Geodesic
An inverse problem for a class of one-dimensional Shrodinger operators with a complex periodic potential
Izvestiya. Mathematics , Tome 37 (1991) no. 3, pp. 611-629.

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A nonselfadjoint Sturm-Liouville operator $L=-d^2/dx^2+q(x)$ $(-\infty$ with a periodic potential which can be extended holomorphically to the upper half plane, is considered. 
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L. A. Pastur; V. A. Tkachenko. An inverse problem for a class of one-dimensional Shrodinger operators with a complex periodic potential. Izvestiya. Mathematics , Tome 37 (1991) no. 3, pp. 611-629. http://geodesic.mathdoc.fr/item/IM2_1991_37_3_a7/

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