Semiclassical Wkb Problem for the Non-Self-Adjoint Dirac Operator Withan Analytic Rapidly Oscillating Potential

Abstract

In this paper we examine the semiclassical behaviorof the scattering data of anon-self-adjoint Dirac operator with a rapidly oscillatingpotential that is complex analytic in some neighborhood of the real line.Some of our results are rigorous and quite general.On the other hand, complete  and concrete understanding requires the  investigation ofthe WKB geometry of specific examples.For such detailed computations we use a particular  example that has been investigated numericallymore than 20 years ago by Bronski and Miller and rely heavily on their numerical computations.Mostly employing the exact WKB method, we provide the complete rigorous uniformsemiclassical analysis of the Bohr-Sommerfeld condition for the location of theeigenvalues across unions of analytic arcs as well as the associated norming constants.For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, weemploy  instead an  older theory that  hasbeen developed in great detail by Olver.  Our analysis is motivatedby the need to understand  the semiclassical behaviour of the focusing cubic NLS equationwith initial data $A\exp\{iS/\epsilon\}$,  in view of the well-known fact discoveredby Zakharov and Shabat that the spectral analysis of the Dirac operator enablesthe solution of the NLS equation via inverse scattering theory.

Keywords: Rigorous WKB, NLS, semiclassical limits, non-self-adjoint differential operators