Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

TL;DR

This paper rigorously analyzes the semiclassical scattering data of a non-self-adjoint Dirac operator with a complex oscillating potential, using the exact WKB method and numerical insights to understand eigenvalues and reflection coefficients.

Contribution

It provides a comprehensive semiclassical analysis of the Dirac operator's spectral data, combining rigorous methods with numerical examples, advancing understanding of non-self-adjoint operators in quantum mechanics.

Findings

Rigorous uniform semiclassical analysis of eigenvalues and norming constants.

Complete WKB-based Bohr-Sommerfeld quantization conditions.

Insights into the spectral behavior near zero energy.

Abstract

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential 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 of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more 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 uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues 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, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/\epsilon\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.