Semiclassical WKB problem for the non-self-adjoint Dirac operator with a decaying potential

In this paper, we examine the semiclassical behavior of scattering data a non-self-adjoint Dirac operator with fairly smooth—but not necessarily analytic—potential decaying at infinity. particular, using ideas and methods from work Langer Olver [Philos. Trans. R. Soc. London, Ser. A 278(1279), 137–174 (1975)], provide rigorous analysis coefficients, Bohr–Sommerfeld condition for location eigenvalues, their corresponding norming constants. Our is motivated by potential applications to focusing cubic NLS equation, in view well-known fact discovered Zakharov Shabat [Sov. Phys. JETP 34(1), 62 (1972)] that spectral basis solution equation via inverse theory. This paper complements extends previous Fujiié Kamvissis [J. Math. 61(1), 011510 (2020)], which considered more restricted problem strictly analytic potential.