Semiclassical WKB Problem for the Non-Self-Adjoint Dirac Operator with a Multi-Humped Decaying Potential

TL;DR

This paper extends the semiclassical analysis of a non-self-adjoint Dirac operator's scattering data to potentials with multiple local extrema, providing rigorous conditions for eigenvalues and scattering parameters.

Contribution

It introduces a rigorous semiclassical framework for analyzing Dirac operators with multi-humped potentials, expanding previous work to more complex potential landscapes.

Findings

Derived Bohr-Sommerfeld conditions for eigenvalues

Analyzed norming constants and reflection coefficients

Extended semiclassical analysis to multi-humped potentials

Abstract

In this paper we continue the study (initiated in arXiv:2003.13584) of the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, fairly smooth but not necessarily analytic potential decaying at infinity; in this paper we allow this potential to have several local maxima and minima. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.