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A discrete Schrödinger spectral problem and associated evolution equations. (English) Zbl 1067.39032

The hierarchies of differential-difference equations related to the spectral problem \[ \psi(n+ 2)+ q(n)\psi(n+1)= \lambda\psi(n) \] which includes the new discrete integrable version found in the discrete sine-Gordon and Liouville equations, are constructed. Finally, a Darboux transformation \(\widetilde\psi= D\psi\), where \(D\) is an opportune shift operator depending on \(q\) and \(\widetilde q\) related to spectral problem \(\widetilde L\widetilde\psi= \lambda\widetilde\psi\) with \(L'(n,t)= E^2+\widetilde q(n,\psi)E'\) is also introduded.

MSC:

39A12 Discrete version of topics in analysis
39A70 Difference operators
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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