Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|⩽|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?