  This paper examines chains of $N$ coupled harmonic oscillators. In isolation,
the $j$th oscillator ($1\leq j\leq N$) has the natural frequency $\omega_j$ and
is described by the Hamiltonian $\frac{1}{2}p_j^2+\frac{1}{2}\omega_j^2x_j^2$.
The oscillators are coupled adjacently with coupling constants that are purely
imaginary; the coupling of the $j$th oscillator to the $(j+1)$st oscillator has
the bilinear form $i\gamma x_jx_{j+1}$ ($\gamma$ real). The complex
Hamiltonians for these systems exhibit {\it partial} $\mathcal{PT}$ symmetry;
that is, they are invariant under $i\to-i$ (time reversal), $x_j\to-x_j$ ($j$
odd), and $x_j\to x_j$ ($j$ even). [They are also invariant under $i\to-i$,
$x_j\to x_j$ ($j$ odd), and $x_j\to- x_j$ ($j$ even).] For all $N$ the quantum
energy levels of these systems are calculated exactly and it is shown that the
ground-state energy is real. When $\omega_j=1$ for all $j$, the full spectrum
consists of a real energy spectrum embedded in a complex one; the
eigenfunctions corresponding to real energy levels exhibit partial
$\mathcal{PT}$ symmetry. However, if the $\omega_j$ are allowed to vary away
from unity, one can induce a phase transition at which {\it all} energies
become real. For the special case $N=2$, when the spectrum is real, the
associated classical system has localized, almost-periodic orbits in phase
space and the classical particle is confined in the complex-coordinate plane.
However, when the spectrum of the quantum system is partially real, the
corresponding classical system displays only open trajectories for which the
classical particle spirals off to infinity. Similar behavior is observed when
$N>2$.
