  This paper explains the systematics of the generation of families of spectra
for the PT-symmetric quantum-mechanical Hamiltonians $H=p^2+x^2(ix)^\epsilon$,
$H=p^2+(x^2)^\delta$, and $H=p^2-(x^2)^\mu$. In addition, it contrasts the
results obtained with those found for a bosonic scalar field theory, in
particular in one dimension, highlighting the similarities and differences to
the quantum-mechanical case. It is shown that the number of families of spectra
can be deduced from the number of noncontiguous pairs of Stokes' wedges that
display PT-symmetry. To do so, simple arguments that use the WKB approximation
are employed, and these imply that the eigenvalues are real. However,
definitive results are in most cases presently only obtainable numerically, and
not all eigenvalues in each family may be real. Within the approximations used,
it is illustrated that the difference between the quantum-mechanical and the
field-theoretical cases lies in the number of accessible regions in which the
eigenfunctions decay exponentially. This paper reviews and implements
well-known techniques in complex analysis and PT-symmetric quantum theory.
