Semi-excited states in nondegenerate potential wells

We consider −h²Δ+V on Rⁿ, when V is smooth, real-valued and has a unique minimum at 0 which is nondegenerate: V(0)=0,V″(0)>0. We also assume that lim inf _|x|→∞V(x)>0. For any fixed δ>0 we get uniform asymptotic formulas for the eigenvalues up to h^δ, when h→0. The proofs use Birkhoff normal forms and pseudodifferential functional calculus.