We propose a gauge-invariant system of the Chern–Simons–Schrödinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.

中文翻译：

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一维格的陈-西蒙-薛定谔理论

我们在一维格子上提出了陈-西蒙-薛定谔型的规范不变系统。通过使用空间规范条件，我们证明了初始值问题在标量场的平方可和序列空间中的局部和全局适定性。我们还研究了静止束缚态的存在区域，这取决于晶格间距和非线性功率。存在问题的一个主要困难与缺乏平稳方程的变分公式有关。我们的方法基于反连续极限中的隐函数定理和连续极限中的可解性约束。