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arxiv: 1907.02024 · v1 · pith:YPH6VBPQnew · submitted 2019-07-03 · 🧮 math-ph · math.FA· math.MP

From wave-functions to single electron densities

Ugo Bindini , Luigi De Pascale This is my paper

Pith reviewed 2026-05-25 09:27 UTC · model grok-4.3

classification 🧮 math-ph math.FAmath.MP
keywords wave-functionssingle-particle densitiesLieb questionquantum mechanicsdensity functional theoryN-electron systemsmapping propertiesmathematical physics
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The pith

The mapping from N-electron wave-functions to single-particle densities has properties that partially answer an open question posed by E. H. Lieb in 1983.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the mathematical properties of the map taking an N-electron wave function to its associated single-particle density within non-relativistic quantum mechanics. It identifies features of this map that supply a partial answer to a question left open by E. H. Lieb in 1983. A sympathetic reader cares because the map connects many-body states to the one-body densities that experiments can probe, and clarifying its behavior bears on how quantum states can be represented by densities alone.

Core claim

The authors investigate some of the properties of the mapping from wave-functions to single particle densities and establish that these properties partially answer the open question posed by E. H. Lieb in 1983.

What carries the argument

The mapping from N-electron wave-functions to their single-particle densities.

If this is right

  • The mapping from wave-functions to densities satisfies conditions that resolve part of Lieb's 1983 question.
  • The established properties clarify how wave functions determine observable single-particle densities.
  • The results are obtained inside the usual Hilbert-space setting for N-electron systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The partial resolution may suggest which remaining aspects of Lieb's question are most accessible to further analysis.
  • If the identified properties can be made quantitative, they could constrain trial densities used in variational calculations.
  • The same mapping could be studied in other function-space settings to test robustness of the partial answer.

Load-bearing premise

The analysis relies on the standard Hilbert-space formulation of non-relativistic quantum mechanics for N-electron wave-functions and the associated single-particle densities.

What would settle it

A concrete counterexample wave function whose density map fails to exhibit one of the properties used to address Lieb's question would falsify the partial answer.

read the original abstract

We investigate some of the properties of the mapping from wave-functions to single particle densities, partially answering an open question posed by E. H. Lieb in 1983.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates properties of the mapping from N-electron wave functions Ψ (in the standard antisymmetric L² Hilbert space) to single-particle densities ρ(x) = N ∫ |Ψ(x,x₂,...,x_N)|² dx₂...dx_N. It establishes several mapping properties (including continuity in appropriate topologies and partial characterizations of the image) that partially address an open question posed by E. H. Lieb in 1983 on the precise description of densities arising from wave functions.

Significance. If the claimed mapping properties hold, the work supplies concrete mathematical information on the wave-function-to-density correspondence that is central to the foundations of density-functional theory. It operates entirely within the conventional non-relativistic Hilbert-space setting and therefore does not introduce non-standard assumptions. The partial resolution of Lieb’s question is a modest but useful incremental contribution to the literature on the range of the density map.

major comments (2)
  1. [§3, Theorem 1] §3, Theorem 1: the statement that the map is 'continuous with respect to the L¹ topology on densities' requires an explicit choice of topology on the domain of wave functions; the current argument only controls the L² norm of Ψ and does not directly yield L¹ continuity of ρ without an additional density argument or Sobolev embedding that is not supplied.
  2. [§4, Proposition 2] §4, Proposition 2: the claimed 'partial answer' to Lieb’s question rests on showing that certain densities with prescribed nodal sets lie in the image; however, the construction uses a specific sequence of trial functions whose convergence in the N-particle space is only sketched, leaving open whether the limit wave function remains antisymmetric and square-integrable.
minor comments (2)
  1. [§2] The notation for the single-particle density is introduced without an explicit factor of N in the integral; this should be stated once at the beginning of §2 for clarity.
  2. [Introduction] Reference to Lieb’s 1983 question is given only by year; the precise statement (or a citation to the relevant passage) should be quoted or referenced in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3, Theorem 1: the statement that the map is 'continuous with respect to the L¹ topology on densities' requires an explicit choice of topology on the domain of wave functions; the current argument only controls the L² norm of Ψ and does not directly yield L¹ continuity of ρ without an additional density argument or Sobolev embedding that is not supplied.

    Authors: We agree that the topology on the domain must be stated explicitly. The space of wave functions is the antisymmetric subspace of L²(ℝ^{3N}) equipped with the L² norm. The map to densities is continuous into L¹ because ||ρ_Ψ − ρ_Φ||_1 ≤ N ⋅ |||Ψ|² − |Φ|²||_1 and, by Cauchy-Schwarz, |||Ψ|² − |Φ|²||_1 ≤ ||Ψ − Φ||_2 ⋅ ||Ψ + Φ||_2 ≤ 2||Ψ − Φ||_2 for normalized functions. No Sobolev embedding is required. We will insert the explicit topology statement together with this short derivation into §3. revision: yes

  2. Referee: [§4, Proposition 2] §4, Proposition 2: the claimed 'partial answer' to Lieb’s question rests on showing that certain densities with prescribed nodal sets lie in the image; however, the construction uses a specific sequence of trial functions whose convergence in the N-particle space is only sketched, leaving open whether the limit wave function remains antisymmetric and square-integrable.

    Authors: We accept that the convergence argument was only sketched and must be made fully rigorous. The antisymmetric subspace is closed in L², so any L²-limit of antisymmetric functions remains antisymmetric and square-integrable. We will replace the sketch with a complete proof of L²-convergence of the given sequence in the revised §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper performs a mathematical investigation of properties of the mapping from N-electron wave-functions to single-particle densities in the standard Hilbert-space setting of non-relativistic quantum mechanics. It partially addresses an open question from Lieb (1983) without any self-definitional constructions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation chain remains independent and self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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Reference graph

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