We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to the one and two-dimensional case, with a set of parameters which implies global well-posedness for the continuous equation. Our proof relies on the use of bilinear estimates for the Shannon interpolation as well as the control of the growth of discrete Sobolev norms that we both prove.

翻译：我们证明了无限格点上离散非线性薛定谔方程的解在任意Sobolev范数下强收敛到全空间上非线性薛定谔方程的解。我们重点关注一维和二维情形，并选取了确保连续方程全局适定性的参数集。我们的证明依赖于香农插值的双线性估计以及离散Sobolev范数增长的控制，这两点均得到了严格证明。