  A popular $PT$-symmetric optical potential (variation of the refractive
index) that supports a variety of interesting and unusual phenomena is the
imaginary exponential, the limiting case of the potential $V_0[\cos(2\pi
x/a)+i\lambda\sin(2\pi x/a)]$ as $\lambda \to 1$, the symmetry-breaking point.
For $\lambda<1$, when the spectrum is entirely real, there is a well-known
mapping by a similarity transformation to an equivalent Hermitian potential.
However, as $\lambda \to 1$, the spectrum, while remaining real, contains
Jordan blocks in which eigenvalues and the corresponding eigenfunctions
coincide. In this limit the similarity transformation becomes singular.
Nonetheless, we show that the mapping from the original potential to its
Hermitian counterpart can still be implemented; however, the inverse mapping
breaks down. We also illuminate the role of Jordan associated functions in the
original problem, showing that they map onto eigenfunctions in the associated
Hermitian problem.
