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Consider the discrete 1D Schrödinger operator on Z with an odd 2k periodic potential q. For small potentials we show that the mapping: q → heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2 k distinct potentials. Finally, the… (More)

We consider the Schrödinger operator on the zigzag and armchair nanotubes (tight-binding models) in a uniform magnetic field B and in an external periodic electric potential. The magnetic and electric fields are parallel to the axis of the nanotube. We show that this operator is unitarily equivalent to the finite orthogonal sum of Jacobi operators. We… (More)

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain. Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby… (More)

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