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Eigenvalue asymptotics for the one-particle density matrix and one-particle kinetic energy density operator

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Pith's one-line read For atomic eigenfunctions whose one-particle density decays fast enough, occupation numbers decay as k^{-8/3}, with the leading constants fixed explicitly by the electron-electron cusp.

desk verdict Explicit occupation-number asymptotics and a faster antisymmetric decay are real, but a concrete index error in Eq. (9.1) breaks the central factorization as written. read the letter →

arxiv 2608.09570 v1 pith:Y2VJZB7I submitted 2026-08-10 math-ph math.FAmath.MPmath.SP

S{o}ren Fournais , Alexander V. Sobolev This is my paper · ORCID
classification math-phmath.FAmath.MPmath.SP MSC 81Q1035J1047G10
keywords eigenvalueasymptoticsone-particledensitymatrixoccupationnumberskineticenergyoperatoratomicSchrödingerpair-coalescencecuspJastrowfactorBirman-Solomyak
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that, for a normalized eigenfunction of an $N$-electron atom whose one-particle density decays fast enough, the eigenvalues (occupation numbers) of the one-particle density matrix decay like $k^{-8/3}$, while those of the kinetic-energy density operator decay like $k^{-2}$, with coefficients given explicitly by integrals of the eigenfunction evaluated at electron-electron coalescence points. For totally antisymmetric (fermionic) eigenfunctions the rates improve to $k^{-10/3}$ and $k^{-8/3}$. The coefficients depend only on the leading cusp singularity $\psi \sim \xi_{j,k}+|x_j-x_k|\eta_{j,k}$ at pair collisions; nuclear singularities and higher-order coalescences do not contribute. This matters because the decay controls the error in finite-basis quantum chemistry computations and gives a universal, cusp-determined tail for occupation numbers.

What carries the argument

The load-bearing object is the pair-coalescence decomposition (3.25): near $x_j=x_k$, $\psi=\xi_{j,k}(x)+|x_j-x_k|\eta_{j,k}(x)$ with real-analytic $\xi_{j,k},\eta_{j,k}$. Combined with the factorization $\Gamma=\mathrm{Int}(\Psi)^*\mathrm{Int}(\Psi)$ and $K=\mathrm{Int}(V)^*\mathrm{Int}(V)$, this reduces the eigenvalue problem to singular values of integral operators whose kernels are $|x|$ or $\nabla|x|$ times smooth factors. Those are pseudodifferential operators with homogeneous symbols, and the Birman-Solomyak asymptotic formula gives $s_k\sim k^{-1-\alpha/3}$ for a kernel homogeneous of order $\alpha$. In the totally antisymmetric case, enhanced smoothness raises the homogeneity order by one, which produces the faster decay.

What would settle it

Take a high-accuracy numerical ground state of helium, evaluate $A$ from (1.9)-(1.10), compute the largest few thousand eigenvalues of $\Gamma$, and test whether $k^{8/3}\lambda_k(\Gamma)$ approaches $A^{8/3}$; a systematic deviation outside the predicted $o(1)$ would disprove the theorem, as would a direct expansion revealing a term such as $|x-y|^{1+\varepsilon}$ with $0<\varepsilon<1$ in the remainder after the leading cusp.

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Extended reading notes

Core claim

Under the decay conditions $\rho_{1,3/8}<\infty$ and $\rho_{1,1/2}<\infty$, Theorem 1.1 establishes $\lim_{k\to\infty} k^{8/3}\lambda_k(\Gamma)=A^{8/3}$ and $\lim_{k\to\infty} k^2\lambda_k(K)=B^2$, with $A=(1/3)(2/\pi)^{5/4}\int H^{3/8}\,dx$ and $B=(4/(3\pi))\int H^{1/2}\,dx$, where $H$ is a sum of pair-coalescence integrals of $|\psi|^2$. Under total antisymmetry, Theorem 1.2 gives the faster limits $k^{10/3}\lambda_k(\Gamma)\to (A_{\mathrm{asym}})^{10/3}$ and $k^{8/3}\lambda_k(K)\to (B_{\mathrm{asym}})^{8/3}$, with coefficients (1.18) built from the coalescence gradient $v=\nabla_x\psi$. The proof isolates the pair-coalescence singularity through $\psi=\xi_{j,k}+|x_j-x_k|\eta_{j,k}$ with real-analytic $\xi,\eta$, replaces each kernel by its leading homogeneous part, and applies Birman-Solomyak spectral asymptotics for pseudodifferential operators with homogeneous symbols.

