This paper explains the systematics of the generation of families of spectra for the
-symmetric quantum-mechanical Hamiltonians
H
=
p
2
+
x
2
(i
x
)
ϵ
,
H
=
p
2
+(
x
2
)
δ
and
H
=
p
2
−(
x
2
)
μ
. In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities to and differences from the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of non-contiguous pairs of Stokes wedges that display
symmetry. To do so, simple arguments that use the Wentzel–Kramers–Brillouin approximation are used, 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
-symmetric quantum theory.
Generation of families of spectra in
PT
-symmetric quantum mechanics and scalar bosonic field theory