Beyond v-representability: Local one-body Hamiltonians for arbitrary densities

The 'ν-representability problem' arising in density functional theory is this: for an script N-particle system of interacting identical particles, can every one-particle density of finite intrinsic energy be realized as a ground-state density upon modifying the Hamiltonian by addition of an appropriate external one-body potential ν? The answer is known to be no. We show that the answer is yes, if the inter-particle interaction is bounded, and allowed modifications are broadened from one-body potentials to local one-body operators. The required operator is obtained from a non-standard potential (in the sense of non-standard analysis) which is a simple function defined on a regular grid of in finitesimal finesse. At a vague conceptual level, therefore, the result is not far from universal ν-representability.