$.
The surface boundary condition can be expressed by a sum of $a_{l}$
up to $l_{max}=$20-30. The results slightly change if the truncation
is $l_{max}<10$, but do not change even if $l_{max}$ is further increased.
Compared with spectral or quasi-spectral methods, small scale structure is
not evident in our finite difference method. Initial configuration contains
$l=$1 or 2 component, so that the truncation is justified.
Another approach without the multipole expansion is
proposed as non-local boundary condition by Green's
formula\citep{2010A&A...513L..12P}, but is not used here.
Finally, for the boundary condition on the symmetric axis, $\theta =0$
and $\pi$ are the regularity for the functions $G$ and $S$, respectively.
In other words, $G=0$ and $S=0$ there.

The functions $G$ and $S$ at the initial time of the numerical
simulation should also be subject to the boundary conditions.
The configuration is easily specified by the functions $g_{l}$
and $s_{l}$ in eqs. (\ref{Gexpd.eqn}) and (\ref{Sexpd.eqn}).
Initial current distribution is chosen as ${\vec j} =0$ at both inner and
outer boundaries.
One simple solution satisfying this is given by
\begin{equation}
s_{l}(r,0)= b_{l} \sin^{2} \left( \frac{\pi (r-r_{1})}{ r_{s}-r_{1} }
\right),
  \label{Sinit.eqn}
\end{equation}
where $b_{l} $ is a constant. Both the function $s_{l}$ and
its derivative $ds_{l}/dr$ approach 0
as $r \to r_{1}$ and $r \to r_{s}$.
For the poloidal field, the function $g_{l}(r,0)$
is numerically given by solving
\begin{equation}
\left( \frac{d^{2}}{dr^{2}} -\frac{l(l+1)}{r^{2}} \right)g_{l}=
 - \sin \left( \frac{\pi (r-r_{1})}{ r_{s}-r_{1} } \right),
  \label{Ginit.eqn}
\end{equation}
with boundary conditions $g_{l}(r_{1},0)=0$
and $g_{l}(r_{s},0)=a_{l}$.
The source term in eq. (\ref{Ginit.eqn}) comes from
$j_{\phi}$, which is chosen to be localized near the
geometrical center $r \approx (r_{1}+r_{s})/2$.
The function $g_{l}(r,0)$ is smoothly connected with the multipole
solution $a_{l}/r^{l}$ at the surface $r_{s}$.

\subsection{Energy and helicity}

In previous works (e.g., \cite{2002MNRAS.337..216H,2004MNRAS.347.1273H}),
coefficients of Legendre polynomials
were utilized to study the system dynamics, and
are useful to understand, for example, energy transfer
among different wavelengths at a given time.
In order to examine the whole dynamics, it is necessary to calculate
the coefficients for many times,
because they are time-dependent.
Our concern in this paper, however, is global aspects,
so that energy $E(t)$ and helicity $H(t)$ are used
to represent the magnetic fields.
These are indicators of the field strength and twisted structure.
Integrating over the entire space, they  are given by
\begin{equation}
  E =\frac{1}{8\pi }\int (\vec{B}\cdot \vec{B})dV,
\end{equation}
\begin{equation}
  H =\int (\vec{A}\cdot \vec{B})dV,
\end{equation}
where $dV$ is the three-dimensional volume element,
and $\vec{A}$ is a vector potential for $\vec{B}$.
The energy is divided to poloidal $E_{P}$ and toroidal $E_{T}$
parts, $E=E_{P}+E_{T}$,  by which the energy transfer between them
is examined.  From eq. ({\ref{DefB.eqn}}),
the explicit forms are given by functions $G$ and $S$:
\begin{equation}
E_{P}=\frac{1}{8\pi }\int
\left( \frac{ \vec{\nabla} G }{R} \right)^2 dV,
~~~
E_{T}=\frac{1}{8\pi }\int \left( \frac{ S}{ R} \right)^2 dV,
~~~
 H =2\int \frac{GS}{R^2}dV.
\end{equation}

From eqs. (\ref{Evl.eqn}) and (\ref{DefJ.eqn}), the evolution of
magnetic energy is given by
\begin{equation}
\frac{d}{dt}E=-\int \frac{1}{\sigma } (\vec{j}\cdot \vec{j}) dV.
  \label{Edecay.eqn}
\end{equation}
This is nothing but Ohmic dissipation.
The Hall drift does not concern the energy dissipation
\citep{1992ApJ...395..250G,1994MNRAS.266..597N},
but may affect the dissipation rate
by modifying the current distribution.
Similarly, the evolution of magnetic helicity $H$ is
\begin{equation}
\frac{d}{dt}H=-\int  2c(\vec{E}\cdot \vec{B})dV=
 -\int \frac{2c}{\sigma }(\vec{j}\cdot \vec{B})dV.
 \label{Hdecay.eqn}
\end{equation}
In this way, both energy and helicity decay through Ohmic
dissipation. Note that $E$ and its dissipation term,
the right-hand side of eq. (\ref{Edecay.eqn}), have a definite sign,
but $H$ and its dissipation term, the right-hand side of
eq. (\ref{Hdecay.eqn}) do not.
The energy should monotonically decrease, but evolution of the helicity
is unknown without numerical integration.
In the numerical calculations, the energy balance equation (\ref{Edecay.eqn})
is used to check the accuracy.  The relative errors to the initial
magnetic energy are less than $10^{-2}$.
They decrease with increase of the grid number.
Relatively large numerical error is produced in the simulation with
large magnetized parameter $\mathcal{R}_{m}$.
It is also prominent in the initial evolution, in which steep
structure is formed. Functions in the late phase are rather smooth,
so that the errors do not accumulate so much, even if numerical
integration continues to a longer time.

\section{Evolution of purely toroidal fields}

This section discusses the evolution of purely toroidal magnetic fields.
The system is similar to that of Burgers' equation, as discussed in Section 2.
The initial configuration of $S$ is specified by a single
component $s_{l}(r,0)$ in eq. (\ref{Sinit.eqn}).
The three models considered here are characterized by
$l=1$ with $b_{1} >0$ in Model A,
$l=2$ with $b_{2} >0$ in Model B, and
$l=2$ with $b_{2} <0$ in Model C.
The toroidal magnetic field $B_{\phi}$ in Model C has the same
amplitude as Model B, but opposite direction.
These initial configurations are shown in
the top to bottom panels of Fig. 1.
The coefficient $b_{1}$ or $b_{2}$ is chosen for the maximum
of $|B_{\phi}|$ to be equal to 1, since the
magnetic fields in the numerical calculation are normalized by the maximum
value at the initial state.
The magnetized parameter for these models is $\mathcal{R}_{m}=100$.

  The top row shows the evolution of $l=1$ with $b_{1} >0$ (Model A).
Snapshots of the configuration are shown at the initial state (left),
at the Hall drift timescale $ \sim 6 \times 10^{-4}\tau_{d}$ (center)
and at the Ohmic decay timescale $ \sim 2 \times 10^{-3}\tau_{d}$ (right).
Note that the amplitude of $s_{1}$ decays
by $\exp(-166t/\tau_{d})$ in the absence of the Hall
term \citep{2002MNRAS.337..216H}, so the
typical decay time is not $\tau_{d}$ but
$ 6\times 10^{-3} \tau_{d}$.
The difference mainly comes from the choice of normalization length.
The stellar radius is used in this paper to compare previous works, but
more appropriate one is the crust size.
Thus, $\tau_{d}$ is rather large to characterize the actual Ohmic decay.
The maximum of $S$ at $t=0$ is located at $r\approx (r_{1}+r_{s})/2$ and
$\theta=\pi/2$, and moves in the negative $Z$-direction,
as shown in close-ups in Fig.1.
The drift stops at the outer boundary in several times the drift timescale.
It is clear that the function $S$ at this time
contains higher multipole components $s_{l}(r,t)$ in addition to
the initial $s_{1}(r,t)$
when it is expanded as in eq. (\ref{Sexpd.eqn}).
Subsequently the shape is nearly fixed, but overall amplitude
gradually decreases with a longer Ohmic decay timescale.
Some numerical calculations were performed longer until $\sim \tau_{d}$,
but the behavior after $ 6 \times 10^{ -2} \tau_{d}$ is well
described by a simple exponential decay.
The results are limited to the early phase since subsequent
evolution is easily inferred from the extrapolation.

