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

Abstract

In the context of a geodesically complete Riemannian manifold 𝑀, we study the self-adjointness of ∇ † ∇ + 𝑉, where ∇ is a metric covariant derivative (with formal adjoint ∇ † ) on a Hermitian vector bundle  over 𝑀, and 𝑉 is a locally square integrable section of End  such that the (fiberwise) norm of the "negative" part 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] and a positive function 𝑞 on 𝑀 satisfying certain growth conditions, such that 𝜀∇ † ∇ + 𝑉 ≥ −𝑞, the inequality being understood in the quadratic form sense over 𝐶 ∞ 𝑐 (). In the first result, which pertains to the case 𝜀 ∈ [0, 1), we use the elliptic equation method. In the second result, which pertains to the case 𝜀 = 1, we use the hyperbolic equation method.