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

Abstract

In this paper, we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr–Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic nonlinear Schrödinger (NLS) equation in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.