  The middle row shows the evolution of Model B ($b_{2} >0$),
and the bottom row shows the evolution of Model C ($b_{2} <0$).
The evolution of the magnetic configuration is quite different
according to the initial sign of $b_{2}$.
The positive and negative regions of $S$
`collide' at the equatorial plane in Model B,
while they `repulse' each other in Model C.
This occurs because the drift is negative in the $Z$-direction for $S>0$,
but positive in the $Z$-direction for $S<0$.
It is also important to note that the shape predominantly moves
in not the $\theta$-direction, but the $Z$-direction
(see eq. (\ref{PureS.eqn})).
The motion in $S$ is clear in close-ups of steep structure in Fig.1.
In any model, the moved shape decays in a longer decay timescale.

These results remarkably show the nature of the Hall drift, where
a different initial sign for $B_{\phi}$ leads to a different fate.
This leads to an interesting question:
Do these different configurations lead to significantly different
dissipation rates?
If so, the direction of $B_{\phi}$ would have a significant effect.
Figure 2 shows magnetic energy time evolutions for three models. The
difference between Model B and Model C is relatively minor.
A more marked difference is shown for
the initial multipole. That is, the energies for
Model B and C decay slightly faster than that of Model A.
The current is swept to the outer boundary in Model C, but
the outer boundary is replaced by the equatorial plane in Model B.
In Fig. 2, a curve with $\exp(-332 t/\tau_{d})$,
corresponding to the energy decay in the absence of the Hall drift,
is also plotted for comparison.

  Figure 3 shows the dependence of the magnetic parameter ${\mathcal R}_{m}$.
The initial configuration is the same as that of Model B,
the model of $l=2$ with $b_{2}>0$. The same figure also shows the magnetic
energy for free decay. The damping is clearly strong for
large ${\mathcal R}_{m}$. The Ohmic decay is enhanced in the presence of
the Hall drift, that is, current is swept into a certain region, where
it is effectively dissipated.

\begin{figure}
  \includegraphics[scale=1.0]{EvlS.eps} %
\hspace{-20mm}
  \includegraphics[scale=1.0]{EvlSzoom.eps} %
\caption{ Snapshots of time evolution for the function $S$,
Model A ($b_{1} >0$) at top, Model B ($b_{2} >0$) in the middle,
and Model C ($b_{2} <0$) at bottom.
Column times are $t/\tau_{d} =0, 6 \times 10^{-4}, 2 \times 10^{-3}$
from left to right.
Whole spherical regions are shown by color contour
in the left nine panels, while their close-ups of the steep region
are shown by contour line with increment of 0.1 in the right nine panels.
The region is limited to
$0.5 \le R/r_{s} \le 1, -1 \le Z/r_{s} \le 0$ in Model A,
$0.5 \le R/r_{s} \le 1, -0.5 \le Z/r_{s} \le 0.5$ in Model B,
and  $0.5 \le R/r_{s} \le 1, -1 \le Z/r_{s} \le 0$ in Model C.
}
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.75]{cmpS1.eps}
\caption{ Normalized magnetic energy as a function of
time $t/\protect\tau_{d}$ for three models with magnetization
parameter $\mathcal{R}_{m} =100$.
Free decay curve is also plotted for comparison. }
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.75]{cmpS2.eps}
\caption{ Normalized magnetic energy as a function of time
$t/\protect\tau_{d}$ for $\mathcal{R}_{m} =20,100, 500$.
Free decay curve is also plotted for comparison. }
\end{figure}

\section{Evolution of mixed fields}

In this section, field evolution is numerically studied for a mixed
magnetic configuration consisting of poloidal and toroidal fields.
The initial configuration is given solely by the $l=1$ component for
both fields, namely, eq. (\ref{Sinit.eqn}) for $s_{1}$
and eq. (\ref{Ginit.eqn}) for $g_{1}$.
The maximum of each field is chosen as the same amplitude,
and the magnetized parameter is $\mathcal{R}_{m}=100$.
Figure 4 shows snapshots of the evolving fields
at representative times. The color contour represents
the function $S$ of the toroidal field, and lines denote
the contour of the magnetic flux function $G$ of the
poloidal field.

  Oscillatory behavior is clearly evident in $G$.
Initially, the function decreases
with the increase in cylindrical distance,
and the maximum is located on the equator $\theta=\pi/2$.
The maximum moves `upward' in the meridian plane,
toward $\theta<\pi/2$, until
$t/\tau_{d} \approx 1.4 \times 10^{-3}$ (second panel).
It then changes direction and goes `downward,' passing through
the equator at $t/\tau_{d} \approx 3.2 \times 10^{-3}$ (third panel)
and reaching a minimum
at $t/\tau_{d} \approx 5.2 \times 10^{-3}$ (fourth panel),
before returning to the initial position at
$t/\tau_{d} \approx 7.8 \times 10^{-3}$ (fifth panel).
During this cycle, the field strength decreases.

  The function $S$ is also oscillatory.
The initial configuration contains only the $l=1$ component
in the angular part ($ S \propto \sin^2 \theta $), which is
symmetric with respect to $\theta=\pi/2$.
The state at
$t/\tau_{d} \approx 1.4 \times 10^{-3}$ (second panel)
markedly differs from the initial state.
The configuration is no longer symmetric, and higher multipoles
can be seen. The field strength itself is weak around this time.
At $t/\tau_{d} \approx 3.2 \times 10^{-3}$ (third panel),
the configuration again becomes symmetric like the initial state,
but the sign of $S$ is reversed. The $l=1$ component is dominated
there. After the direction of $B_{\phi}(=S/(r\sin\theta))$ again
changes, the configuration returns to the initial one at
$t/\tau_{d} \approx 7.8 \times 10^{-3}$ (fifth panel).
The directional change occurs around
$t/\tau_{d} \approx 1.4 \times 10^{-3}$ (second panel)
and $5.2 \times 10^{-3}$ (fourth panel),
which correspond to a local minimum of toroidal field strength.
The overall toroidal field strength also decreases during
this cycle.

\begin{figure}
\centering
\includegraphics[scale=1.0]{EvlGS.eps} %
\caption{Snapshots of time evolution for functions $G$ and $S$.
Contour lines outwardly represent the level of $G$
for $0.02 \times n \times (B_{0} r_{s}^{3}), n=1,2,\cdots$.
Color contour represents $S$ normalized by
$B_{0} r_{s}$.
Note that different color scales are used,
since $S$ becomes very small at the bounces in the second
and fourth panels.
Those panels use the color scale on the left; the others, the scale on the right.
}
\end{figure}

Figure 5 clearly shows the oscillatory behavior of the
magnetic energy, which is divided into poloidal and toroidal
parts, $E_{P}$ and $E_{T}$, respectively.
The magnitude of the amplitudes at $t=0$ are nearly the same.
\footnote{
Values are not exactly the same, because the maximum amplitude of
each field is fixed at same value, the distributions are slightly different.}
The curve of toroidal energy $E_{T}$ represents a damped oscillation.
Local minima can be seen at $t/\tau_{d} \approx 1.4 \times 10^{-3}$
and $5.2 \times 10^{-3}$,
corresponding to the time of the second and fourth panels in Fig. 4.
The configurations of the third and fifth panels in Fig. 4 are those of
the local maxima at $t/\tau_{d} \approx 3.2 \times 10^{-3}$
and $7.8 \times 10^{-3}$.
There is remarkable energy transfer between the toroidal and poloidal parts
during the initial phase. Initially $E_{P}$ increases
until $t/\tau_{d} \approx 1.4 \times 10^{-3}$, although
the total energy $E_{sum}=E_{P}+E_{T}$ decreases.
The sum always decreases due to Ohmic decay
(see eq. (\ref{Edecay.eqn})).
The initial rapid decay of $E_{T}$ is thus partially due to this
transfer. Energy is subsequently transferred between the two components
in turn, but the behavior becomes less clear.
The magnetic field decays on the timescale
$t/\tau_{d} \approx 10^{-2}$, so the coupling becomes weak.
The oscillation period gradually becomes longer,
since the drift timescale increases.
Figure 5 also shows the evolution of magnetic helicity $H$,
which exhibits oscillatory damping with Ohmic decay timescale
$t/\tau_{d} \approx 10^{-2}$.
The change of sign in $H$ denotes an inversion of the toroidal
field $B_{\phi }$, which occurs around the local minima of $E_{T}$.

