Expansions over the ‘‘squared’’ solutions and difference evolution equations

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Abstract
The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived. It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results. These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action-angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum version of the IST.

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