Strictly correlated electrons in a quantum ring: from Kohn-Sham to Kantorovich potentials

Thiago Carvalho Corso

arxiv: 2604.09908 · v1 · submitted 2026-04-10 · 🧮 math-ph · cond-mat.str-el· math.AP· math.MP· quant-ph

Strictly correlated electrons in a quantum ring: from Kohn-Sham to Kantorovich potentials

Thiago Carvalho Corso This is my paper

Pith reviewed 2026-05-10 15:57 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.str-elmath.APmath.MPquant-ph

keywords optimal transportdensity functional theoryKantorovich potentialquantum ringsemiclassical limitSeidl conjecturestrictly correlated electronsLieb functional

0 comments

The pith

For electrons on a quantum ring, the Lieb density functional converges to the optimal transport functional and its potential to a Kantorovich potential in the semiclassical limit.

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

The paper first characterizes the class of pairwise interactions for which the Seidl conjecture holds in the symmetric multimarginal optimal transport problem on the line, extending prior results to include physically relevant cases for electrons on a ring. It then proves that in one dimension, for electrons in a quantum ring or on an interval, both the Lieb functional and its representing potential converge to the optimal transport functional and a regular Kantorovich potential in the semiclassical limit. This provides a rigorous bridge between Kohn-Sham density functional theory and the strictly correlated electrons model. The functional convergence alone is also extended to periodic systems in arbitrary dimensions.

Core claim

In the semiclassical limit for electrons in a quantum ring or one-dimensional interval, not only does the Lieb density functional converge to the optimal transport functional, but the representing potential converges to a regular Kantorovich potential.

What carries the argument

The adiabatic connection potential representing the Lieb functional, shown to converge to the Kantorovich potential arising from optimal transport plans for the characterized interactions.

If this is right

The strictly correlated electron functional is the exact leading-order term for the Lieb functional in the strong-interaction limit on a quantum ring.
Kohn-Sham potentials for strongly interacting one-dimensional systems can be directly replaced by Kantorovich potentials from optimal transport.
The functional limit result applies to periodic systems in any dimension, not just the ring geometry.

Where Pith is reading between the lines

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

If one-dimensional models approximate real systems, Kantorovich potentials might serve as practical initial guesses for Kohn-Sham calculations in ring-like molecular structures.
The characterization of interactions could be used to test new pairwise potentials in quantum chemistry software for 1D confinement.
Extensions to time-dependent or excited-state functionals might follow by applying the same semiclassical analysis to the dynamic adiabatic connection.

Load-bearing premise

The pairwise interactions must belong to the class for which the Seidl conjecture holds, and the semiclassical limit uses the scaling where the interaction dominates.

What would settle it

Numerical computation of the adiabatic connection potential for electrons on a quantum ring with an interaction outside the characterized class, showing failure to approach a Kantorovich potential as the semiclassical parameter goes to zero.

Figures

Figures reproduced from arXiv: 2604.09908 by Thiago Carvalho Corso.

**Figure 1.** Figure 1: Visual illustration of the swapping procedure to reduce the cost in the case 𝑛 = 7 and 𝐴 = {1, 3, 4, 8, 9, 11, 12}. Here 𝐴1 = 𝐵(𝐴) and 𝐴2 = 𝐵(𝐴1). Moreover, the blue markers represent the elements in 𝐴, 𝐴1 and 𝐴2, while the red markers represent the elements in the complementary sets [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of the convex quadrilaterals with vertices (𝑥𝑗 , 𝑓 (𝑥𝑗)) for {𝑥1, 𝑥2, 𝑥3, 𝑥4} = {0.5, 1.5, 3, 4} (in red) with diagonals (in yellow) on the graph of different convex and non-increasing functions 𝑓 (in blue). where 𝑑𝑖𝑗 := |𝑣𝑖 − 𝑣𝑗 | = √︃ (𝑥𝑖 − 𝑥𝑗) 2 + [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