The coefficients $a_{l}$
of the multi-moments in eq. (\ref{Vac.eqn}) describe the exterior poloidal field.
The right panel of Fig. 5 shows the evolution of a few of the lowest values.
Higher multipoles are induced until
$t/\tau_{d} \approx 1.4 \times 10^{-3}$,
the configuration shown in the second panel of Fig. 4.
The poloidal field is no longer dipole at this time.
Interestingly, the coefficient $a_{1}$ is significantly decreased,
as compared with the initial value,
although a large amount of energy is stored in the poloidal part,
as shown in the left panel of Fig. 5.
Dipole field strength at this time is not a good indicator of
overall magnetic energy.
The poloidal field returns back to the dipole at
$t/\tau_{d} \approx 3.2 \times 10^{-3}$
(the time of the third panel in Fig. 4),
at which higher multipoles are temporarily zero.
After sinusoidal oscillation, higher multipoles decay rather rapidly,
so the polar dipole remains for the later time
$t/\tau_{d} > 10^{-2}$.
The behavior after $10^{ -2}\tau_{d}$ is rather simple.
Magnetic energy is significantly dissipated, so that the Hall drift
becomes less important. Subsequent evolution in the late phase is described
by a free decay due to Ohmic dissipation, and is omitted here.

\begin{figure}
\centering
\includegraphics[scale=1.0]{cmpGSax.eps}
\caption{ Time evolution of energy $E$, helicity $H$ (left panel),
and multipole moments $a_{l}$ at the surface (right panel).
Energy is normalized by the initial total energy, and helicity
by the initial value. Coefficient $a_{l}$ of the $l$-th moment
is normalized by the initial dipole value $a_{1}r_{s}^{l-1} $.}
\end{figure}

  Figures 6--9 compare evolutions of energy, helicity, and polar multipole
coefficients for four models.
The initial magnetic configuration is the same as that of Figs. 4 and 5,
but the strength and magnetized parameter $\mathcal{R}_{m}$
vary. The ratio of poloidal to toroidal fields is $1/4,$ or $4$ in amplitude,
so $16(=4^{2})$ or $1/16$ in energy,
and $\mathcal{R}_{m}$ is $\mathcal{R}_{m}=20$, or $100$.
The right panel shows energy plotted on a logarithmic scale, $\log_{10}(E)$,
and the left helicity $H$ and a few lowest multipole coefficients $a_{l}$.
Figure 6 shows the results of a model with $\mathcal{R}_{m}=20$,
in which the energy is initially dominated by the poloidal part.
Polar multipole components are induced through the coupling to
the toroidal part, but the dipole field is barely affected.
The decay curve is well described by $\exp(-55 t/\tau_{d})$,
which coincides with free decay of the dipole.
The energy is always dominated by the polar dipole,
and the evolution is described by $\exp(-110 t/\tau_{d})$,
a line in the logarithmic scale. The curve of toroidal energy is
nonlinear on the logarithmic scale, but is oscillatory.
The minor component is highly affected by the poloidal one, but
is almost neglected in the dynamics of the whole system.
No consequence of the toroidal field comes from the
fast decay rate $\exp(-166 t/\tau_{d})$ in the amplitude
as seen in Section 4, or from the initial small strength.

  Figure 7 shows the results of an evolution in which the toroidal
component is dominated at the initial state. Compared with the decay
curve of total energy in Fig. 6, the damping is much faster.
Most of the magnetic energy is initially stored in the toroidal part,
but rapidly decays. The poloidal component decays
rather slowly, and dominates at the later time. The higher
multipole moments are induced
and have larger amplitudes than those of Fig. 6.
The overall dipole decay curve is nonetheless very similar to that
of the large poloidal case.

  Figure 8 shows the results for large a magnetization
parameter $\mathcal{R}_{m}=100$ and an initially large poloidal field.
The evolution of the total energy is very similar to that of Fig. 6,
although the oscillatory behavior in the toroidal energy is evident.
The timescale determined by the Hall drift is approximately 1/5
that of Fig. 6, in which $\mathcal{R}_{m}=20$.
The oscillatory behavior is also clear in the coefficients $a_{l}$
and helicity $H$. However, except for the initial wavy
structure seen in the Hall drift timescale,
the decay of the dipole field is similar to that of Fig. 6.
The dominant component, the polar dipole, is thus not affected by
the toroidal field,
which never plays an important role because of its rapid decay.

  Figure 9 shows the results for a large magnetization
parameter $\mathcal{R}_{m}=100$ and an initially large toroidal field.
In this case, a large amount of energy is transferred from the
toroidal to poloidal components.
Higher multipoles in the poloidal field are induced
around $ t/\tau_{d}\approx 10^{-3}$.
The configuration of the poloidal field at this time is significantly
distorted as in the second panel in Fig. 4.
The toroidal energy becomes much smaller than the poloidal energy
around $ t/\tau_{d}\approx 10^{-2}$, and
both components are subsequently decoupled. After that, the dipole
field evolution is determined by free decay.
Despite the disorder at $ t/\tau_{d}\approx 10^{-3}$,
the amplitude of the free decay phase, for example
at $ t/\tau_{d} = 2 \times 10^{-2}$,
does not differ so much from that of Figs. 6--8.

\begin{figure}
\centering
 \includegraphics[scale=0.75]{Lpd1.eps}
\caption{ Time evolution of energy in logarithmic scale $\log_{10}(E)$
(left panel) and helicity $H$ and multipole moments $a_{l}$
at the surface (right panel) for an initially large poloidal field
with ${\mathcal R}_m=20$.
These normalizations are the same as in Fig. 5.}
\end{figure}
\begin{figure}
\centering
 \includegraphics[scale=0.75]{Ltd1.eps}
\caption{ Same as Fig. 6, but for an initially large toroidal field
with ${\mathcal R}_m=20$. }
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.75]{Hpd1.eps}
\caption{ Same as Fig. 6, but for an initially large poloidal field
with ${\mathcal R}_m=100$. }
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.75]{Htd1.eps}
\caption{ Same as Fig. 6, but for an initially large toroidal field
with ${\mathcal R}_m=100$. }
\end{figure}

\section{Summary and Discussion}

  Numerical simulations demonstrate how the Hall drift
changes the current and magnetic configuration.
In a purely toroidal evolution, Ohmic dissipation is enhanced
by accumulated currents elsewhere.
The spatial location depends on the initial data,
but the energy dissipation rate does not so significantly depend
on the accumulation position in our uniform conductivity model.
In a realistic case the conductivity will decrease with the radius,
so most of the magnetic energy may be effectively dissipated
near the surface. The results become more sensitive to the initial
condition.
Energy is, in principle, transferred between poloidal and toroidal
components if both are initially involved.
It is important for understanding the evolution
that the free decay rate of the polar dipole is the least.
When the polar dipole dominates, it is rarely affected by
the toroidal field, which rapidly decays.
On the other hand, a significant amount of energy is transferred
to the poloidal field until almost equipartition
when the toroidal component is initially dominant.
Global nonlinear coupling is manifest in the Hall drift timescale
only when the corresponding energies are of the same order.
Moreover, the poloidal field at the surface and the exterior
is highly distorted, and is no longer described by a pure dipole
field. During this phase, the dipole is not
a good indicator of overall field strength.
In a longer Ohmic dissipation timescale the toroidal field
decays rapidly, so the coupling vanishes.
The polar dipole eventually survives.

  The early evolution highly depends on the choice of the
initial data, i.e, the configuration and strength.
As discussed by \cite{2007A&A...472..233R},
stationary conditions in the presence of Hall drift
for an axisymmetric magnetic field means that
the isosurface of $S$ coincides with that of $G$; in other words, $S=S(G)$,
for which the coupling term ${\vec \nabla G}\times {\vec \nabla S}$
vanishes.
This condition is not satisfied in the whole shell region.
The outer boundary condition requires that the toroidal component is
concentrated in the interior ($S=0$), whereas
the poloidal one may leak out to the exterior, meaning
that $G \ne 0$ at the surface.
The interior poloidal field is described by the value at the surface
if the topology is simply connected.
The condition $S(G)$ therefore means that $S=0$ everywhere,
irrespective of $G$.
Our numerical calculation starts with an initial
configuration different from the `Hall equilibrium',
and shows the behavior to the state $S=0$.
It may be necessary as a next step to study plausible initial
configurations, since little known about them.

  The timescale is less accurate in our simplified model
with uniform density and conductivity.
Realistic models are necessary, but the distributions depend on
equation of state and neutron star mass.
Assuming that our uniform model is obtained as a result of
spatial average of a certain model with stratified number density
and conductivity, the timescale is estimated.
The overall normalization constant $\tau _{d} $ is given by
$\tau _{d}= 4\pi \sigma r_{s}^{2}/c^{2}$
$\approx 4.4\times 10^{9}$
$( \bar{\sigma} /10^{25} \mathrm{s}^{-1})$
$(r_{s}/10 \mathrm{km}) ^{2}$ years, where
$\bar{\sigma} $ is an averaged value of the conductivity.
The decay timescale of the dipole is
$2\times10^{-2}\tau _{d}\approx 9\times 10^{7}$ years,
and that of the toroidal one is
$6\times10^{-3}\tau _{d}\approx 3\times 10^{7}$ years.
These numbers are slightly larger, since the crust size
in our present model is thick, $L= r_{s}/4$.
The timescale is proportional to $L^2$, so that
the actual values may be smaller by a factor of
$\sim 10^{-2}$--$10^{-1}$.
A typical decay timescale is around $10^{5}$--$10^{7}$ years.

