Corollary to the Hohenberg-Kohn Theorem

TL;DR

This paper presents a corollary to the Hohenberg-Kohn theorem, showing that certain degenerate Hamiltonians with different physical systems can share the same ground state density, extending the theorem's applicability.

Contribution

It introduces a corollary demonstrating the limitations of the Hohenberg-Kohn theorem when Hamiltonians differ by an intrinsic constant, including an extension to the time-dependent case.

Findings

Identifies degenerate Hamiltonians with identical densities

Extends the corollary to time-dependent Hohenberg-Kohn theorem

Clarifies the theorem's applicability limits

Abstract

In this paper we construct such a set of `degenerate' Hamiltonians $\hat{H}$, which differ by an `intrinsic' constant but represent different physical systems yet possess the same ground state density. . Thus, although the proof of Hohenberg-Kohn (HK) theorem is independent of whether the constant $C$ is additive or intrinsic, its applicability is restricted to excluding the case of the latter. This constitutes the corollary to the theorem.The corollary has also been extended to the time-dependent version of the HK theorem.