Inverse problem for the discrete periodic Schrödinger operator

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.