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Let ( M ,  g ) be a complete non‐compact Riemannian manifold. We consider operators of the form Δ  g   +  V , where Δ  g   is the non‐negative Laplacian associated with the metric  g , and  V  a locally integrable function. Let    ρ     :      (    M  ∘   ,      g  ∘    )   →   (   M  ,     g   )be a Riemannian covering, with Laplacian Δ  ĝ   and potentialV  ∘   =  V  ∘  ρ   . If the operator Δ +  V  is non‐negative on ( M ,  g ), then the operatorΔ   g  ∘    +   V  ∘is non‐negative on(    M  ∘   ,   g  ∘    )    . In this note, we show that the converse statement is true provided thatπ  1    (   M  ∘   )is a co‐amenable subgroup of π 1 ( M ).

Spectral positivity and Riemannian coverings