- 05-392 Steve Clark and Fritz Gesztesy
- On Self-adjoint and J-self-adjoint Dirac-type Operators: A Case Study
Nov 15, 05
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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.