read the original abstract

Our goal in this paper is twofold. First, we characterize the class of pairwise interactions for which the Seidl conjecture on the structure of optimal plans for the symmetric multimarginal optimal transport problem with one-dimensional marginal holds. This extends previous results by Colombo, De Pascale, and Di Marino [Can. Jou. Math., 67 (2015), https://doi.org/10.4153/CJM-2014-011-x], which treated the case of translation-invariant, convex and decreasing interactions. In particular, our results apply to physically relevant interactions for electrons living on a quantum ring. The second main goal of the paper is to rigorously derive the leading order asymptotics of the adiabatic connection potential for strongly interacting systems. More precisely, we show that for electrons in a quantum ring (or one-dimensional interval), not only the Lieb density functional converges to the optimal transport (or strictly correlated) functional in the semiclassical limit, but also the representing potential converges to a regular Kantorovich potential. As an intermediate step, we also extend previous results on the strongly interacting limit of the Lieb functional to periodic systems in arbitrary dimensions.

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.

Desk Editor's Note private letter to a colleague

The paper characterizes a broader class of interactions for the Seidl conjecture on the circle and proves that the representing potential converges to a Kantorovich potential.

read the letter

The paper's main advance is characterizing the class of pairwise interactions on a circle or interval that make the Seidl conjecture true for the symmetric multimarginal optimal transport problem, and then showing that this structure implies the Kohn-Sham potential converges to a Kantorovich potential in the semiclassical limit for electrons on a quantum ring. This extends the results of Colombo, De Pascale, and Di Marino by relaxing the translation-invariant convex decreasing condition to include cases relevant to physical electron interactions in one dimension. They also prove an extension of the strongly interacting limit for the Lieb functional to periodic systems in any dimension. The work is solid on the mathematical side. It delivers rigorous proofs that build directly on cited theorems without introducing circular steps or fitted parameters. The separation between the characterization step and the potential asymptotics step makes the logic easy to follow, and the periodic extension is a clean addition that could stand alone. A minor soft spot is that the new conditions on the interaction are somewhat technical, so it takes some work to see exactly which potentials qualify. The paper asserts that the quantum ring case is covered, and on reading the details the membership holds for the interactions they consider, so the potential convergence result goes through. The stress-test concern about the ring interaction failing the hypotheses does not apply here because the authors verify the structure for their setting. This is aimed at mathematicians and theoretical physicists working on density functional theory, optimal transport, and strongly correlated electrons. Someone looking for precise asymptotic results in the strong-interaction regime will find the potential convergence useful and citable. The paper deserves a serious referee. The claims are well-supported by the derivations, and the extensions are meaningful without overclaiming. I would recommend sending this to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript characterizes the class of pairwise interactions on the one-dimensional circle or interval for which the Seidl conjecture holds regarding the structure of optimal plans in the symmetric multimarginal optimal transport problem. This extends the translation-invariant convex decreasing case treated by Colombo, De Pascale, and Di Marino to include interactions relevant for electrons on a quantum ring. Using this characterization, the paper derives the leading-order asymptotics of the adiabatic connection potential, proving that the Kohn-Sham potential converges to a regular Kantorovich potential in the semiclassical limit as the Lieb functional converges to the strictly correlated functional. It also extends prior results on the strongly interacting limit of the Lieb functional to periodic systems in arbitrary dimensions.

Significance. If the characterizations and limit derivations hold, the work provides a rigorous link between Kohn-Sham DFT and optimal transport for strongly correlated electrons in low-dimensional periodic settings. The broadened class of interactions where the Seidl structure applies is a useful technical advance, and the potential convergence supplies concrete asymptotics for the adiabatic connection that could inform approximations in strongly interacting regimes. The periodic extension of the Lieb limit broadens the result's scope beyond non-periodic cases.

major comments (2)

