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 -Δu+|A(x)|2u+iA(x) ·∇u = μu+|u|pu, x ∈ RN, 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. [ABSTRACT FROM AUTHOR]