Geodesic
Matrix finite-gap operators
Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya, Tome 23 (1983), pp. 33-78

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A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of $J$-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such $J$-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves. 
@article{INTD_1983_23_a1,
     author = {B. A. Dubrovin},
     title = {Matrix finite-gap operators},
     journal = {Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya},
     pages = {33--78},
     publisher = {mathdoc},
     volume = {23},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTD_1983_23_a1/}
}
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B. A. Dubrovin. Matrix finite-gap operators. Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya, Tome 23 (1983), pp. 33-78. http://geodesic.mathdoc.fr/item/INTD_1983_23_a1/