[Characterization theorem (Section 3)] The central application to quantum rings requires that the physical interaction (periodic Coulomb or similar) belongs to the newly characterized class where the symmetric multimarginal OT plan retains the monotonicity and support properties needed for the Seidl structure. The manuscript should include an explicit verification or membership test for this interaction in the statement of the main characterization theorem, as this is load-bearing for the claim that the Kohn-Sham potential converges to a regular Kantorovich potential.
[Asymptotics of the adiabatic connection potential (Section 5)] In the derivation of the potential asymptotics, the passage from the adiabatic connection to the Kantorovich potential limit relies on the explicit structure of the optimal plan from the first part. More precise control on how the monotonicity properties pass to the limit (including error estimates) would be needed to confirm the convergence holds uniformly for the periodic case.

minor comments (2)

[Introduction] The introduction could clarify the precise meaning of a 'regular' Kantorovich potential and how it differs from the general case, to help readers connect the result to prior OT literature.
[Notation and preliminaries] Notation for the representing potential and the adiabatic connection could be made more uniform across sections to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. The suggestions help clarify the applicability of our characterization to physical interactions and strengthen the rigor of the asymptotic analysis. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Characterization theorem (Section 3)] The central application to quantum rings requires that the physical interaction (periodic Coulomb or similar) belongs to the newly characterized class where the symmetric multimarginal OT plan retains the monotonicity and support properties needed for the Seidl structure. The manuscript should include an explicit verification or membership test for this interaction in the statement of the main characterization theorem, as this is load-bearing for the claim that the Kohn-Sham potential converges to a regular Kantorovich potential.

Authors: We agree that an explicit check would make the link to the quantum-ring application more transparent. In the revised version we will add a corollary immediately following the main characterization theorem (currently Theorem 3.2) that verifies the periodic Coulomb interaction satisfies the required monotonicity, convexity, and periodicity conditions on the circle. This verification uses the explicit form of the interaction and confirms that the Seidl structure applies, thereby justifying the subsequent convergence statements without altering the general theorem. revision: yes
Referee: [Asymptotics of the adiabatic connection potential (Section 5)] In the derivation of the potential asymptotics, the passage from the adiabatic connection to the Kantorovich potential limit relies on the explicit structure of the optimal plan from the first part. More precise control on how the monotonicity properties pass to the limit (including error estimates) would be needed to confirm the convergence holds uniformly for the periodic case.

Authors: We acknowledge that the current argument establishes convergence via the structure of the optimal plan but does not supply quantitative error estimates for the monotonicity properties in the periodic setting. In the revision we will insert a new lemma in Section 5 that provides explicit bounds on the deviation of the approximating plans from monotonicity, using the stability of optimal transport plans with respect to the interaction. These estimates will be used to upgrade the pointwise convergence of the potentials to uniform convergence on the circle, completing the proof for the periodic case. revision: yes

Circularity Check

0 steps flagged

Derivations rely on new proofs extending external results; no reduction to self-definition or fitted inputs.

full rationale

The paper's two main goals are addressed via explicit mathematical characterization of a class of interactions (extending Colombo-De Pascale-Di Marino) for which the Seidl conjecture holds, followed by limit arguments showing convergence of the Lieb functional and its representing potential to the strictly-correlated and Kantorovich objects. These steps are presented as rigorous derivations with intermediate extensions to periodic systems; they do not presuppose the target conclusions by definition, rename known patterns, or invoke self-citations as the sole justification for load-bearing premises. The quantum-ring applicability follows from verifying the new hypotheses on the interaction, which is an independent check rather than a tautology. The overall chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from optimal transport theory and semiclassical analysis in density functional theory. No free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Standard properties of symmetric multimarginal optimal transport problems in one dimension
Invoked for the structure of optimal plans and the Seidl conjecture.
standard math Existence and regularity of Kantorovich potentials under suitable interaction assumptions
Used for the convergence of the representing potential.

pith-pipeline@v0.9.0 · 5506 in / 1367 out tokens · 52392 ms · 2026-05-10T15:57:11.244933+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We characterize the class of pairwise interactions for which the Seidl conjecture... well-ordering interaction... convex and decreasing
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

periodic systems in arbitrary dimensions... quantum ring... 8-tick not mentioned

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

Ambrosio,N

[AFP00] L. Ambrosio,N. Fusco, andD. Pallara,Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (2000). MR 1857292, Zbl 0957.49001. [Amb00] L. Ambrosio,Lecture notes on optimal transport problemsIn: Mathematical aspects of evolving interfaces (Funchal, 2000)...

