Abstract
We discuss the Bose--Einstein condensation (BEC for short) for the
pure hopping model on some nonhomogeneous amenable graphs, and in
particular for the comb graphs $G_d$ with backbone $Z^d$ and
fingers $Z^d$--copies of $Z$. It is shown that the BEC phenomena
are connected with the transience of the graphs (relative to the
adjacence operator), and not only with the finiteness of the critical
density, the last properties being not equivalent for nonhomogeneous
graphs considered here. The situation differs to that relative to
$Z^d$,
and $Z^d$--periodic lattices (Matsui) where the transience and
the finiteness of the critical density are known to be equivalent, and
determined by the dimension $d$ of the lattices. Namely,
the comb graphs considered here, exhibit hidden spectrum near zero,
thus the critical density is finite. But, they are transient iff
$d\geq3$. By considering periodic boundary condition along the
backbone for the finite--volume theories, it is proven that whenever
$dłeq2$ and the
sequence of chemical potential $\mu_Łambda_n\uparrow0$, it is
impossible to obtain any infinite--volume state describing BEC and
having a reasonable local description (i.e.
non locally normal states for which $\displaystylełim_Łambda_n\uparrow
G_d
ømega_Łambda_n\big(a^+(\delta_j)a(\delta_j)\big)=+ınfty$). If
$d\geq3$, a careful choice of the sequence $\mu_Łambda_n\uparrow0$
should be done in order to obtain locally normal states describing
BEC. For all these infinite volume states $ømega$, the density
$$
\rho(ømega):=łim_Łambda_nG_d1|Łambda_n|
\sum_jınŁambda_nømega\big(a^+(\delta_j)a(\delta_j)\big)
$$
coincides with the critical density $\rho_c$. The situation behaves as
follows.
Below the critical density, the system describes (pure hopping)
Bosons allocated on a $d+1$--dimensional system (i.e. the dimension
described by the vertices of the comb $G_d$). Above the critical
density, a considerable portion of particles condensates in
momentum space as well as in configuration space, due to
nonhomogeneity. Such a condensate is essentially localized along the
backbone. The system undergoes a ``dimensional transition''. If
$dłeq2$ we cannot exhibit any locally normal state describing BEC. If
$d\geq3$,
we can construct locally normal states $ømega$ describing BEC only if the
sequence $\mu_Łambda_n\uparrow0$ is carefully chosen such that
$\rho(ømega)=\rho_c$.
Other relevant examples are finite additive
perturbations of periodic graphs, where the shape of the wave function
of the condensate (coinciding with the Perron--Frobenious eigenvector for
the
adjacence) describes a condensate essentially localized in a finite
portion of the
perturbed graph under consideration, and we have again no locally
normal states describing BEC. Finally, we easily exhibit locally
normal states $ømega$ describing the BEC for $N$, which is transient
but $\rho_c=+ınfty$, thus $\rho(ømega)=+ınfty$, and for the comb graph
with backbone $N$ and fingers $N$--copies of $Z^2$, where
locally normal states can be constructed with any $\rho(ømega)$ including
$\rho(ømega)\geq\rho_c$.
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