Self‐adjointness of non‐semibounded covariant Schrödinger operators on Riemannian manifolds

In the context of a geodesically complete Riemannian manifold M, we study the self‐adjointness of ∇†∇+V$\nabla ^{\dagger }\nabla +V$, where ∇ is a metric covariant derivative (with formal adjoint ∇†$\nabla ^{\dagger }$) on a Hermitian vector bundle V$\mathcal {V}$ over M, and V is a locally square integrable section of EndV$\operatorname{End}\mathcal {V}$ such that the (fiberwise) norm of the “negative” part V−$V^{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number ε∈[0,1]$\varepsilon \in [0,1]$ and a positive function q on M satisfying certain growth conditions, such that ε∇†∇+V≥−q$\varepsilon \nabla ^{\dagger }\nabla +V\ge -q$, the inequality being understood in the quadratic form sense over Cc∞(V)$C_{c}^{\infty }(\mathcal {V})$. In the first result, which pertains to the case ε∈[0,1)$\varepsilon \in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case ε=1$\varepsilon =1$, we use the hyperbolic equation method.