Abstract: We give an account of a recent joint work with Petr Siegl
in which we apply semiclassical methods to throw doubt on
the concept of PT-symmetric quantum mechanics due to Bender et al.
It is commonly accepted in that community that parity-time symmetric
albeit non-Hermitian observables give rise to a consistent
quantum mechanics through a similarity to self-adjoint operators.
As a prominent example there is the imaginary cubic oscillator
that stayed at the advent of PT-symmetric quantum mechanics
in the turn of the millennium. We show that this example,
as an element of a large class of non-self-adjoint
Schroedinger operators, is not similar to a self-adjoint operator
via physically relevant transformations. Our argument is based on
known semiclassical results which involve a direct construction
of a continuous family of approximate eigenstates
of complex energies far from the spectrum.