Load-bearing premise

The argument assumes that every atomic eigenfunction under study has exactly the Coulombic cusp structure $\psi=\xi+|x_j-x_k|\eta$ with real-analytic $\xi,\eta$ in a neighborhood of every pair-coalescence point, together with the uniform derivative bounds quoted from earlier regularity work; an additional non-analytic singularity at pair coalescence would change the explicit constants and the error estimates.

Editorial extensions

If this is right

  • For any atomic eigenfunction with moderate density decay, the natural-orbital occupation numbers have a universal algebraic tail $k^{-8/3}$ whose leading constant is fixed solely by the electron-electron cusp.
  • The kinetic-energy density operator's eigenvalues decay as $k^{-2}$ with an explicit constant, giving a quantitative convergence rate for finite-dimensional approximations to kinetic-energy functionals.
  • Totally antisymmetric (spinless-fermion) eigenfunctions decay faster: $k^{-10/3}$ for occupation numbers and $k^{-8/3}$ for kinetic-energy eigenvalues, so fermionic exchange measurably hardens the spectrum.
  • Nuclear coalescence points and three-particle coalescences contribute only lower-order terms; the pair-coalescence behaviour alone controls the leading asymptotics.
  • The same formulas extend to fixed-nucleus molecules, so the cusp-determined constants apply beyond atoms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the constants $A$ and $B$ are effectively the first coefficients of a Weyl-type expansion for reduced density matrices, so basis-set extrapolation schemes in quantum chemistry could be calibrated against (1.10) rather than fitted numerically.
  • Inference: the same machinery should apply to the thermal one-particle density matrix of $e^{-\beta H}$, whose kernel inherits the same cusp; a testable prediction is that the occupation-number tail keeps the exponent $k^{-8/3}$ with a $\beta$-dependent constant.
  • Inference: because the antisymmetric speed-up comes from the vanishing of $\psi$ at $x_j=x_k$, any eigenfunction sector with a Pauli-type zero of order one at coalescence should exhibit the same faster decay, not only the fully antisymmetric case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper establishes explicit asymptotic formulas for the eigenvalues of the one-particle density matrix Γ and the one-particle kinetic energy density operator K associated with a normalized eigenfunction ψ of the N-particle atomic Schrödinger operator. In the general case, it proves k^{8/3} λ_k(Γ) → A^{8/3} and k^2 λ_k(K) → B^2, where A and B are explicit integrals of the wavefunction at pair-coalescence points. Under total antisymmetry, faster rates k^{10/3} for Γ and k^{8/3} for K are obtained with coefficients A_asym and B_asym expressed through ∇ψ. The proof uses a factorization of Γ and K, Besov-space estimates, Birman–Solomyak spectral asymptotics, and a model operator, and it relies on previously established regularity results for Coulombic eigenfunctions.

Significance. If the results are correct, they provide a parameter-free, explicit description of the universal algebraic decay of natural occupation numbers and of kinetic-energy eigenvalues for atomic eigenfunctions. The antisymmetric case is entirely new, and the explicit constants (1.10) and (1.18) are integrals of ψ and ∇ψ rather than fitted quantities. The paper's strengths include a detailed factorization argument, a transparent reduction to a model pseudodifferential operator, and careful use of Besov-space techniques and Birman–Solomyak asymptotics. The reliance on published regularity results from [21] and [28] is clearly indicated, and the lattice-quasi-norm conditions make the decay assumptions precise and checkable.

