Equally spaced delta functions on a line, with a periodically
repeated N-plet of couplings $e_j$ (or rather associated lengths
$a_j$) is known to form an exactly solvable model at any N. The
authors allow for complex couplings and keep the model only
parity-times-complex-conjugation (i.e., PT-) invariant. They
discover the ``fragile" appearance and disappearance of forbidden
and allowed energy bands (``Brillouine zones"). They study this
new and exciting phenomenon (impossible in the ``robust" Hermitian
standard models) in detail up to N=6. Quite impressive a result,
having in mind that the previous similar model (studied in ref.
[11] in 1999, using a sine curve in place of the present delta
functions AND requiring an enormous precision of the necessary
evaluation of all the necessary band-edge-boundary roots) proves
only able to compete (i.e., to guarantee similar assertions
concerning the gaps) at N=1. Moreover, several new phenomena
(e.g., an emergence of certain ``special" spectral points where
the gradient of the energy diverges, or the possibility of an
extreme sensitivity of the wavefunction-localization to the
imaginary parts of the couplings) have been revealed and announced
to wait for a deeper interpretation, say, in the condensed-matter
physics (where one can speak about ``chains of atoms") or physics
of quantum wires (in this context, the standard transport
phenomena become significantly changed) etc.



MR2100329 Cerveró, José M. ; Rodríguez, Alberto . The band
spectrum of periodic potentials with $\scr P\scr T$-symmetry. J.
Phys. A 37 (2004), no. 43, 10167--10177