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Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential. (English) Zbl 1431.81059
Summary: In this paper, we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic nonlinear Schrödinger (NLS) equation in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.
©2020 American Institute of Physics

81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81R25 Spinor and twistor methods applied to problems in quantum theory
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
81U40 Inverse scattering problems in quantum theory
Full Text: DOI
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[19] As we explain in Refs. 3 and 26, this corresponds to different sheets of the logarithmic kernel in this integral. Different approximations are required in different sheets for best results.
[20] We could still ignore the improvement in Ref. 6 and give a different argument involving different circles in different steps of the Riemann-Hilbert sequence. We feel that the argument would become a bit more cumbersome.
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