A spectral result for Hardy inequalities

Let P be a linear, elliptic second order symmetric operator, with an associated quadratic form q, and let W be a potential such that the Hardy inequality λ0 ∫ Ω Wu 2 ≤ q(u) holds with (non-negative) best constant λ0. We give sufficient conditions so that the spectrum of the operator 1 W P is [λ0,∞). In particular, we apply this to several well-known Hardy inequalities: (improved) Hardy inequalities on a bounded convex domain of R with potentials involving the distance to the boundary, and Hardy inequalities for minimal submanifolds of R.