Spectral positivity and Riemannian coverings

Let (M,g) be a complete non-compact Riemannian manifold. We consider operators of the form ∆g + V , where ∆g is the non-negative Laplacian associated with the metric g, and V a locally integrable function. Let ρ : (M̂ , ĝ) → (M,g) be a Riemannian covering, with Laplacian ∆ĝ and potential V̂ = V ◦ ρ. If the operator ∆ + V is non-negative on (M,g), then the operator ∆ĝ + V̂ is non-negative on (M̂, ĝ). In this note, we show that the converse statement is true provided that π1(M̂) is a co-amenable subgroup of π1(M).