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Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

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    *Corresponding author

1Present address: Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden

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  • Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the unit circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.

    Mathematics Subject Classification: Primary: 34L40; Secondary: 81Q20.


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  • Figure 1.  The configuration of Stokes lines for V (x) = cos x, with legends describing the size of Re z(x; 0, λ) for λ = 1 (left panel) and λ = 1 + 10−1i (right panel)

    Figure 2.  The configuration of Stokes lines and the behavior of Re z(x; x1; λ) for V (x) = cos x, where λ = and x1 satisfies Re x1 > 0, cos x1 = µ (the turning point in the middle). The top panel describes the situation for µ = 1/2 and the bottom panel for µ = 1/2 + i/10. Branch cuts are located along (the curved edges of) the white regions.

    Figure 3.  The location of amplitude base points relative the neighboring turning points $x_k(\mu)$ over a partial period for generic $V$ and $\lambda = i\mu\in B_\varepsilon(\lambda_0)$. Branch cuts are indicated by dashed lines.

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