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

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