Analogue of Inverse Scattering Theory for the Discrete Hill's Equation and Exact Solutions for the Periodic Toda Lattice

An analogue of the inverse scattering theory for the discrete Hill's equation is developed. A method consists of the introduction of abelian integrals and the solution of Jacobi's inversion problem. The equation of motion of the periodic Tod lattice is integrated exactly by means of Riemann theta functions.

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