1},n_{2},n_{3}\rangle
\;,
\eeq
where we have omitted the occupation number of the fourth well $n_4= N-(n_{1}+n_2+n_3)$.

\begin{figure}[h]
\centerline{\includegraphics[width=5cm,clip]{i0ter.eps}
\includegraphics[width=5cm,clip]{i1ter.eps}
\includegraphics[width=5cm,clip]{i2ter.eps}}
\centerline{
\includegraphics[width=5cm,clip]{i3ter.eps}
\includegraphics[width=5cm,clip]{i4ter.eps}
\includegraphics[width=5cm,clip]{i5ter.eps}
}
\caption{(Color online). Horizontal axis: kets $|n_1,n_2,n_3\rangle$ [the fourth well occupation being $n_4=N-(n_1+n_2+n_3)$]. Vertical axis: the squared modulus of ground-state coefficients $|c_{n_1,n_2,n_3}|^2$. Here: $U_0=1$, $J=0.02$ and $N=32$. Top panels: $0<U_1<U_0/2$. First (from the left): $U_1=0.00001$. Second: $U_1=0.1$. Third: $U_1=0.499$. Bottom panels: $U_0/2<U_1<U_0$. First (from the left): $U_1=0.50251$. Second: $U_1=0.5030$. Third: $U_1=0.5038$. In the bottom panels, parameters lie in the filled triangle plotted in Fig.~\ref{fig2}.}
\label{fig1}
\end{figure}





\begin{figure}
\epsfig{file=triangolo.eps,width=0.92\linewidth,clip=}
\caption{The shaded area is the region where the solutions $x_1$ and $x_2$ 
(given by Eq. (\ref{sol1a})) are defined: the inequality (\ref{ineq1a}) 
is satisfied. Dashed line: left-hand-side of Eq. (\ref{ineq1a}). 
Solid line: right-hand-side of Eq. (\ref{ineq1a}). $U_0=1$, $J=0.02$, $N=32$.}
\label{fig2}
\end{figure}

\begin{figure}[h]
\centerline{\includegraphics[width=5cm,clip]{i6ter.eps}
\includegraphics[width=5cm,clip]{i7ter.eps}
}
\centerline{\includegraphics[width=5cm,clip]{i8ter.eps}
\includegraphics[width=5cm,clip]{i9ter.eps}
}
\centerline{\includegraphics[width=5cm,clip]{i10ter.eps}
\includegraphics[width=5cm,clip]{i11ter.eps}
}
\caption{(Color online). Horizontal axis: kets $|n_1,n_2,n_3\rangle$ 
[the fourth well occupation being $n_4=N-(n_1+n_2+n_3)$]. Vertical axis: 
the squared modulus of ground-state coefficients $|c_{n_1,n_2,n_3}|^2$. 
Here: $U_0=1$ and $N=32$. Top panels ($J=0.03$): $U_1=0.499$ (left);
$U_1=0.50438$ (right). Middle panels ($J=0.035$): $U_1=0.499$ (left); 
$U_1=0.5038$ (right). Bottom panels ($J=0.04$): $U_1=0.499$ (left); 
$U_1=0.5051$ (right).}
\label{fig3}
\end{figure}


Different ground states $|\Psi\rangle$ are sustained by the Hamiltonian (\ref{ham4})
depending on the relative magnitude of the parameters $U_1$ and $U_0$. To show how this causes
changes in the ground state structure, we have studied the probabilities $|c_{n_1,n_2,n_3}|^2$
on varying of $U_1$ in suitable ranges of values (see the discussion below) by keeping fixed
both $J$ and $U_0$. The results of this analysis, performed for a total boson number $N=32$, are shown in Fig.~\ref{fig1} (where, for convenience of representation, we have reported only $|c_{n_1,n_2,n_3}|^2 \gtrsim 1\times 10^{-2}$). Moreover, in Figs. ~\ref{fig1} and ~\ref{fig2} we assume $U_0$ as the energy scale ($U_0=1$).
Let us give a look to plots of Fig.~\ref{fig1} starting from the top panels. Here $0<U_1<U_0/2$. This corresponds to the ultraweak dipolar-interaction regime. By inspecting these distributions, two things can be clearly observed.
As a first, $|c_{n_1,n_2,n_3}|^2$ attains its maximum value for $n_1=n_2=n_3=8$ that is equivalent,
in the CVP language, to $x_1=x_2=x_3=x_4$: the uniform solution. The second observation is that
increasing $U_1$ has the effect to produce a progressive depletion of state $|8,8,8\rangle$,
i.e. state $|N/4,N/4,N/4,N/4\rangle$ remains the maximally populated one,
thus confirming the predictions of the CVP approach.

The bottom panels of Fig.~\ref{fig1}, instead, represent what happens in the weak
dipolar-interaction regime characterized by $U_0/2<U_1<U_0$. We have chosen the values of $U_1$
in accordance with the analysis of the previous section so that to satisfy the inequality
(\ref{ineq1a}). We have studied, in other words, the ground state corresponding to the parameter
values in the (lower) shaded area of Fig.~\ref{fig2}. The panel corresponding to $U_1=0.50251$ reveals that a transition occurs
in the ground state: the bosons populate with the highest probability the states $|0,16,0\rangle$
and $|16,0,16\rangle$ (i.e. the states $|0,N/2,0,N/2\rangle$ and $|N/2,0,N/2,0\rangle$, respectively)
that, in the CVP fashion, correspond to $x_3=x_1$ and $x_4=x_2$.

The ground state of the Hamiltonian (\ref{ham4}) is therefore a symmetric superposition of the states $|0,N/2,0,N/2\rangle$ and
$|N/2,0,N/2,0\rangle$. This result corroborates the CVP studies that predict a change in the ground state structure from the uniform state to a macroscopic two-pulse state when $U_1$ crosses $U_0/2$. By further increasing $U_1$ the above superposition is still the ground state
and the $|c|^2$'s  pertaining to $|0,16,0\rangle$ and $|16,0,16\rangle$ becomes larger,
as it can be seen from the fifth and sixth panels corresponding to $U_1=0.5030$ and $U_1=0.5038$, respectively.

 
As a conclusive remark, we note that the CVP-predicted localization-delocalization transition 
is captured by numerics in wider terms.
In fact, we have found numerically the ground-state of the four-site BH Hamiltonian (\ref{ham4}) 
with $N=32$ in correspondence to $J=0.03$ (top panels of Fig.~\ref{fig3}) and 
$J=0.035$ (middle panels of Fig.~\ref{fig3}), and $J=0.04$ (bottom panels of Fig.~\ref{fig3}).  
From these plots, one can clearly see the change experienced by the ground-state 
when the boundary $U_0/2$ is crossed