  We consider a second-order differential equation $$
-y''(z)-(iz)^{N+2}y(z)=\lambda y(z), \quad z\in \Gamma $$ with an eigenvalue
parameter $\lambda \in \mathbb{C}$. In $\mathcal{PT}$ quantum mechanics $z$
runs through a complex contour $\Gamma\subset \mathbb{C}$, which is in general
not the real line nor a real half-line. Via a parametrization we map the
problem back to the real line and obtain two differential equations on
$[0,\infty)$ and on $(-\infty,0].$ They are coupled in zero by boundary
conditions and their potentials are not real-valued. The main result is a
classification of this problem along the well-known limit-point/ limit-circle
scheme for complex potentials introduced by A.R.\ Sims 60 years ago. Moreover,
we associate operators to the two half-line problems and to the full axis
problem and study their spectra.