  The characteristic age of the low-field magnetar SGR0418+5729
is more than $ 2.4\times 10^{7} $ years \citep{2010Sci...330..944R}.
The dipole field ($<7.5\times 10^{12} $G) may decay or survive
within this period. However, the internal toroidal field is
likely to dissipate more quickly.
It is therefore difficult to understand why the toroidal field has
a much larger field strength, $>10^{13}$-$10^{14}$G,
which is required for the activity.
One possible explanation is that current age estimates are inaccurate.
The characteristic age is normally estimated by
assuming a constant dipole magnetic field.
The Hall drift significantly affects the surface value,
especially at the initial epoch with strong field.
As demonstrated in the second panel of Fig. 4, the surface field is
highly distorted from the pure dipole.
SGR0418+5729 may correspond to a young phase of
oscillatory evolution
in which the surface dipole temporarily decreases, but
there is a strong internal toroidal component.

\section*{Acknowledgements}
This work was supported in part by a Grant-in-Aid for Scientific Research
(No.21540271) from the Japanese Ministry of Education, Culture, Sports,
Science and Technology(Y.K.), and from the Japan Society for Promotion
of Science (S.K.).


\section{Introduction}

The first-order Duffin-Kemmer-Petiau (DKP) formalism \cite{pet}-\cite{kem}
describes spin-0 and spin-1 particles and has been used to analyze
relativistic interactions of spin-0 and spin-1 hadrons with nuclei as an
alternative to their conventional second-order Klein-Gordon and Proca
counterparts. The DKP formalism enjoys a richness of couplings not capable
of being expressed in the Klein-Gordon and Proca theories \cite{gue}-\cit
{vij}. Although the formalisms are equivalent in the case of minimally
coupled vector interactions \cite{mr}-\cite{lun}, the DKP formalism opens
news horizons as far as it allows other kinds of couplings which are not
possible in the Klein-Gordon and Proca theories. Nonminimal vector
potentials, added by other kinds of Lorentz structures, have already been
used successfully in a phenomenological context for describing the
scattering of mesons by nuclei \cite{cla1}-\cite{cla2}. Nonminimal vector
coupling with a quadratic potential \cite{Ait}, with a linear potential \cit
{kuli}, and mixed space and time components with a step potential \cite{ccc3
-\cite{ccc2} and a linear potential \cite{jpa}-\cite{cascas} have been
explored in the literature. See also Ref. \cite{jpa} for a comprehensive
list of references on other sorts of couplings and functional forms for the
potential functions. In Refs. \cite{jpa}-\cite{cascas} it was shown that
when the space component of the coupling is stronger than its time component
the linear potential, a sort of vector DKP oscillator, can be used as a
model for confining bosons.

The Schr\"{o}dinger equation with a quadratic plus inversely quadratic
potential, known as singular oscillator, is an exactly solvable problem \cit
{lan}-\cite{pr} which works for constructing solvable models of $N$
interacting bodies \cite{cal1}-\cite{cal2} as well as a basis for
perturbative expansions and variational analyses for spiked harmonic
oscillators \cite{hal1}-\cite{hal7}. Generalizations for finite-difference
relativistic quantum mechanics \cite{nag} as well as for time-dependent
parameters in the nonrelativistic version have also been considered \cit
{cam}-\cite{dod}.

The main purpose of the present article is to report on the properties of
the DKP theory with nonminimal vector linear plus inversely linear
potentials for spin-0 and spin-1 bosons in a unified way. The problem is
mapped into an exactly solvable Sturm-Liouville problem of a Schr\"{o
dinger-like equation. The effective potential resulting from the mapping has
the form of the singular oscillator potential. The Schr\"{o}dinger equation
with quadratic plus inversely quadratic potential is indeed an exactly
solvable problem. Nevertheless, only positive coefficients are involved in
the well-known solution. Because we need solutions involving a repulsive as
well as an attractive inverse-square term in the effective potential, the
calculation including this generalization with proper boundary conditions at
the origin is presented. The closed form solution for the bound states is
uniquely determined with boundary conditions which emerge as a direct
consequence of the equation of motion and the normalization condition, and
do not have to be imposed. It is shown that the spectrum exhibits
degeneracy. It is also shown that the existence of bound-state solutions for
vector bosons depends on a too restrictive condition on the potential
parameters.

It should be mentioned that a somewhat less general sort of problem, only
with the space component of a nonminimal vector potential (erroneously
called pseudoscalar potential), has already appeared in the literature \cit
{cha}.

\section{Nonminimal vector couplings in the DKP equation}

The DKP equation for a free boson is given by \cite{kem} (with units in
which $\hbar =c=1$
\begin{equation}
\left( i\beta ^{\mu }\partial _{\mu }-m\right) \psi =0  \label{dkp}
\end{equation
\noindent where the matrices $\beta ^{\mu }$\ satisfy the algebra $\beta
^{\mu }\beta ^{\nu }\beta ^{\lambda }+\beta ^{\lambda }\beta ^{\nu }\beta
^{\mu }=g^{\mu \nu }\beta ^{\lambda }+g^{\lambda \nu }\beta ^{\mu }$
\noindent and the metric tensor is $g^{\mu \nu }=\,$diag$\,(1,-1,-1,-1)$.
That algebra generates a set of 126 independent matrices whose irreducible
representations are a trivial representation, a five-dimensional
representation describing the spin-0 particles and a ten-dimensional
representation associated to spin-1 particles. The second-order Klein-Gordon
and Proca equations are obtained when one selects the spin-0 and spin-1
sectors of the DKP theory. A well-known conserved four-current is given by
J^{\mu }=\bar{\psi}\beta ^{\mu }\psi $\noindent $/2$ where the adjoint
spinor $\bar{\psi}$ is given by $\bar{\psi}=\psi ^{\dagger }\eta ^{0}$ with
\eta ^{0}=2\beta ^{0}\beta ^{0}-1$. The time component of this current is
not positive definite but it may be interpreted as a charge density. Then
the normalization condition $\int d\tau \,J^{0}=\pm 1$ can be expressed a
\begin{equation}
\int d\tau \,\bar{\psi}\beta ^{0}\psi =\pm 2  \label{norm}
\end{equation
where the plus (minus) sign must be used for a positive (negative) charge.

With the introduction of nonminimal vector interactions, the DKP equation
can be written a
\begin{equation}
\left( i\beta ^{\mu }\partial _{\mu }-m-i[P,\beta ^{\mu }]A_{\mu }\right)
\psi =0  \label{dkp2}
\end{equation
where $P$ is a projection operator ($P^{2}=P$ and $P^{\dagger }=P$) in such
a way that $\bar{\psi}[P,\beta ^{\mu }]\psi $ behaves like a vector under a
Lorentz transformation as does $\bar{\psi}\beta ^{\mu }\psi $. Once again
\partial _{\mu }J^{\mu }=0$ \cite{jpa}. If the potential is time-independent
one can write $\psi (\vec{r},t)=\phi (\vec{r})\exp (-iEt)$, where $E$ is the
energy of the boson, in such a way that the time-independent DKP equation
become
\begin{equation}
\left[ \beta ^{0}E+i\beta ^{i}\partial _{i}-\left( m+i[P,\beta ^{\mu
}]A_{\mu }\right) \right] \phi =0  \label{DKP10}
\end{equation
In this case \ $J^{\mu }=\bar{\phi}\beta ^{\mu }\phi /2$ does not depend on
time, so that the spinor $\phi $ describes a stationary state.

