This is a joint work with Bruno Colbois. Consider a compact Riemannian manifold $M$ with an involutive isometry $\gamma$, and assume that the distance of any point to its image under $\gamma$ is bounded below by a positive constant $\beta$ (the smallest displacement). We observe that this simple geometric situation has a strong consequences on the spectrum of a large class of $\gamma$-invariant operators $D$ (including the Schrödinger operator acting on functions and the Hodge Laplacian acting on forms): roughly speaking, the gap $\lambda_2(D)-\lambda_1(D)$ between the first and the second eigenvalue of $D$ is uniformly bounded above by a constant depending only on the displacement $\beta$ (in particular, not depending on $D$).