Optimal constant in the Heisenberg inequality with Riemannian distance in Schwarzschild geometry

Establish the precise optimal constant C(n) in the Heisenberg-type inequality (1/2) ∫_ℰ |ψ|^2 dv_g ≤ (∫_ℰ d^2 |ψ|^2 dv_g)^{1/2} (∫_ℰ |∇_g ψ|^2 dv_g)^{1/2} on the Riemannian manifold (ℰ, g), where ℰ = {x ∈ ℝ^n : |x| > 1} is the exterior of the Schwarzschild black hole, g is the reduced Schwarzschild metric g = (r^{n−2}/(r^{n−2}−1)) dr ⊗ dr + r^2 g_{S^{n−1}}, d is the Riemannian distance to the event horizon given by d(r) = ∫_1^r √(ξ^{n−2}/(ξ^{n−2}−1)) dξ, and ψ ranges over locally absolutely continuous functions satisfying ψ^2(x) |x|^n → 0 as |x| → ∞; in particular, ascertain whether the constant 1/2 is optimal.

Background

The authors prove a sharp Heisenberg-type inequality on the Schwarzschild exterior using an induced distance s(r), with optimal constant 1/2 and an explicit family of minimizers. Since s(r) ≤ d(r), they derive a corollary Heisenberg inequality with the Riemannian distance d(r) that also has constant 1/2.

Because s(r) ~ d(r)/n for large r, they note that the inequality with d(r) may admit a larger optimal constant. Their current method cannot establish or refute the optimality of 1/2 for the d-based inequality, leaving the exact constant unresolved.