PhysicsLogic / Hohenberg-Kohn 1964

What a Ground-State Density Determines

Follow the variational comparison and the eigenvalue equations to an actual potential-gauge classification.

Definitions and assumptions

Scope and physical limitations

This is an exact one-particle, three-site hopping diagnostic of the Hohenberg-Kohn argument, not a continuum Coulomb proof or a time-dependent Runge-Gross result. Potentials are arbitrary real functions on the sites, waves are complex, and density is their squared modulus. Admissibility requires normalization, the eigenvalue equation and a minimum unique up to one global phase. Nonzero amplitude at every site follows from these conditions for this specific hopping matrix; it is no longer an extra premise. A checked equivalence proves that the simpler domain is identical to the explicitly supported domain. This is not continuum unique continuation. The domain is inhabited by a computed ground state. No surjectivity onto arbitrary densities is claimed.

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Examples and distinctions

Constructed examples of this model; each answer includes the Lean statement.

Exact root statement

potentialDetermination.ReducedClassifiesFullUpToAt ()
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Notes and questions

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Definitions, structures, and assumptions

Structures, classes, and inductive alternatives

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Checked statements: parameters, hypotheses, and interpretation

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Constructors, projections, and other meaning dependencies

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External mathematical declarations

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