We investigate the effects of competition between two complex,
PT-symmetric potentials on the PT-symmetric phase of a
"particle in a box". These potentials, given by VZ​(x)=iZsign(x) and
Vξ​(x)=iξ[δ(x−a)−δ(x+a)], represent long-range and localized
gain/loss regions respectively. We obtain the PT-symmetric phase in
the (Z,ξ) plane, and find that for locations ±a near the edge of the
box, the PT-symmetric phase is strengthened by additional losses to
the loss region. We also predict that a broken PT-symmetry will be
restored by increasing the strength ξ of the localized potential. By
comparing the results for this problem and its lattice counterpart, we show
that a robust PT-symmetric phase in the continuum is consistent
with the fragile phase on the lattice. Our results demonstrate that systems
with multiple, PT-symmetric potentials show unique, unexpected
properties.Comment: 7 pages, 3 figure