On Self-adjoint and J-self-adjoint Dirac-type Operators: A Case Study

TL;DR

This paper compares spectral properties of self-adjoint and J-self-adjoint Dirac operators, introduces Weyl-Titchmarsh coefficients in the non-self-adjoint setting, and demonstrates how spectral arc crossings relate to spectral projection blowup.

Contribution

It introduces the concept of Weyl-Titchmarsh coefficients for non-self-adjoint Dirac operators and analyzes spectral arc crossings and their implications.

Findings

Crossing spectral arcs lead to blowup of spectral projection norms.

Weyl-Titchmarsh coefficients are extended to non-self-adjoint Dirac operators.

Spectral theory for J-self-adjoint operators is developed and compared to self-adjoint cases.

Abstract

We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schr\"odinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 x 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoint context imply the blowup of the norm of spectral projections in the limit where the crossing point is approached.