  We analyze the eigenvalue problem for the semiclassical Dirac (or
Zakharov-Shabat) operator on the real line with general analytic potential. We
provide Bohr-Sommerfeld quantization conditions near energy levels where the
potential exhibits the characteristics of a single or double bump function.
From these conditions we infer that near energy levels where the potential (or
rather its square) looks like a single bump function, all eigenvalues are
purely imaginary. For even or odd potentials we infer that near energy levels
where the square of the potential looks like a double bump function,
eigenvalues split in pairs exponentially close to reference points on the
imaginary axis. For even potentials this splitting is vertical and for odd
potentials it is horizontal, meaning that all such eigenvalues are purely
imaginary when the potential is even, and no such eigenvalue is purely
imaginary when the potential is odd.
