Nelson’s essential skew-adjointness conjecture

Establish that, for every compactly supported divergence-free vector field v in L²(Rᵈ), the operator A₀ defined on C_c^∞(Rᵈ) by A₀ρ = v · ∇ρ is essentially skew-adjoint on L²(Rᵈ).

Background

The paper studies the relationship between essential skew-adjointness of the differentiation-along-the-vector-field operator A₀ρ = v * ∇ρ and well-posedness properties for the continuity equation associated with a compactly supported divergence-free square-integrable vector field v. The authors prove that essential skew-adjointness is equivalent to renormalization of square-integrable solutions and to uniqueness of square-integrable solutions of the continuity equation both forward and backward in time.

The conjecture attributed to Nelson and Aizenman asserts essential skew-adjointness of A₀. The paper does not establish this property for every vector field in the stated class; instead, it characterizes its equivalence with analytic properties of the continuity equation and constructs higher-dimensional examples showing that forward uniqueness alone does not imply backward uniqueness.