Paper ``Analytic Continuation of Eigenvalue Problems" by C. M.
Bender and A. V. Turbiner [Phys. Lett. Vol. A 173 (1993), p.
442-445] might be cited as one of many half-forgotten proposals of
the phenomenologically motivated replacement of the usual real line
of coordinates $x \in \mathbb{R}$ (say, in some one-dimensional
Schroedinger equation with an analytic potential, say,
$(p^2+(gx)^\alpha)\psi=E\psi$) by a complex contour ${\cal S}
\subset \mathbb{C}$. The specific choice of the left-right symmetric
(often called PT-symmetric) contours ${\cal S} \subset \mathbb{C}$
as made in Ref. [1] proved most successful and particularly
satisfactory by giving, unexpectedly, the strictly real
bound-state-like spectrum for $g=i$ (imaginary unit) and, in the
case of an ``optimal" contour, for {\em all} $\alpha \geq 2$. In
ref. [7], the ``less optimal" contours were declared to pose a
``quantum-toboggan" problem: ``what changes when the
cut-complex-plane choice of ${\cal S} \subset \mathbb{C}$ is
replaced by the general full-Riemann-surface (i.e., multisheeted)
choice of ${\cal S} \subset {\cal R}$?". One of the best existing
partial answers (cf. also H. B\'{\i}la, Spectra of PT-symmetric
Hamiltonians on tobogganic contours, Pramana - J. Phys.  73 (2009),
307 – 314, or M. Znojil, Topology-controlled spectra of imaginary
cubic oscillators in the large-L approach, Phys. Lett. A 374 (2010)
807–812) is provided by the presented paper. The characteristic
graphical samples of the overall pattern are displayed in Figures 4
and 6 and their very nice classical and semiclassical non-numerical
explanation is provided.

MR2749106 Bender, Carl M.; Jones, Hugh F. Quantum counterpart of
spontaneously broken classical $\scr{PT}$ symmetry. J. Phys. A 44
(2011), no. 1, 015301, 9 pp. 81Qxx