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 V0[cos(2πx/a) + iλ sin(2πx/a)] as λ → 1, the symmetrybreaking
point. For λ < 1, when the spectrum is entirely real, there is a well-known mapping
by a similarity transformation to an equivalent Hermitian potential. However, as λ → 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.