Geodesic
Inverse problem for the discrete periodic Schr\"odinger operator
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 96-101

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We study the isospectral sets for the discrete 1D Schrödinger operator on $\mathbb Z$ with a N+1 periodic potential. We show that for small odd potentials the isospectral set consists of $2^{(N+1)/2}$ elements, while for the large potentials the isospectral set consists of $(N+1)!$ elements. Moreover, the asymptotics of the end of the spectrum of the Schrödinger operator for small (and large) potentials are determined. 
@article{ZNSL_2004_315_a6,
     author = {E. Korotyaev and A. Kutsenko},
     title = {Inverse problem for the discrete periodic {Schr\"odinger} operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {96--101},
     publisher = {mathdoc},
     volume = {315},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a6/}
}
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E. Korotyaev; A. Kutsenko. Inverse problem for the discrete periodic Schr\"odinger operator. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 96-101. http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a6/