Some remarks on the magnetic field operators ±iA and its applications

In the present paper, we give some remarks on the magnetic field operators iA. As its applications, we study the Schr\"odinger equation with a magnetic field equation* - u+|A (x) |^2u+iA (x) u= u+|u|^pu, ~x R^N, equation* where u is a complex-valued function and R. When N>2, for 2 p+2 2NN-2 or N=2, for 2 p+2 +, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition divA (x) =0 plays a crucial role in our study.