Geodesic
An inverse problem for periodic Jacobi matrices
Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 297-307

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The problem of determining a matrix from its spectrum is solved for infinite periodic Jacobi matrices in the case in which the spectrum consists of $n-1$ characteristic values and $n$ intervals on the real axis. 
@article{MZM_1970_8_3_a2,
     author = {I. V. Stankevich},
     title = {An inverse problem for periodic {Jacobi} matrices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {297--307},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a2/}
}
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I. V. Stankevich. An inverse problem for periodic Jacobi matrices. Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 297-307. http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a2/