The DKP equation is invariant under the parity operation, i.e. when
\overrightarrow{r}\rightarrow -\overrightarrow{r}$, if $\overrightarrow{A}$
changes sign, whereas $A_{0}$ remains the same. This is because the parity
operator is $\mathcal{P}=\exp (i\delta _{P})P_{0}\eta ^{0}$, where $\delta
_{P}$ is a constant phase and $P_{0}$ changes $\overrightarrow{r}$ into $
\overrightarrow{r}$. Because this unitary operator anticommutes with $\beta
^{i}$ and $[P,\beta ^{i}]$, they change sign under a parity transformation,
whereas $\beta ^{0}$ and $[P,\beta ^{0}]$, which commute with $\eta ^{0}$,
remain the same. Since $\delta _{P}=0$ or $\delta _{P}=\pi $, the spinor
components have definite parities. The charge-conjugation operation can be
accomplished by the transformation $\psi \rightarrow \psi _{c}=\mathcal{C
\psi =CK\psi $, where $K$ denotes the complex conjugation and $C$ is a
unitary matrix such that $C\beta ^{\mu }=-\beta ^{\mu }C$. Meanwhile $C$
anticommutes with $[P,\beta ^{\mu }]$. The matrix that satisfies these
relations is $C=\exp \left( i\delta _{C}\right) \eta ^{0}\eta ^{1}$. The
phase factor $\exp \left( i\delta _{C}\right) $ is equal to $\pm 1$, thus
E\rightarrow -E$. Note also that $J^{\mu }\rightarrow -J^{\mu }$, as should
be expected for a charge current and the charge-conjugation operation
entails no change on $A_{\mu }$. The invariance of the nonminimal vector
potential under charge conjugation means that it does not couple to the
charge of the boson. In other words, $A_{\mu }$ does not distinguish
particles from antiparticles. Hence, whether one considers spin-0 or spin-1
bosons, this sort of interaction can not exhibit Klein's paradox.

For the case of spin 0, we use the representation for the $\beta ^{\mu }$\
matrices given by \cite{ned1
\begin{equation}
\beta ^{0}
\begin{pmatrix}
\theta & \overline{0} \\
\overline{0}^{T} & \mathbf{0
\end{pmatrix
,\quad \beta ^{i}
\begin{pmatrix}
\widetilde{0} & \rho _{i} \\
-\rho _{i}^{T} & \mathbf{0
\end{pmatrix
,\quad i=1,2,3  \label{rep}
\end{equation
\noindent wher
\begin{eqnarray}
\ \theta &=
\begin{pmatrix}
0 & 1 \\
1 &
\end{pmatrix
,\quad \rho _{1}
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 0 &
\end{pmatrix}
\notag \\
&&  \label{rep2} \\
\rho _{2} &=
\begin{pmatrix}
0 & -1 & 0 \\
0 & 0 &
\end{pmatrix
,\quad \rho _{3}
\begin{pmatrix}
0 & 0 & -1 \\
0 & 0 &
\end{pmatrix}
\notag
\end{eqnarray
\noindent $\overline{0}$, $\widetilde{0}$ and $\mathbf{0}$ are 2$\times $3,
$\times $2 \ and 3$\times $3 zero matrices, respectively, while the
superscript T designates matrix transposition. Here the projection operator
can be written as \cite{gue} $P=\left( \beta ^{\mu }\beta _{\mu }-1\right)
/3=\mathrm{diag}\,(1,0,0,0,0)$. In this case $P$ picks out the first
component of the DKP spinor. The five-component spinor can be written as
\psi ^{T}=\left( \psi _{1},...,\psi _{5}\right) $ in such a way that the
time-independent DKP equation for a boson constrained to move along the $X
-axis, restricting ourselves to potentials depending only on $x$, decomposes
int
\begin{equation*}
\left( \frac{d^{2}}{dx^{2}}+k^{2}\right) \phi _{1}=0
\end{equation*
\begin{equation}
\phi _{2}=\frac{1}{m}\left( E+iA_{0}\right) \,\phi _{1}  \label{dkp4}
\end{equation
\begin{equation*}
\phi _{3}=\frac{i}{m}\left( \frac{d}{dx}+A_{1}\right) \phi _{1},\quad \phi
_{4}=\phi _{5}=0
\end{equation*
wher
\begin{equation}
k^{2}=E^{2}-m^{2}+A_{0}^{2}-A_{1}^{2}+\frac{dA_{1}}{dx}  \label{k}
\end{equation
Meanwhile,
\begin{equation}
J^{0}=\frac{E}{m}\,|\phi _{1}|^{2},\quad J^{1}=\frac{1}{m}\text{Im}\left(
\phi _{1}^{\ast }\,\frac{d\phi _{1}}{dx}\right)  \label{corrente4}
\end{equation}

For the case of spin 1, the $\beta ^{\mu }$\ matrices are \cite{ned2
\begin{equation}
\beta ^{0}
\begin{pmatrix}
0 & \overline{0} & \overline{0} & \overline{0} \\
\overline{0}^{T} & \mathbf{0} & \mathbf{I} & \mathbf{0} \\
\overline{0}^{T} & \mathbf{I} & \mathbf{0} & \mathbf{0} \\
\overline{0}^{T} & \mathbf{0} & \mathbf{0} & \mathbf{0
\end{pmatrix
,\quad \beta ^{i}
\begin{pmatrix}
0 & \overline{0} & e_{i} & \overline{0} \\
\overline{0}^{T} & \mathbf{0} & \mathbf{0} & -is_{i} \\
-e_{i}^{T} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\overline{0}^{T} & -is_{i} & \mathbf{0} & \mathbf{0
\end{pmatrix}
\label{betaspin1}
\end{equation
\noindent where $s_{i}$ are the 3$\times $3 spin-1 matrices $\left(
s_{i}\right) _{jk}=-i\varepsilon _{ijk}$, $e_{i}$ are the 1$\times $3
matrices $\left( e_{i}\right) _{1j}=\delta _{ij}$ and $\overline{0}
\begin{pmatrix}
0 & 0 &
\end{pmatrix
$, while\textbf{\ }$\mathbf{I}$ and $\mathbf{0}$\textbf{\ }designate the 3
\times $3 unit and zero matrices, respectively. In this representation
P=\,\beta ^{\mu }\beta _{\mu }-2=\mathrm{diag}\,(1,1,1,1,0,0,0,0,0,0)$, i.e.
$P$ projects out the four upper components of the DKP spinor. \noindent With
the spinor written as $\psi ^{T}=\left( \psi _{1},...,\psi _{10}\right) $,
and partitioned a
\begin{equation*}
\psi _{8}=0
\end{equation*
\begin{equation*}
\psi _{I}^{\left( +\right) }=\left(
\begin{array}{c}
\psi _{3} \\
\psi _{4
\end{array
\right) ,\quad \psi _{I}^{\left( -\right) }=\psi _{5}
\end{equation*
\begin{equation}
\psi _{II}^{\left( +\right) }=\left(
\begin{array}{c}
\psi _{6} \\
\psi _{7
\end{array
\right) ,\quad \psi _{II}^{\left( -\right) }=\psi _{2}  \label{part}
\end{equation
\begin{equation*}
\psi _{III}^{\left( +\right) }=\left(
\begin{array}{c}
\psi _{10} \\
-\psi _{9
\end{array
\right) ,\quad \psi _{III}^{\left( -\right) }=\psi _{1}
\end{equation*
the one-dimensional time-independent DKP equation can be expressed as
\begin{equation*}
\left( \frac{d^{2}}{dx^{2}}+k_{\sigma }^{2}\right) \phi _{I}^{\left( \sigma
\right) }=0
\end{equation*
\begin{equation}
\phi _{II}^{\left( \sigma \right) }=\frac{1}{m}\left( E+i\sigma A_{0}\right)
\,\phi _{I}^{\left( \sigma \right) }  \label{spin1-ti}
\end{equation
\begin{equation*}
\phi _{III}^{\left( \sigma \right) }=\frac{i}{m}\left( \frac{d}{dx}+\sigma
A_{1}\right) \phi _{I}^{\left( \sigma \right) },\quad \phi _{8}=0
\end{equation*
where $\sigma $ is equal to $+$ or $-$, an
\begin{equation}
k_{\sigma }^{2}=E^{2}-m^{2}+A_{0}^{2}-A_{1}^{2}+\sigma \frac{dA_{1}}{dx}
\end{equation
Now the components of the four-current ar
\begin{equation}
J^{0}=\frac{E}{m}\sum\limits_{\sigma }|\phi _{I}^{\left( \sigma \right)
}|^{2},\quad J^{1}=\frac{1}{m}\text{Im}\sum\limits_{\sigma }\phi
_{I}^{\left( \sigma \right) \dagger }\,\frac{d\phi _{I}^{\left( \sigma
\right) }}{dx}  \label{CUR2}
\end{equation
\qquad

It is of interest to note that $\phi _{I}^{\left( +\right) }$ (in the vector
sector) obeys the same equation obeyed by the first component of the DKP
spinor in the scalar sector and, taking account of (9) and (14), $\phi _{1}$
and $\phi _{I}^{\left( \sigma \right) }$ are square-integrable functions.
Given that the interaction potentials satisfy certain conditions, we have a
well-defined Sturm-Liouville problem and hence a natural and definite method
for determining the possible discrete or continuous eigenvalues of the
system. We also note that there is only one independent component of the DKP
spinor for the spin-0 sector instead of the three required for the spin-1
sector, and that the presence of a space component of the potential might
compromise the existence of solutions for spin-1 bosons when compared to the
solutions for spin-0 bosons with the very same potentials ($A_{0}$ and $A_{1}
$). This might happen because the solution for this class of problem
consists in searching for bounded solutions for two Schr\"{o}dinger
equations. It should not be forgotten, though, that the equations for $\phi
_{I}^{\left( +\right) }$\ and $\phi _{I}^{\left( -\right) }$ are not
independent because the energy of the boson, $E$, appears in both equations.
Therefore, one has to search for bound-state solutions for $\phi
_{I}^{\left( +\right) }$\ and $\phi _{I}^{\left( -\right) }$ with a common
energy. This amounts to say that the solutions for the spin-1 sector of the
DKP theory can be obtained from a restricted class of solutions of the
spin-0 sector.