major comments (2)
  1. [Section 4.1, Theorem 4.1] The bound (4.4) is announced with the proof dismissed in a single sentence: 'The proof of (4.4) follows the same plan. Omitting the details...'. This is a load-bearing step: (4.4) is used in Section 4.2 to obtain (4.8), which is needed for Lemma 4.3, Theorem 4.6, Lemma 4.7, and ultimately for the representation (4.36)–(4.37) used in the proof of Theorem 9.5 and hence Theorem 1.2. Please supply the full argument, including the treatment of derivatives in all 3N variables and the use of antisymmetry in the term involving B^m_x e^F φ, so that the proof of Theorem 1.2 can be checked.
  2. [Section 9.2.1, Eq. (9.1)] Equation (9.1) is a load-bearing definition, but as written it is ambiguous. For j<N, the tuple (x_1,...,x_{j-1},x,x_j,...,x_{N-1}) appears to omit x_N and duplicate x_j. It is correct only when the variables x_j,...,x_{N-1} on the right are understood as the relabeled integration variables x_{j+1},...,x_N after a change of variables in the integral defining γ. Please add an explicit sentence stating this relabeling convention, since a literal reading would make the factorization (9.2) fail for N≥3.
minor comments (3)
  1. [Section 9.3, Lemma 9.6] The notation M(\hat{x};b) is used in (9.12) and in the proof of Lemma 9.6, but it is not defined before its first use. Please define it explicitly, e.g., as M(\hat{x};b) = (\int_{|\hat{y}-\hat{x}|<1} |b(\hat{y})|^2 d\hat{y})^{1/2}.
  2. [Section 10.2, Lemma 10.2] In the paragraph after (10.13), the equality φ_{j,k}(\tilde{x}_{k,N},x,x)=ψ_j(\tilde{x}_{k,N},x,x) is asserted. This uses G(0)=0 in (10.3); please state this explicitly to avoid confusion.
  3. [General] There are a few minor typographical issues, such as the use of \varkappa in the abstract versus \kappa in the main text, and the phrase 'sect.' instead of 'Section'. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic coefficients are explicit integrals of the eigenfunction, and the cited regularity inputs are independent theorems that do not assume the target eigenvalue asymptotics.

full rationale

No circularity found. The asymptotic coefficients A, B, A_asym and B_asym are defined as explicit integrals of the eigenfunction and its gradient (Eqs. (1.10) and (1.18)), and are not fitted to the eigenvalue sequences. The derivation factorizes the one-particle operators as Γ = Int(Ψ)* Int(Ψ) and K = Int(V)* Int(V) (Eq. (9.2)), isolates the leading homogeneous pair-coalescence singularities of the kernels using the analytic cusp representation (3.25) from [20, Theorem 1.4] and the derivative bounds from [21,28], and then computes the constants directly from Birman–Solomyak asymptotics for integral and pseudodifferential operators with homogeneous symbols (Proposition 7.6 and Theorem 8.1). The cited regularity results are published theorems with stated assumptions that do not include the target eigenvalue asymptotics; they therefore count as independent evidence, not self-citation circularity. The paper's earlier results [42,43] are cited as background and as the source of the previously known rates (1.8), but the proofs of Theorems 1.1 and 1.2 do not invoke (1.8); they are re-derived with explicit coefficients. The structural representation (3.25) and the bounds (3.16)–(3.18) are imported rather than proved here, and the proof of (4.4) is only sketched, but these are dependency and completeness limitations, not circular reductions: none of these inputs is the conclusion being derived. No fitted parameter is renamed as a prediction, and no load-bearing argument reduces by definition to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: A, B, A_asym, and B_asym are explicit integrals of ψ and ∇ψ. The results inherit external deep tools: real-analytic cusp structure, pointwise derivative bounds, and Birman-Solomyak asymptotics. No new physical entity is introduced.

