In this paper, we study normalized solutions to the Chern-Simons-Schrodinger system, which is a gauge-covariant nonlinear Schoridnger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the $L^2$-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, the orbital stability of the set of minimizers to the problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solution. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, the instability of ground states is explored

中文翻译：

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陈-西蒙-薛定谔系统的归一化解

在本文中，我们研究了 Chern-Simons-Schrodinger 系统的归一化解，该系统是非相对论量子力学理论中出现的具有长程电磁场的规范协变非线性 Schoridger 系统。解决方案对应于受 $L^2$-norm 约束的基础能量泛函的临界点。我们的研究涵盖了几个方面。首先，在质量次临界情况下，我们建立任何最小化序列对相关全局最小化问题的紧致性。作为任何最小化序列的紧凑性的副产品，实现了该问题的最小化器集的轨道稳定性。此外，我们讨论了该问题的极小值的径向对称性和唯一性。其次，在大规模临界情况下，我们研究归一化解的存在和不存在。最后，在质量超临界情况下，我们证明了基态和无限多个径向对称解的存在。此外，探索了基态的不稳定性