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arxiv: 2604.18386 · v1 · submitted 2026-04-20 · 🧮 math-ph · math.AP· math.MP· math.SP

Eigenvalue asymptotics of M\"uller minimizers for atoms and molecules

Rupert L. Frank , Long Meng , Phan Th\`anh Nam , Heinz Siedentop This is my paper

Pith reviewed 2026-05-10 03:15 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MPmath.SP
keywords Müller functionaleigenvalue asymptoticsone-particle density matrixatoms and moleculesquantum chemistryintegral kernel
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The pith

The k-th eigenvalue of a Müller minimizer behaves asymptotically as A_* k^{-8/3}.

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

This paper establishes the large-k asymptotic behavior of the eigenvalues of the one-particle density matrix that minimizes the Müller functional for atoms and molecules. Under suitable conditions on the total nuclear charge Z and electron number N, the k-th eigenvalue of the minimizer γ_* is shown to scale as A_* k^{-8/3}, where the prefactor A_* is positive and can be expressed explicitly in terms of the density associated to γ_*. The analysis adapts techniques from the study of Schrödinger ground states but introduces new estimates to handle the singular diagonal behavior of the integral kernel and its decay at infinity. This matters for understanding the structure of approximate ground states in quantum chemistry because it quantifies how the minimizer encodes high-frequency information.

Core claim

We study the spectral properties of minimizers of the Müller functional for atoms and molecules with N electrons and total nuclear charge Z. We prove that under some suitable assumptions on Z and N, the k-th eigenvalue of a Müller minimizer γ_* behaves as A_* k^{-8/3} when k→∞, with a constant A_*>0 determined explicitly by the density of γ_*. In particular, in the atomic case V=Z|x|^{-1} our assumption holds if Z is sufficiently large and N≤Z−C_0 Z^{1/3}.

What carries the argument

The Müller minimizer γ_*, a positive trace-class operator minimizing the Müller energy, whose eigenvalues' asymptotics are derived from its density via new estimates on the integral kernel's singularity near the diagonal and decay at infinity.

If this is right

  • The prefactor A_* is determined explicitly by the density of the minimizer.
  • The asymptotic holds for both atoms and molecules under the stated conditions on Z and N.
  • The leading large-k term in the eigenvalue expansion is identified.
  • New estimates on the kernel singularity and far-field decay are required beyond the Schrödinger case.

Where Pith is reading between the lines

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

  • This decay could guide efficient numerical truncations of the basis when computing Müller minimizers.
  • The rate invites comparison with semiclassical Thomas-Fermi predictions for large-Z systems.
  • Similar asymptotics may hold for other non-local functionals used in quantum chemistry.

Load-bearing premise

The suitable assumptions on Z and N, specifically that Z is large enough and N is bounded away from Z by order Z^{1/3} in atoms, which ensure the minimizer exists with the required kernel properties.

What would settle it

Numerically computing a Müller minimizer for an atom with large Z, extracting its eigenvalues λ_k for large k, and checking whether λ_k scales proportionally to k^{-8/3} with a prefactor consistent with the density.

read the original abstract

We study the spectral properties of minimizers of the M\"uller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a M\"uller minimizer $\gamma_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $\gamma_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schr\"odinger ground states, the analysis in M\"uller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.

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

0 major / 2 minor

Summary. The paper proves that, under suitable assumptions on the total nuclear charge Z and the number of electrons N, the k-th eigenvalue of a Müller minimizer γ_* for atoms and molecules satisfies λ_k ∼ A_* k^{-8/3} as k → ∞, where A_* > 0 is explicitly determined by the density of γ_*. The proof adapts Sobolev's method for Schrödinger ground states but requires new estimates on the singularity of the integral kernel near the diagonal and the decay at infinity. For atoms with potential V = Z/|x|, the assumptions hold when Z is sufficiently large and N ≤ Z − C_0 Z^{1/3}.

Significance. If the result holds, it provides a precise asymptotic description of the spectrum of Müller minimizers, extending classical results to a more general functional setting in quantum chemistry. The explicit link between A_* and the density of γ_* is particularly useful, as it allows the constant to be computed from the minimizer itself. The technical innovations in handling the kernel estimates represent a non-trivial adaptation of existing techniques and could inspire similar analyses for other density-matrix functionals.

minor comments (2)
  1. The introduction could benefit from a more detailed outline of the new estimates required for the kernel singularity and decay at infinity, to clarify how they differ from Sobolev's original arguments.
  2. The constant A_* is said to be determined explicitly by the density; including its precise formula (or a reference to the relevant equation) already in the abstract or introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending minor revision. The summary accurately reflects the main theorem on the asymptotic decay of eigenvalues of Müller minimizers and the conditions under which it holds for atoms. We are pleased that the explicit link to the density and the new kernel estimates are viewed as valuable contributions.

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained mathematical proof

full rationale

The paper establishes the eigenvalue asymptotic A_* k^{-8/3} for Müller minimizers by adapting Sobolev's external method on one-particle density matrices, supplemented by new estimates on kernel singularity near the diagonal and decay at infinity. The constant A_* is explicitly constructed from the density of γ_* as part of the derived relation, not presupposed or fitted. The result is conditional on stated assumptions on Z and N with no reduction of the central claim to self-citation chains, self-definitional loops, or renamed inputs. The argument remains independent of the present paper's fitted values and relies on verifiable analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard functional-analytic tools (Sobolev inequalities, spectral theory) plus new estimates specific to the Müller kernel; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard results from functional analysis and spectral theory for integral operators on L2 spaces.
    Invoked to control the eigenvalue decay and kernel properties of the minimizer.
  • domain assumption Existence of Müller minimizers under the stated conditions on Z and N.
    The paper assumes the minimizer γ_* exists before deriving its spectral asymptotics.

pith-pipeline@v0.9.0 · 5488 in / 1338 out tokens · 26171 ms · 2026-05-10T03:15:50.367850+00:00 · methodology

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Reference graph

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