assumptions (5)
  • domain assumption Atomic Schrödinger eigenfunction setting: H=-Δ+V in (1.1), ψ∈D(H), normalized, eigenvalue E, with decay conditions ρ_{1,q}<∞ stated in Theorems 1.1 and 1.2.
    The theorems are conditional on these hypotheses; they delimit the class of wavefunctions covered.
  • domain assumption Pair-coalescence cusp representation (3.25): ψ=ξ_{j,k}+|x_j-x_k|η_{j,k} with real-analytic ξ,η near x_j=x_k, cited from [20, Theorem 1.4].
    The proof isolates the leading homogeneous singularity from this representation; it is not derived in the paper.
  • domain assumption Pointwise derivative bounds (3.16)-(3.18) from [21,28], including bounds on the Jastrow factor φ.
    These external regularity estimates control the remainders in Lemmas 10.1 and 11.1; the current paper relies on them without reproving them.
  • standard math Birman-Solomyak spectral asymptotics for pseudodifferential operators with asymptotically homogeneous symbols (Proposition 7.6, citing [4],[6]).
    Converts singular-value asymptotics of the model operator into the surface integrals defining A and B.
  • domain assumption Total antisymmetry condition (1.15) for Theorem 1.2.
    This is an additional symmetry hypothesis; it makes the coefficients A and B vanish and produces the faster decay.

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Pith. "Pith review of Eigenvalue asymptotics for the one-particle density matrix and one-particle kinetic energy density operator." pith.science (2026). https://pith.science/paper/Y2VJZB7I

@misc{pith2026260809570,
  author       = {Pith},
  title        = {Pith review of: Eigenvalue asymptotics for the one-particle density matrix and one-particle kinetic energy density operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2VJZB7I}},
  note         = {Machine review of arXiv:2608.09570}
}
abstract

Let $\psi({\mathbf x})$, ${\mathbf x} \in{\mathbb R}^{3N}$, be an eigenfunction of the $N$-particle atomic Schr\"odinger operator. We consider the one-particle density matrix $\gamma(x, y)$ and one-particle kinetic energy density $\varkappa(x, y)$, $x, y\in {\mathbb R}^3$, associated with the eigenfunction $\psi$. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators ${\sf{\Gamma}}$ and ${\sf{K}}$ with kernels $\gamma(x, y)$ and $\varkappa(x, y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues $\lambda_k({\sf{\Gamma}})>0$ and $\lambda_k({\sf{K}})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf{\Gamma}}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,\lambda_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $\psi$. These asymptotics are determined by the singularities of the function $\psi$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $\psi$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $\psi$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues $\lambda_k(\sf{\Gamma})$ and $\lambda_k(\sf{K})$. The asymptotic formulas take the form \[ \lim_{k\to \infty} k^{\frac{10}{3}} \,\lambda_k({\sf\Gamma}) = \big(A_{asym}\big)^{\frac{10}{3}},\quad \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf K}) = \big(B_{asym}\big)^{\frac{8}{3}}, \] where $A_{asym}$ and $B_{asym}$ are non-negative constants given explicitly in terms of the gradient of $\psi$.

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Works this paper leans on

46 extracted references · 45 canonical work pages

  1. [42]

    A. V. Sobolev,Eigenvalue asymptotics for the one-particle density matrix. Duke Math. J.171(17): 3481–3513, 2022

    work page 2022

  2. [43]

    A. V. Sobolev,Eigenvalue asymptotics for the one-particle kinetic energy density operator. J. Funct. Anal.283(8): Paper No. 109604, 35, 2022

    work page 2022

  3. [20]

    Fournais, M

    S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ø. Sørensen,Analytic structure of many-body Coulombic wave functions. Comm. Math. Phys.289(1): 291–310, 2009

    work page 2009

  4. [21]

    Fournais and T

    S. Fournais and T. Østergaard Sørensen,Estimates on derivatives of Coulombic wave functions and their electron densities. Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) 2021(775): 1–38, 2021

    work page 2021

  5. [28]

    Hearnshaw and A

    P. Hearnshaw and A. V. Sobolev,The diagonal behaviour of the one-particle Coulombic density matrix. Probab. Math. Phys.4(4): 935–964, 2023

    work page 2023

  6. [1]

    R. A. Adams and J. J. F. Fournier,Sobolev spaces,Pure and Applied Mathematics (Amsterdam), vol. 140. Second edn., Elsevier/Academic Press, Amsterdam, 2003

    work page 2003

  7. [2]

    A. B. Aleksandrov, S. Janson, V.V. Peller and R. Rochberg,An interesting class of operators with unusual Schatten-von Neumann behavior.Function spaces, interpolation theory and related topics (Lund, 2000), 61–149, de Gruyter, Berlin, 2002

    work page 2000

  8. [3]

