Complex eigenvalue splitting for the Dirac operator

TL;DR

This paper studies the eigenvalues of the semiclassical Dirac operator with analytic potentials, deriving quantization conditions and analyzing eigenvalue splitting near specific energy levels, revealing symmetry-dependent eigenvalue behaviors.

Contribution

It provides Bohr-Sommerfeld quantization conditions for the Dirac operator and characterizes eigenvalue splitting and symmetry effects near potential bump features.

Findings

Eigenvalues are purely imaginary near single bump potential levels.

Eigenvalues split exponentially near double bump levels, with splitting type depending on potential symmetry.

Eigenvalue behavior is characterized as purely imaginary or not based on potential parity.

Abstract

We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov-Shabat) operator on the real line with general analytic potential. We provide Bohr-Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.