work page arXiv 2000
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Colombo, L

[CDD15] M. Colombo, L. De Pascale,andS. Di Marino, Multimarginal optimal transport maps for one-dimensional repulsive costs,Canadian Journal of Mathematics67(2015), no. 2, pp.350–368. MR 3314838 Zbl 1312.49052 [CDS19] M. Colombo,S. Di MarinoandF. Stra, Continuity of multimarginal optimal transport with repulsive cost,SIAM Journal on Mathematical Analysis5...

work page arXiv 2015
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MR 4983685, Zbl 8124730 [CFK13] C. Cotar,G. Friesecke, andC. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Communications on Pure and Applied Mathematics66(2013), no. 4, pp. 548–599. MR 3020313 Zbl 1266.82057 [CFK18] C. Cotar,G. Friesecke, andC. Klüppelberg, Smoothing of transport plans with fixed marginals and rigor...

work page arXiv 2013
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[CKG+17] L. Cort,D. Karlsson,L. Giovanna, andR. van Leeuwen, Time-dependent density-functional theory for strongly interacting electrons,Physical Review A95(2017), no

work page 2017
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[CL25] T. C. CorsoandA. LaestadiusA quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension,arXiv 2503.18440(2025). [DFM08] J. Dolbeault,P. Felmer, andJ. Mayorga-Zambrano, Compactness properties for trace-class operators and applications to quantum mechanics,Monatshefte für Mathematik155(2008...

work page internal anchor Pith review Pith/arXiv arXiv 2025
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Friesecke,Optimal transport—a comprehensive introduction to modeling, analysis, simulation, applications, Society for Industrial and Applied Mathematics, Philadelphia, (2025)

[Fri25] G. Friesecke,Optimal transport—a comprehensive introduction to modeling, analysis, simulation, applications, Society for Industrial and Applied Mathematics, Philadelphia, (2025). MR 4904240, Zbl 1564.49001. [Gir60] M. GirardeauRelationship between systems of impenetrable bosons and fermions in one dimension,Journal of Mathematical Physics1(1960), ...

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Gori-Giorgi,M

MR 3957876, [GSV09] P. Gori-Giorgi,M. Seidl, andG. Vignale, Density-Functional Theory for Strongly Interacting Electrons,Physical Review Letters103(2009), no

work page 2009
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[Lev79] M. Levy, Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the 𝑣-representability problem,Proceedings of the National Academy of Sciences of the United States of America76(1979), no. 12, pp. 6062–6065. MR 554891. [Lew18] M. Lewin, Semi-classical limit of the Levy–Lieb f...

work page arXiv 1979
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[LG13] P.-F. LoosandP. M. W. Gill, Uniform electron gases. I. Electrons on a ringThe Journal of Chemical Physics138(2013), no

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[Lie83] E. H. Lieb, Density functionals for Coulomb systems,International Journal of Quantum Chemistry24(1983), pp. 243–277. [LL01] E. H. LiebandM. Loss,Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI (2001). MR 1817225, Zbl 0966.26002 [MMC+13] F. Malet,A. Mirtschink,J. C. Cremon,S. M. Reimann, andP. Gori-Giorg...

work page arXiv 1983
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Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,SIAM Journal on Mathematical Analysis43(2011), no

[Pas11] B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,SIAM Journal on Mathematical Analysis43(2011), no. 6, pp. 2758-2775. year = 2011, MR 2873240, Zbl 1248.49061 [PKF19] V. D. Pham,K. Kanisawa, andS. Fölsch, Quantum Rings Engineered by Atom Manipulation,Physical Review Letters 123(2019), no

work page arXiv 2011
[12]

Räsänen,M

[RSG11] E. Räsänen,M. Seidl, andP. Gori-Giorgi, Strictly correlated uniform electron droplets,Physical Review B83(2011), no

work page 2011
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Santambrogio,Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications87, Birkhäuser/Springer, Cham (2015)

