A semiclassical WKB problem for the Dirac operator with a decaying potential

An important step in the process of solving the initial value problem for the (1+1) focusing NLS equation with cubic nonlinearity and with rapidly decaying initial data -using the inverse scattering transform- is finding the scattering data (direct scattering problem). In the semiclassical regime, we are interested in finding the nature of solutions when the semiclassical parameter tends to zero. In order to do so, the first step is the investigation of the semiclassical behavior of the scattering data of the Lax operator which in our case is the (non self-adjoint) Dirac operator with a fairly smooth (but not necessarily analytic) potential decaying (fast) at infinity. In particular, in https://arxiv.org/abs/2003.13584, using ideas and methods going back to Olver, we were able to provide a rigorous semiclassical analysis of the scattering coefficients, the Bohr-Sommerfeld condition for the location of the EVs and their corresponding norming constants. In this talk -due to time restrictions- we only focus on the case of the eigenvalues that lie away from zero and the reflection coefficient. But the strategy we shall present, can be applied (with some alterations) to the remaining cases.