Semi‐classical analysis for the eigenvalues of a Schrödinger operator with magnetic field

We discuss semi‐classical analysis for the eigenvalues of the Schrödinger operator on ${\mathbb{R}}$ ^d with magnetic and electoric potentials. We consider the case where the electric potential vanishes precisely of even order at the wells. Our final aim is to clarify that when the operator has a non‐degenerate eigenvalue and then to get the complete asymptotics of the eigenvalue in the semi‐classical sense. Moreover, we give an example which seems to be new.