\section{The linear plus inversely linear potential}

Now we are in a position to use the DKP equation with specific forms for
vector interactions. Let us focus our attention on potentials in the linear
plus inversely linear form, viz.
\begin{equation}
A_{0}=m^{2}\omega _{0}|x|+\frac{g_{0}}{|x|},\quad A_{1}=m^{2}\omega _{1}x
\frac{g_{1}}{x}  \label{sc1a}
\end{equation
where the coupling constants, $\omega _{0},\omega _{1},g_{0}$ and $g_{1}$,
are real dimensionless parameters. Our problem is to solve (\ref{dkp4}) and
\ref{spin1-ti}) for $\phi _{1}$ (in the scalar sector) and $\phi
_{I}^{\left( \sigma \right) }$ (in the vector sector), and to determine the
allowed energies. Although the absolute value of $x$ in $A_{0}$ is
irrelevant in the effective equations for $\phi _{1}$ and $\phi _{I}^{\left(
\sigma \right) }$, it is there for ensuring the covariance of the DKP theory
under parity. It follows that the DKP spinor will have a definite parity and
$A^{\mu }$ will be a genuine four-vector. In this case the first equations
of (\ref{dkp4}) and (\ref{spin1-ti}) transmutes int
\begin{equation}
H_{\sigma }\,\Phi _{\sigma }=\varepsilon _{\sigma }\Phi _{\sigma }
\label{sc1b}
\end{equation
where $\Phi _{\sigma }$ is equal to $\phi _{1}$ for the scalar sector, and
to $\phi _{I}^{\left( \sigma \right) }$ for the vector sector, with

\begin{equation}
H_{\sigma }=-\frac{1}{2m}\frac{d^{2}}{dx^{2}}+V_{\sigma }\,  \label{Heff}
\end{equation
\begin{equation}
\varepsilon _{\sigma }=\frac{E^{2}-m^{2}+m^{2}\left[ \sigma \omega
_{1}+2\left( \omega _{0}g_{0}-\omega _{1}g_{1}\right) \right] }{2m}
\label{sc2}
\end{equation
an
\begin{equation}
V_{\sigma }=\frac{1}{2}m\Omega ^{2}x^{2}+\frac{\alpha _{\sigma }}{x^{2}}
\label{effpot}
\end{equation
with
\begin{equation}
\Omega ^{2}=m^{2}\left( \omega _{1}^{2}-\omega _{0}^{2}\right) ,\quad \alpha
_{\sigma }=\frac{g_{1}\left( g_{1}+\sigma \right) -g_{0}^{2}}{2m}
\label{sc3}
\end{equation}

\noindent The set (\ref{sc1b})-(\ref{sc3}), with $\alpha _{\sigma }=0$ (
\alpha _{\sigma }\neq 0$) and $\Omega ^{2}>0$, is precisely the Schr\"{o
dinger equation for the nonrelativistic nonsingular (singular) harmonic
oscillator. For $\alpha _{\sigma }<0$ and $\Omega ^{2}=0$ the effective
potential has also a form that would make allowance for bound-state
solutions with $\varepsilon _{\sigma }<0$. All the remaining possibilities
for $\alpha _{\sigma }$ and $\Omega ^{2}$ make the effective potential
either everywhere repulsive or there appears an attractive region about $x=0$
between potential barriers ($\alpha _{\sigma }<0$ and $\Omega ^{2}<0$) which
lead to boson tunneling. It is well known that the singularity at $x=0$ for
\alpha _{\sigma }<0$ menaces the boson to collapse to the center \cite{lan}
so that the condition $\alpha _{\sigma }\geq \alpha _{\mathtt{crit
}=-1/\left( 8m\right) $ (with $\Omega ^{2}>0$) is required for the formation
of bound-state solutions. We shall limit ourselves to study the bound-state
solutions.

Since $V_{\sigma }$ is invariant under reflection through the origin (
x\rightarrow -x$), eigenfunctions with well-defined parities can be built.
Thus, one can concentrate attention on the positive half-line and impose
boundary conditions on $\Phi _{\sigma }$ at $x=0$ and $x=\infty $.
Continuous eigenfunctions on the whole line with well-defined parities can
be constructed by taking symmetric and antisymmetric linear combinations of
\Phi _{\sigma }$. These new eigenfunctions possess the same eigenenergy,
then, in principle, there is a two-fold degeneracy. As $x\rightarrow 0$, the
solution behaves as $Cx^{s_{\sigma }}$, where $C\neq 0$ is a constant and
s_{\sigma }$ is a solution of the algebraic equation

\begin{equation}
s_{\sigma }(s_{\sigma }-1)-2m\alpha _{\sigma }=0  \label{17}
\end{equation
viz.

\begin{equation}
s_{\sigma }=\frac{1}{2}\left( 1\pm \sqrt{1+8m\alpha _{\sigma }}\right)
\label{18}
\end{equation}

\noindent where $s_{\sigma }$ is not necessarily a real quantity.
Nevertheless, $J^{1}$ for a stationary state, as expressed by (\re
{corrente4}) and (\ref{CUR2}), is the same at all points of the $X$-axis and
vanishes for a bound-state solution (because $\Phi _{\sigma }$ vanishes as
x\rightarrow \infty $), so we demand that $s_{\sigma }\in
\mathbb{R}
$ and so $\alpha _{\sigma }\geq \alpha _{\mathtt{crit}}$. Normalizability of
$\Phi _{\sigma }$ also requires $s_{\sigma }>-1/2$ and due to the two-fold
possibility of values of $s_{\sigma }$ for $\alpha _{\mathtt{crit}}<\alpha
_{\sigma }<3/(8m)$, it seems that the solution of our problem can not be
uniquely determined. This ambiguity can be overcome by a regularization of
the potential. Following the steps of Ref. \cite{lan}, we replace $V_{\sigma
}(x)$ by $V_{\sigma }(x_{0})$ \ for $x<x_{0}\approx 0$ and after using the
continuity conditions for $\Phi _{\sigma }$ and $d\Phi _{\sigma }/dx$ in the
cutoff \ we take the limit $x_{0}\rightarrow 0$. It turns out that the
solution with the lesser value of $s_{\sigma }$ is suppressed relative to
that one involving the greater value as $x_{0}\rightarrow 0$. Thus, the
minus sign in (\ref{18}) must be ruled out for $\alpha _{\sigma }\neq 0$ in
such a way that $\Phi _{\sigma }(0)=0$. Hence, $s_{\sigma }=0$ or $s_{\sigma
}\geq 1/2$. The homogeneous Dirichlet boundary condition ($\Phi _{\sigma
}\left( 0\right) =0$) is essential whenever $\alpha _{\sigma }\neq 0$,
nevertheless it also develops for $\alpha _{\sigma }=0$ when $s_{\sigma }=1$
but not for $s_{\sigma }=0$. The continuity of $\Phi _{\sigma }$ at the
origin excludes the possibility of an odd-parity eigenfunction for $s=0$,
and effects on the even-parity eigenfunction for $s=1$ are due to the
continuity of its first derivative. Effects due to the potential on $d\Phi
_{\sigma }/dx$ in the neighbourhood of the origin can be evaluated by
integrating (\ref{sc1b}) from $-\delta $ to $+\delta $ and taking the limit
\delta \rightarrow 0$. The connection formula between $d\Phi _{\sigma }/dx$
at the right and $d\Phi _{\sigma }/dx$ at the left can be summarized a
\begin{equation}
\lim_{\delta \rightarrow 0}\left. \frac{d\Phi _{\sigma }}{dx}\right\vert
_{x=-\delta }^{x=+\delta }=2m\alpha _{\sigma }\lim_{\delta \longrightarrow
0}\int_{-\delta }^{+\delta }dx\;\frac{\Phi _{\sigma }}{x^{2}}
\end{equation
so tha
\begin{equation}
\lim_{\delta \rightarrow 0}\left. \frac{d\Phi _{\sigma }}{dx}\right\vert
_{x=-\delta }^{x=+\delta }=\left\{
\begin{array}{c}
2mC\alpha _{\sigma }\left. \frac{|x|^{s_{\sigma }-1}}{s_{\sigma }-1
\right\vert _{x=-\delta }^{x=+\delta }\text{, for }\Phi _{\sigma }\text{
symmetric with }\alpha _{\sigma }\neq 0 \\
\\
0\text{, otherwise
\end{array
\right.
\end{equation
Therefore, the eigenfunctions have a first derivative continuous at the
origin, except for $s_{\sigma }<1$ when the eigenfunction is not
differentiable at the origin. Thus, the bound-state solutions for the
singular oscillator potential are two-fold degenerate: for a same energy
there is an odd-parity solution with a continuous first derivative, and an
even-parity solution with a continuous first derivative if $s_{\sigma }>1$
or a discontinuous first derivative if $s_{\sigma }<1$. There is no
odd-parity solution for $s_{\sigma }=0$ and there develops the homogeneous
Neumann condition ($\left. d\Phi _{\sigma }/dx\right\vert _{x=0}=0$), and
there is no even-parity solution for $s_{\sigma }=1$. Thus, the bound-state
solutions for the nonsingular oscillator potential are nondegenerate.

