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We study the Cauchy problem of the Chern-Simons-Schrodinger equations with a neutral field, under the Coulomb gauge condition, in energy space . We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.

Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field