In E. O. Kane's original work on Zener Tunneling, he uses a two-band $k\cdot p$ model for the semiconductor bandstructure:
Moving along real $k$, one sees a conduction and valence band repelling each other via the momentum matrix element. But if you step onto the imaginary $k$ axis, there is a curve stretching through the band-gap to connect the extrema of the conduction and valence bands. These solutions for imaginary $k$ can be used to represent surface states, and they provide a smooth path for a semiclassical wavepacket to "tunnel" between bands while always having a well-defined position and wavevector.
But the above Hamiltonian is non-Hermitian when one plugs in complex values for $k$. (And if you make it Hermitian by conjugating one of the off-diagonal terms, the connection between the bands disappears.) So these spatially decaying states (which all have real energies within a certain $k$ range) are not actually eigenvectors of a Hermitian Hamiltonian...what are they?
I suppose I'm missing something fundamental about this procedure, but I don't see why this famous model assumes and works with a non-Hermitian Hamiltonian and what that means for the computed eigenvalues*. Has anyone come across any physics papers that address the subtle aspects of this?
.* For instance, I noticed that, at least in the range where all eigenvalues are real, the eigenvectors must be non-orthogonal (otherwise this non-Hermitian matrix could be real-diagonalized, which is a contradiction.)