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\"{o}dinger equation with a magnetic field {equation*} -Δ u+|A(x)|²u+iA(x)· ∇ u=μ u+|u|ᵖu,~x∈ RN, {equation*} where u is a complex-valued function and μ∈ R. When $N>2$, for 2 p+2 2N/N-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.