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

Abstract

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