    M. S. Birman and M. Z. Solomyak,Asymptotics of the spectrum of weakly polar integral operators. Izv. Akad. Nauk SSSR Ser. Mat.34: 1142–1158, 1970. 76 SØREN FOURNAIS AND ALEXANDER V. SOBOLEV

    work page 1970

Show all 46 references
  1. [4]

    M. S. Birman and M. Z. Solomyak,Asymptotic behavior of the spectrum of pseudodifferential oper- ators with anisotropically homogeneous symbols. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13(3): 13–21, 169, 1977

    1977

  2. [5]

    M. S. Birman and M. Z. Solomyak,Estimates for the singular numbers of integral operators. (Rus- sian). Uspehi Mat. Nauk32(1(193)): 17–84, 1977

    1977

  3. [6]

    M. S. Birman and M. Z. Solomyak,Asymptotic behavior of the spectrum of pseudodifferential opera- tors with anisotropically homogeneous symbols. II. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13(3): 5–10, 121, 1979

    1979

  4. [7]

    M. S. Birman and M. Z. Solomyak,Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and its Applications (Soviet Series), D. Reidel, 1987. Translated from the 1980 Russian original by S. Khrushch¨ ev and V. Peller

    1987

  5. [8]

    V. I. Burenkov,Sobolev spaces on domains,Teubner-Texte zur Mathematik [Teubner Texts in Math- ematics], vol. 137. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998

    1998

  6. [9]

    Cioslowski,Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems

    J. Cioslowski,Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems. The Journal of Chemical Physics153(15): 154108, 2020

    2020

  7. [10]

    Cioslowski and F

    J. Cioslowski and F. Pratnicki,Universalities among natural orbitals and occupation numbers per- taining to ground states of two electrons in central potentials. The Journal of Chemical Physics 151(184107), 2019

    2019

  8. [11]

    Cioslowski and K

    J. Cioslowski and K. Strasburger,Angular-Momentum Extrapolations to the Complete Basis Set Limit: Why and When They Work. Journal of Chemical Theory and Computation17(6): 3403– 3413, 2021

    2021

  9. [12]

    Coleman and V

    A. Coleman and V. Yukalov,Reduced Density Matrices,Lecture Notes in Chemistry, vol. 72. Springer-Verlag Berlin Heidelberg, 2000

    2000

  10. [13]

    J. M. Combes and L. Thomas,Asymptotic behaviour of eigenfunctions for multiparticle Schr¨ odinger operators. Comm. Math. Phys.34: 251–270, 1973

    1973

  11. [14]

    Davidson,Reduced Density Matrices in Quantum Chemistry

    E. Davidson,Reduced Density Matrices in Quantum Chemistry. Academic Press, 1976

    1976

  12. [15]

    Deift, W

    P. Deift, W. Hunziker, B. Simon and E. Vock,Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems. IV. Comm. Math. Phys.64(1): 1–34, 1978/79

    1978

  13. [16]

    L. C. Evans,Partial differential equations,Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998

    1998

  14. [17]

    Fournais, M

    S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ø. Sørensen,The electron den- sity is smooth away from the nuclei. Comm. Math. Phys.228(3): 401–415, 2002

    2002

  15. [18]

    Fournais, M

    S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ø. Sørensen,Analyticity of the density of electronic wavefunctions. Ark. Mat.42(1): 87–106, 2004

    2004

  16. [19]

    Fournais, M

    S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ø. Sørensen,Sharp regularity results for Coulombic many-electron wave functions. Comm. Math. Phys.255(1): 183–227, 2005

    2005

  17. [22]

    Friesecke,The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions

    G. Friesecke,The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal.169(1): 35–71, 2003

    2003

  18. [23]

    Friesecke,On the infinitude of non-zero eigenvalues of the single-electron density matrix for atoms and molecules

    G. Friesecke,On the infinitude of non-zero eigenvalues of the single-electron density matrix for atoms and molecules. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.459(2029): 47–52, 2003

    2003

  19. [24]

