  Paraxial linear propagation of light in an optical waveguide with material
gain and loss is governed by a Schr\"odinger equation with a complex potential.
Properties of parity-time-symmetric complex potentials have been heavily
studied before. In this article, new classes of non-parity-time-symmetric
complex potentials featuring conjugate-pair eigenvalue symmetry in its spectrum
are constructed by operator symmetry methods. Due to this eigenvalue symmetry,
it is shown that the spectrum of these complex potentials is often all-real.
Under parameter tuning in these potentials, phase transition can also occur,
where pairs of complex eigenvalues appear in the spectrum. A peculiar feature
of the phase transition here is that, the complex eigenvalues may bifurcate out
from an interior continuous eigenvalue inside the continuous spectrum, in which
case a phase transition takes place without going through an exceptional point.
In one spatial dimension, this class of non-parity-time-symmetric complex
potentials is of the form $V(x)=h'(x)-h^2(x)$, where $h(x)$ is an arbitrary
parity-time-symmetric complex function. These potentials in two spatial
dimensions are also derived. Diffraction patterns in these complex potentials
are further examined, and unidirectional propagation behaviors are
demonstrated.