We shall now consider separately two different possibilities for $\Omega $,
namely $\Omega =0$ and $\Omega \neq 0$.

\subsection{$\Omega =0$}

Definin
\begin{equation}
z_{\sigma }=2\sqrt{-2m\varepsilon _{\sigma }}\,x  \label{z}
\end{equation
\noindent for $x>0$, and using the set (\ref{sc1b})-(\ref{sc3}), one obtains
a special case of Whittake
\'{
s differential equation \cite{abr
\begin{equation}
\frac{d^{2}\Phi _{\sigma }}{dz_{\sigma }^{2}}+\left( -\frac{1}{4}-\frac
2m\alpha _{\sigma }}{z_{\sigma }^{2}}\right) \Phi _{\sigma }=0
\end{equation
The normalizable asymptotic form of the solution as $z_{\sigma }\rightarrow
\infty $ is $e^{-z_{\sigma }/2}$ with $z_{\sigma }>0$. Notice that this
asymptotic behaviour rules out the possibility $\varepsilon _{\sigma }>0$,
as has been pointed out already based on qualitative arguments. The exact
solution can now be written a
\begin{equation}
\Phi _{\sigma }=z_{\sigma }^{s_{\sigma }}w(z_{\sigma })e^{-z_{\sigma }/2}
\label{comb1}
\end{equation
where $w$ is a regular solution of the confluent hypergeometric equation
(Kummer's equation) \cite{abr}
\begin{equation}
z_{\sigma }\frac{d^{2}w}{dz_{\sigma }^{2}}+(b_{\sigma }-z_{\sigma })\frac{d
}{dz_{\sigma }}-a_{\sigma }w=0  \label{35}
\end{equation
\noindent with the definition
\begin{equation}
a_{\sigma }=s_{\sigma },\quad b_{\sigma }=2s_{\sigma }  \label{36}
\end{equation
The general solution of \ (\ref{35}) is expressed in terms of the confluent
hypergeometric functions (Kummer's functions) $_{1}F_{1}(a_{\sigma
},b_{\sigma },z_{\sigma })$ (or $M(a_{\sigma },b_{\sigma },z_{\sigma })$)
and $_{2}F_{0}(a_{\sigma },1+a_{\sigma }-b_{\sigma },-1/z_{\sigma })$ (or
U(a_{\sigma },b_{\sigma },z_{\sigma })$):
\begin{equation}
w=A_{\sigma }\,\,_{1}F_{1}(a_{\sigma },b_{\sigma },z_{\sigma })+B_{\sigma
}z_{\sigma }^{-a_{\sigma }}\,_{2}F_{0}(a_{\sigma },1+a_{\sigma }-b_{\sigma
},-\frac{1}{z_{\sigma }}),\quad b_{\sigma }\neq -\tilde{n}_{\sigma }
\label{W}
\end{equation
where $\tilde{n}_{\sigma }$ is a nonnegative integer. Due to the singularity
of the second term at $z_{\sigma }=0$, only choosing $B_{\sigma }=0$ gives a
behavior at the origin which can lead to square-integrable solutions.
Furthermore, the requirement of finiteness for $\Phi _{\sigma }$ at
z_{\sigma }=\infty $ implies that the remaining confluent hypergeometric
function ($_{1}F_{1}(a_{\sigma },b_{\sigma },z_{\sigma })$) should be a
polynomial. This is because $_{1}F_{1}(a_{\sigma },b_{\sigma },z_{\sigma })$
goes as $e^{z_{\sigma }}$ as $z_{\sigma }$ goes to infinity unless the
series breaks off. This demands that $a_{\sigma }=-n_{\sigma }$, where
n_{\sigma }$ is also a nonnegative integer. This requirement combined with
\ref{36}) implies that the existence of bound-state solutions for pure
inversely quadratic potentials ($|\omega _{1}|=|\omega _{0}|$) is out of
question.

\subsection{$\Omega \neq 0$}

As for $\Omega \neq 0$, it is convenient to define the dimensionless
quantity $\xi $,

\negthinspace
\begin{equation}
\xi =m\sqrt{\Omega ^{2}}\,x^{2}  \label{15}
\end{equation
\noindent \noindent and using (\ref{sc1b})-(\ref{sc3}), one obtains the
complete form for Whittake
\'{
s equation \cite{abr}

\begin{equation}
\,\xi \frac{d^{2}\Phi _{\sigma }}{d\xi ^{2}}+\frac{1}{2}\frac{d\Phi _{\sigma
}}{d\xi }+\left( \frac{\varepsilon _{\sigma }}{2\sqrt{\Omega ^{2}}}-\frac
\xi }{4}-\frac{m\alpha _{\sigma }}{2\xi }\right) \Phi _{\sigma }=0
\label{16}
\end{equation}

\noindent The normalizable asymptotic form of the solution as $\xi
\rightarrow \infty $ is given by $e^{-\xi /2}$ only for $\Omega ^{2}>0$ (
|\omega _{1}|>|\omega _{0}|$). The solution for all $\xi $ can now be
written a
\begin{equation}
\Phi _{\sigma }=\xi ^{s_{\sigma }/2}w(\xi )e^{-\xi /2}  \label{comb2}
\end{equation
where $w(\xi )$ is a regular solution from
\begin{equation}
\xi \frac{d^{2}w}{d\xi ^{2}}+(b_{\sigma }-\xi )\frac{dw}{d\xi }-a_{\sigma
}w=0
\end{equation
wit
\begin{eqnarray}
a_{\sigma } &=&\frac{b_{\sigma }}{2}-\frac{\varepsilon _{\sigma }}{2|\Omega
}  \notag \\
&&  \label{036} \\
b_{\sigma } &=&s_{\sigma }+1/2  \notag
\end{eqnarray
\noindent Then, $w$ is expressed as $_{1}F_{1}(a_{\sigma },b_{\sigma },\xi )$
and in order to furnish normalizable $\Phi _{\sigma }$, the confluent
hypergeometric function must be a polynomial. This demands that $a_{\sigma
}=-n_{\sigma }$ and $b_{\sigma }\neq -\tilde{n}_{\sigma }$. Note that
_{1}F_{1}(-n_{\sigma },b_{\sigma },\xi )$ is proportional to the generalized
Laguerre polynomial $L_{n_{\sigma }}^{\left( b_{\sigma }-1\right) }(\xi )$,
a polynomial of degree $n_{\sigma }$. The requirement $a_{\sigma
}=-n_{\sigma }$ combined with the top line of (\ref{036}), also implies into
quantized effective eigenvalues:
\begin{equation}
\varepsilon _{\sigma }=\left( 2n_{\sigma }+s_{\sigma }+\frac{1}{2}\right)
|\Omega |,\qquad n_{\sigma }=0,1,2,\ldots   \label{37}
\end{equation
\noindent with
\begin{equation}
\Phi _{\sigma }\propto \xi ^{s_{\sigma }/2}e_{\;}^{-\xi /2}\;L_{n_{\sigma
}}^{\left( s_{\sigma }-1/2\right) }\left( \xi \right)   \label{38}
\end{equation
\noindent\

