Ground-state-energy universality of noninteracting fermionic systems

When noninteracting fermions are confined in a D-dimensional region of volume O(LD) and subjected to a continuous (or piecewise-continuous) potential V which decays sufficiently fast with distance, in the thermodynamic limit, the ground-state energy of the system does not depend on V. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in D dimensions. We find that this universality property holds under a quite mild condition on V, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.