Complex Eigenvalue Splitting for the Dirac Operator

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