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

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

2005-11-15

arxiv.org/abs/math/0511369v3

Mathematics

Spectral Theory

38 pages

Scientific paper

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.

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