WKB Spectral Asymptotics for a One-Dimensional Dirac Operator with a Slowly Varying Mass Profile

We study the semiclassical spectral theory of a one-dimensional Dirac operator describing waves at the interface between topologically distinct media. We derive a modified Bohr-Sommerfeld quantization condition for the squared operator via a systematic formal WKB construction producing approximate eigenpairs. Our result differs from the standard result by the half-integer shift depending on the pseudo-spin index which allows for recovering the topologically protected zero mode. We verify our result for the solvable Pöschl–Teller potential and provide numerical computations confirming the convergence of eigenvalues and eigenfunctions to their WKB approximations.