    Froese and I

    R. Froese and I. Herbst,Exponential bounds and absence of positive eigenvalues forN-body Schr¨ odinger operators. Comm. Math. Phys.87(3): 429–447, 1982/83. EIGENV ALUE ASYMPTOTICS 77

    1982

  20. [25]

    I. M. Gel’fand and G. E. Shilov,Generalized Functions, Volume 1: Properties and Operations. AMS Chelsea Publishing, 1964

    1964

  21. [26]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

    2001

  22. [27]

    H¨ attig, W

    C. H¨ attig, W. Klopper, A. K¨ ohn and D. P. Tew,Explicitly Correlated Electrons in Molecules. Chem. Rev112(1): 4–74, 2012

    2012

  23. [29]

    Hoffmann-Ostenhof, T

    M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen,Electron wavefunc- tions and densities for atoms. Ann. Henri Poincar´ e2(1): 77–100, 2001

    2001

  24. [30]

    Kato,On the eigenfunctions of many-particle systems in quantum mechanics

    T. Kato,On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. Math.10: 151–177, 1957

    1957

  25. [31]

    O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and quasilinear elliptic equations. Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc, Translation editor: Leon Ehrenpreis

    1968

  26. [32]

    J. Leray,Sur les solutions de l’´ equation de Schr¨ odinger atomique et le cas particulier de deux ´ electrons.Trends and applications of pure mathematics to mechanics (Palaiseau, 1983),Lecture Notes in Phys., vol. 195, 235–247, Springer, Berlin, 1984

    1983

  27. [33]

    Lewin,Solutions of the multiconfiguration equations in quantum chemistry

    M. Lewin,Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal.171(1): 83–114, 2004

    2004

  28. [34]

    Lewin,Geometric methods for nonlinear many-body quantum systems

    M. Lewin,Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal.260(12): 3535–3595, 2011

    2011

  29. [35]

    Lewin, E

    M. Lewin, E. H. Lieb, and R. Seiringer,Universal Functionals in Density Functional Theory. 2019. 1912.10424

    2019 arXiv

  30. [36]

    E. H. Lieb and R. Seiringer,The stability of matter in quantum mechanics. Cambridge University Press, Cambridge, 2010

    2010

  31. [37]

    Nikol’skii,Approximation of Functions of Several Variables and Imbedding Theorems

    S. Nikol’skii,Approximation of Functions of Several Variables and Imbedding Theorems. Die Grundlehren der mathematischen Wissenschaften, Springer, Berlin, Heidelberg, 1975

    1975

  32. [38]

    Ponge,Connes’ integration and Weyl’s laws

    R. Ponge,Connes’ integration and Weyl’s laws. J. Noncommut. Geom.17(2): 719–767, 2023

    2023

  33. [39]

    Reed and B

    M. Reed and B. Simon,Methods of modern mathematical physics. II. Fourier analysis, self- adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975

    1975

  34. [40]

    Simon,Exponential decay of quantum wave functions,

    B. Simon,Exponential decay of quantum wave functions,. http://www.math.caltech.edu/simon/Selecta/ExponentialDecay.pdf, Online notes, part of B. Simon’s Online Selecta at http://www.math.caltech.edu/simon/selecta.html

  35. [41]

    Sobolev,Eigenvalue estimates for the coulombic one-particle density matrix and the kinetic energy density matrix

    A. Sobolev,Eigenvalue estimates for the coulombic one-particle density matrix and the kinetic energy density matrix. Functional Analysis and Its Applications59(3): 162–184, 2025

    2025

  36. [44]

    E. M. Stein,Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970

    1970

  37. [45]

    Szabo and N

    A. Szabo and N. S. Ostlund,Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Dover Publications, 1996

    1996

  38. [46]

    Triebel,Interpolation theory, function spaces, differential operators,North-Holland Mathematical Library, vol

    H. Triebel,Interpolation theory, function spaces, differential operators,North-Holland Mathematical Library, vol. 18. North-Holland Publishing Co., Amsterdam-New York, 1978. 78 SØREN FOURNAIS AND ALEXANDER V. SOBOLEV Department of Mathematics, University of Copenhagen, Univers...

    1978

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