TOPICAL REVIEW: Periodic and almost-periodic potentials in inverse problems

An updated and detailed survey of basic ideas of the finite-gap theory is presented. That theory, developed to construct periodic and quasi-periodic solutions of the soliton equations, combines the Bloch-Floquet spectral theory of linear periodic operators, the theory of completely integrable Hamiltonian systems, the classical theory of Riemann surfaces, and theta-functions.

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