In summary, only for $\Omega ^{2}>0$ ($\varepsilon _{\sigma }>0$) and
\alpha _{\sigma }\geq \alpha _{\mathtt{crit}}$ the potentials (\ref{sc1a})
are able to furnish bound states. The spectrum is purely discrete and the
solution is expressed in an exact closed form. It is instructive to note
that $s_{\sigma }=0$ or $s_{\sigma }=1$ for the case $\alpha _{\sigma }=0$
and the associated Laguerre polynomial $L_{n_{\sigma }}^{\left( -1/2\right)
}(\xi )$ and $L_{n_{\sigma }}^{\left( +1/2\right) }(\xi )$ are proportional
to $H_{2n_{\sigma }}\left( \sqrt{\xi }\right) $ or $\xi ^{-1/2}H_{2n_{\sigma
}+1}\left( \sqrt{\xi }\right) $, respectively \cite{abr}. Therefore, the
solution for the effective nonsingular harmonic oscillator can be succinctly
written in the customary form in terms of Hermite polynomials
\begin{equation}
\varepsilon _{\sigma }=\left( n_{\sigma }+\frac{1}{2}\right) |\Omega
|,\qquad n_{\sigma }=0,1,2,\ldots  \label{410a}
\end{equation
with $\Phi _{\sigma }$ proportional to
\begin{equation}
e^{-\xi /2}H_{n_{\sigma }}\left( \sqrt{\xi }\right)  \label{410b}
\end{equation}

The preceding analyses shows that the effective eigenvalues for the scalar
sector are equally spaced with a step given by $2|\Omega |$ when the
effective potential is singular at the origin. It is remarkable that the
level stepping is independent of the sign and intensity of the parameter
responsible for the singularity of the potential. We also note that the
effective spectrum varies continuously with $\alpha _{+}$ as far as $\alpha
_{+}\neq 0$. When the effective potential becomes nonsingular ($\alpha
_{+}=0 $) the step switches abruptly to $|\Omega |$. There is a clear phase
transition when $\alpha _{+}\rightarrow 0$ due the disappearance of the
singularity. In the limit as $\alpha _{\sigma }\rightarrow 0$ the Neumann
boundary condition, in addition to the Dirichlet boundary condition always
present for $\alpha _{+}\neq 0$, comes to the scene. This occurrence permits
the appearance of even Hermite polynomials and their related eigenvalues,
which intercalate among the pre-existent eigenvalues related to odd Hermite
polynomials. The appearance of even Hermite polynomials makes $\Phi
_{+}(0)\neq 0$ and this boundary condition is never permitted when the
singular potential is present, even though $\alpha _{+}$ can be small. You
might also understand the lack of such a smooth transition by starting from
a nonsingular potential ($\alpha _{+}=0$), when the solution of the problem
involves even and odd Hermite polynomials, and then turning on the singular
potential as a perturbation of the nonsingular potential. Now the
\textquotedblleft perturbative singular potential\textquotedblright\ by
nature demands, if is either attractive or repulsive, that $\Phi _{+}(0)=0$
so that it naturally kills the solution involving even Hermite polynomials.
Furthermore, there is no degeneracy in the spectrum for $\alpha _{+}=0$.

Now we move on to match a common energy to the spin-1 boson problem. The
compatibility of the solutions for $\Phi _{+}=\phi _{I}^{\left( +\right) }$
and $\Phi _{-}=\phi _{I}^{\left( -\right) }$ demands that the quantum
numbers $n_{+}$ and $n_{-}$ must satisfy the relatio
\begin{equation}
n_{+}-n_{-}=R(\omega _{0},\omega _{1})
\end{equation
for the nonsingular potential, an
\begin{equation}
n_{+}-n_{-}=\frac{1}{2}\left[ R(\omega _{0},\omega _{1})-\sqrt{1+8m\alpha
_{+}}+\sqrt{1+8m\alpha _{-}}\right]
\end{equation
for the singular potential. Her
\begin{equation*}
R(\omega _{0},\omega _{1})=\frac{\mathrm{sgn}(\omega _{1})}{\sqrt{1-\left(
\frac{\omega _{0}}{\omega _{1}}\right) ^{2}}}
\end{equation*
and $\mathrm{sgn}(\omega _{1})$ stands for the sign function. For the
nonsingular potential $|n_{+}-n_{-}|$ varies from 1 to infinity (as $|\omega
_{0}/\omega _{1}|$ varies from 0 to 1), whereas for the singular potential
|n_{+}-n_{-}|$ varies from the minor to the major value between $\left\vert
R(\omega _{0},\omega _{1})\right\vert /2$ and $\left\vert R(\omega
_{0},\omega _{1})\pm \sqrt{8g_{1}}\right\vert /2$ (the $\pm $ sign
corresponding to $\alpha _{\pm }=\alpha _{\mathtt{crit}}$). These
constraints on the nodal structure of $\phi _{I}^{\left( +\right) }$ and
\phi _{I}^{\left( -\right) }$ dictate that acceptable solutions only occur
for a restricted number of possibilities for the potential parameters and
that no solution should be expected for $g_{1}<0$. More than this, these
constraints exclude a few low-lying quantum numbers for a given set of
possible solutions for the simple reason that $n_{+}$ and $n_{-}$ are
nonnegative integers. In conclusion, the possible bound-state solutions for
spin-1 bosons exist only for a finite set of potential parameters.

To have a better understanding of the spectrum, we plot in Figure \ref{Fig1}
the effective spectrum for the four low-lying levels as a function of
\alpha _{+}$, for fixed $\omega _{0}$ and $g_{0}$. From this figure we see
the phase transition for the spin-0 spectrum at $\alpha _{+}=0$ and the
severe restriction on the solutions for the spin-1 spectrum. Notice that no
bound-state solution for the spin-1 spectrum exists if $\alpha _{+}$ exceeds
$|R(\omega _{0},\omega _{1})|/2$ and that not all the bound states have a
nodeless $\Phi _{+}$ for the ground state.

\section{Conclusions}

We approached the problem of a particle in a (3+1)-dimensional world despite
the restriction to the one-dimensional movement. The motion on axis allow us
to explore the physical consequences of the nonminimal vector coupling in a
mathematically simpler and more physically transparent way. There is no
angular momentum so that there is no spin-orbit coupling. The more general
potential matrix ensuring a conserved four-current can be written in terms
of well-defined Lorentz structures. For the spin-0 sector there are two
scalar, two vector and two tensor terms \cite{gue}, whereas for the spin-1
sector there are two scalar, two vector, a pseudoscalar, two pseudovector
and eight tensor terms \cite{vij}. There is no pseudoscalar potential in the
spin-0 sector of the DKP theory. In fact, the space component of a
nonminimal vector potential is used in Ref. \cite{cha}. We described spin-0
particles by a five-component spinor, of which only one component is
independent instead the two required in Ref. \cite{cha}. Also, the number of
degrees of freedom described by the ten-component spinor in the spin-1
sector reduces correctly to three. By considering space plus time components
of a nonminimal vector potential, we showed that scalar and vector bosons in
the background of one-dimensional linear plus inversely linear potential is
mapped into a Sturm-Liouville problem with an effective singular oscillator.
The existence of solutions for spin-0 bosons requires $|\omega _{1}|>|\omega
_{0}|$ and $g_{1}\left( g_{1}+\sigma \right) -g_{0}^{2}\geq -1/4$. In words,
the time component of the nonminimal vector potential ($A_{0}$) alone can
not hold bound states, and the presence of the time component in the
inversely linear potential lessens the interval of parameters which provides
confinement. As for spin-1 bosons, confining solutions requires $g_{1}>0$
and when compared to the solutions for spin-0 bosons with the very same
potentials ($A_{0}$ and $A_{1}$) they can only be obtained from a restricted
class of solutions of the spin-0 sector. It may be salutary to note that,
because the differential equation (\ref{sc1b}) has a singularity at the
origin for $\alpha _{\sigma }\neq 0$, one could expect the existence of
singular solutions for $\Phi _{\sigma }$ so that we are not free to suppose
that $s_{\sigma }>0$. Due to the fact that $\Phi _{\sigma }$, whether they
are even or odd, disappears at the origin for $\alpha _{\sigma }\neq 0$ this
one-dimensional quantum-mechanical problem is degenerate and $H_{\sigma }$
in (\ref{Heff}) is a self-adjoint operator.

\bigskip \bigskip\bigskip \bigskip

\noindent {\textbf{Acknowledgments}}

This work was supported in part by means of funds provided by Coordena\c{c
\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'{\i}vel Superior\ (CAPES) and
Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'{o}gico
(CNPq). The authors gratefully acknowledge the anonymous referees for their
valuable comments and suggestions.

\newpage