[San15] F. Santambrogio,Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications87, Birkhäuser/Springer, Cham (2015). MR 3409718, Zbl 1401.49002. [Sei99]M. Seidl, Strong-interaction limit of density-functional theory,Physical Review A60(1999), no. 6, pp. 4387–4395. [SGS07] M. Seidl,P. Gori-Giorgi, a...

work page arXiv 2015
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[SPL98] M. Seidl,J. P. PerdewandM. Levy, Strictly correlated electrons in density-functional theory,Physical Review A59 (1999), n. 1, pp. 51–54. [SPR+24] S. M. Sutter,M. Penz,M. Ruggenthaler,R. van Leuween, andK. J. H. Giesbertz, Solution of the 𝑣-representability problem on a one-dimensional torus,Journal of Physics. A. Mathematical and Theoretical57(2024), no

work page 1999
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Zbl 07945786

MR 4840211. Zbl 07945786. [SPR+25] Sutter, S. M.,Penz, M.,Ruggenthaler, M.,van Leuween, R., andGiesbertz, K. J. H., 𝑣-representability on a one-dimensional torus at elevated temperatures,Journal of Physics A: Mathematical and Theoretical59(2025). [VGD+23] S. Vuckovic,A. Gerolin,K. J. Daas,H. Bahmann,G. Friesecke, andP. Gori-Giorgi, Density functionals bas...

work page 2025
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Villani,Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI (2003)

[Vil03] C. Villani,Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI (2003). MR 1964483, Zbl 1106.90001. [VKS+04] S. Viefers,P. Koskinen,P. Singha Deo, andM. Manninen,Quantum rings for beginners: energy spectra and persistent currents,Physica E: Low-dimensional Systems and Nanostructures21...

work page arXiv 2003

[1] [1]

Ambrosio,N

[AFP00] L. Ambrosio,N. Fusco, andD. Pallara,Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (2000). MR 1857292, Zbl 0957.49001. [Amb00] L. Ambrosio,Lecture notes on optimal transport problemsIn: Mathematical aspects of evolving interfaces (Funchal, 2000)...

work page arXiv 2000

[2] [2]

Colombo, L

[CDD15] M. Colombo, L. De Pascale,andS. Di Marino, Multimarginal optimal transport maps for one-dimensional repulsive costs,Canadian Journal of Mathematics67(2015), no. 2, pp.350–368. MR 3314838 Zbl 1312.49052 [CDS19] M. Colombo,S. Di MarinoandF. Stra, Continuity of multimarginal optimal transport with repulsive cost,SIAM Journal on Mathematical Analysis5...

work page arXiv 2015

[3] [3]

MR 4983685, Zbl 8124730 [CFK13] C. Cotar,G. Friesecke, andC. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Communications on Pure and Applied Mathematics66(2013), no. 4, pp. 548–599. MR 3020313 Zbl 1266.82057 [CFK18] C. Cotar,G. Friesecke, andC. Klüppelberg, Smoothing of transport plans with fixed marginals and rigor...

work page arXiv 2013

[4] [4]

[CKG+17] L. Cort,D. Karlsson,L. Giovanna, andR. van Leeuwen, Time-dependent density-functional theory for strongly interacting electrons,Physical Review A95(2017), no

work page 2017

[5] [5]

[CL25] T. C. CorsoandA. LaestadiusA quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension,arXiv 2503.18440(2025). [DFM08] J. Dolbeault,P. Felmer, andJ. Mayorga-Zambrano, Compactness properties for trace-class operators and applications to quantum mechanics,Monatshefte für Mathematik155(2008...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[6] [6]

Friesecke,Optimal transport—a comprehensive introduction to modeling, analysis, simulation, applications, Society for Industrial and Applied Mathematics, Philadelphia, (2025)

