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In this paper we study the time-dependent discrete nonlinear Schrödinger equation with complex, not necessarily bounded, potential and sufficiently general nonlinearity on a multi dimensional lattice. Under natural assumptions we prove the global well-posedness in weightedspaces. In the dissipative case we obtain a result on the existence of global compact attractor in such spaces.

Initial value problem of the discrete nonlinear Schrödinger equation with complex potential