[Fri25] G. Friesecke,Optimal transport—a comprehensive introduction to modeling, analysis, simulation, applications, Society for Industrial and Applied Mathematics, Philadelphia, (2025). MR 4904240, Zbl 1564.49001. [Gir60] M. GirardeauRelationship between systems of impenetrable bosons and fermions in one dimension,Journal of Mathematical Physics1(1960), ...

work page arXiv 2025

[7] [7]

Gori-Giorgi,M

MR 3957876, [GSV09] P. Gori-Giorgi,M. Seidl, andG. Vignale, Density-Functional Theory for Strongly Interacting Electrons,Physical Review Letters103(2009), no

work page 2009

[8] [8]

[Lev79] M. Levy, Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the 𝑣-representability problem,Proceedings of the National Academy of Sciences of the United States of America76(1979), no. 12, pp. 6062–6065. MR 554891. [Lew18] M. Lewin, Semi-classical limit of the Levy–Lieb f...

work page arXiv 1979

[9] [9]

LoosandP

[LG13] P.-F. LoosandP. M. W. Gill, Uniform electron gases. I. Electrons on a ringThe Journal of Chemical Physics138(2013), no

work page 2013

[10] [10]

[Lie83] E. H. Lieb, Density functionals for Coulomb systems,International Journal of Quantum Chemistry24(1983), pp. 243–277. [LL01] E. H. LiebandM. Loss,Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI (2001). MR 1817225, Zbl 0966.26002 [MMC+13] F. Malet,A. Mirtschink,J. C. Cremon,S. M. Reimann, andP. Gori-Giorg...

work page arXiv 1983

[11] [11]

Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,SIAM Journal on Mathematical Analysis43(2011), no

[Pas11] B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,SIAM Journal on Mathematical Analysis43(2011), no. 6, pp. 2758-2775. year = 2011, MR 2873240, Zbl 1248.49061 [PKF19] V. D. Pham,K. Kanisawa, andS. Fölsch, Quantum Rings Engineered by Atom Manipulation,Physical Review Letters 123(2019), no

work page arXiv 2011

[12] [12]

Räsänen,M

[RSG11] E. Räsänen,M. Seidl, andP. Gori-Giorgi, Strictly correlated uniform electron droplets,Physical Review B83(2011), no

work page 2011

[13] [13]

Santambrogio,Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications87, Birkhäuser/Springer, Cham (2015)

[San15] F. Santambrogio,Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications87, Birkhäuser/Springer, Cham (2015). MR 3409718, Zbl 1401.49002. [Sei99]M. Seidl, Strong-interaction limit of density-functional theory,Physical Review A60(1999), no. 6, pp. 4387–4395. [SGS07] M. Seidl,P. Gori-Giorgi, a...

work page arXiv 2015

[14] [14]

[SPL98] M. Seidl,J. P. PerdewandM. Levy, Strictly correlated electrons in density-functional theory,Physical Review A59 (1999), n. 1, pp. 51–54. [SPR+24] S. M. Sutter,M. Penz,M. Ruggenthaler,R. van Leuween, andK. J. H. Giesbertz, Solution of the 𝑣-representability problem on a one-dimensional torus,Journal of Physics. A. Mathematical and Theoretical57(2024), no

work page 1999

[15] [15]

Zbl 07945786

MR 4840211. Zbl 07945786. [SPR+25] Sutter, S. M.,Penz, M.,Ruggenthaler, M.,van Leuween, R., andGiesbertz, K. J. H., 𝑣-representability on a one-dimensional torus at elevated temperatures,Journal of Physics A: Mathematical and Theoretical59(2025). [VGD+23] S. Vuckovic,A. Gerolin,K. J. Daas,H. Bahmann,G. Friesecke, andP. Gori-Giorgi, Density functionals bas...

work page 2025

[16] [16]

Villani,Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI (2003)

[Vil03] C. Villani,Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI (2003). MR 1964483, Zbl 1106.90001. [VKS+04] S. Viefers,P. Koskinen,P. Singha Deo, andM. Manninen,Quantum rings for beginners: energy spectra and persistent currents,Physica E: Low-dimensional Systems and Nanostructures21...

work page arXiv 2003