The oscillatory fourth term in the ground-state energy of large atoms: open quantitative problems on the Weyl sums of Córdoba–Fefferman–Seco (1995)

The ground-state energy $E_S(Z)$ of a neutral atom of nuclear charge $Z$ satisfies $E_S(Z) = E_1^{\rm TF} Z^{7/3} + \frac{q}{4} Z^2 - C_{\rm DS} Z^{5/3} + o(Z^{5/3})$, where $q$ is the number of electron spin states, $E_1^{\rm TF}$ is the Thomas–Fermi energy, and $-C_{\rm DS}Z^{5/3}$ is the Dirac–Schwinger correction. Fefferman and Seco conjectured that the expansion continues with a fourth, possibly oscillating term of order $Z^{3/2}$. Córdoba, Fefferman and Seco (Rev. Mat. Iberoamericana 11 (1995) 165–226) reduced the analysis of this conjectural term to oscillatory Weyl-type exponential sums with a phase determined by the Thomas–Fermi potential, and proved that the resulting oscillatory quantity is bounded above and below by constants times $Z^{3/2}$ — sharp in order of magnitude, but leaving its arithmetic, pointwise, and dynamical structure undetermined. The Problem Statement below formulates the resulting quantitative questions.

Background

According to nonrelativistic quantum mechanics, a neutral atom with nuclear charge $Z$ has a ground-state energy $E_S(Z)$ — the lowest eigenvalue of the $Z$-electron Schrödinger operator with Coulomb attraction to the nucleus — whose behavior as $Z \to \infty$ is a central object of mathematical physics. It is known that

$$E_S(Z) = E_1^{\rm TF} Z^{7/3} + \frac{q}{4} Z^2 - C_{\rm DS} Z^{5/3} + o(Z^{5/3}),$$

where $q$ is the number of electron spin states ($q=2$ for physical electrons), $E_1^{\rm TF}$ is the energy of the Thomas–Fermi model of a unit-charge atom (Thomas–Fermi theory is the semiclassical 'statistical' approximation in which the electrons are described by a continuous density rather than by occupied orbitals), the Scott term $\frac{q}{4}Z^2$ accounts for the strongly bound electrons very close to the nucleus, where the semiclassical approximation fails, and the Dirac–Schwinger term $-C_{\rm DS}Z^{5/3}$, with an explicitly computable constant $C_{\rm DS}>0$, captures exchange effects and the fine semiclassical asymptotics of the sum of eigenvalues of the one-particle operator $-\frac{1}{2}\Delta - \Phi^{\rm TF}$ in the Thomas–Fermi potential. The last expansion is the cumulative achievement of the Scott-conjecture program; its final term, the Dirac–Schwinger correction, was made rigorous in the monumental Fefferman–Seco series of papers in Advances in Mathematics (see Fefferman–Seco 1994).

Fefferman and Seco conjectured that the expansion continues with a fourth, possibly oscillating term of order $Z^{3/2}$ (documented in Fefferman–Seco 1994, pp. 13–14; recounted in the 2023 review of Frank, Merz and Siedentop). 'Oscillating' means that the normalized difference between $E_S(Z)$ and the first three terms, scaled by $Z^{-3/2}$, need not converge to a constant but may keep oscillating as $Z \to \infty$, so that no fourth constant coefficient exists. The term's interest is sharpened by an analogy with lattice-point problems, such as the Gauss circle problem of counting integer lattice points inside a dilated circle, whose error term is likewise governed by oscillatory exponential sums.

Córdoba, Fefferman and Seco (1994–1996) reduced the analysis of this conjectural term to the study of oscillatory exponential sums of Weyl type — finite sums $\sum_n e^{i\varphi(n)}$ with a smooth, nonlinear phase function $\varphi$, named after H. Weyl's classical estimates for polynomial phases. In the 1995 Rev. Mat. Iberoamericana paper the phase is determined by the radial Thomas–Fermi potential: writing $\mu$ for the spherical measure generating the potential and $\hat{\mu}(n)$ for its $n$-th Fourier coefficient, the analysis assumes the strong decay condition $\sum_n \sqrt{|n|}\,|\hat{\mu}(n)| < \infty$. The paper proves upper and lower bounds of order $Z^{3/2}$ for the oscillatory quantity (in a mean-square sense and along a subsequence of integers), a closed-orbit trace formula — an identity expressing the oscillatory part of the spectral counting function as a sum over classical periodic orbits of the Thomas–Fermi flow, in the spirit of the Selberg trace formula — and sharp estimates for the boundary contributions to the sum, whose remainder terms depend on the Diophantine properties of the phase derivative $\varphi'(b)$ (that is, on how well the irrational number $\varphi'(b)$ is approximated by rationals). The companion papers provided related inputs: the aperiodicity of the classical Hamiltonian flow in the Thomas–Fermi potential, that is, the fact that periodic trajectories form, in a suitable sense, an exceptional set (Fefferman–Seco 1993), and a number-theoretic estimate for the Thomas–Fermi density (Córdoba–Fefferman–Seco 1996).

Problem Statement

The problem is to advance the quantitative theory of the oscillatory term of order $Z^{3/2}$ in the expansion of the ground-state energy of large atoms — the term governed by the Weyl sums $\sum_n e^{i\varphi(n)}$ with phase $\varphi$ determined by the Thomas–Fermi potential. The 1995 paper leaves five specific open questions on the quantitative theory of this term:

(i) Arithmetic classification of the remainders. In the paper's evaluation of the sum through boundary contributions, the remainders in the $o$-terms depend, when the phase derivative $\varphi'(b)$ is irrational, on its Diophantine properties (that is, on how well $\varphi'(b)$ is approximated by rationals). Give a complete classification of the remainder behavior — rates and dependence on the quality of rational approximation — in terms of the arithmetic of $\varphi'(b)$.

(ii) Pointwise large values at integers. The paper proves mean-square estimates and non-vanishing of the normalized oscillatory sum along a subsequence of integers, but gives no characterization of the set $\{Z \in \mathbb{Z} : |\Psi_Q(Z)| \gtrsim Z^{3/2}\}$ of integers at which the oscillatory sum attains its maximal order; here $\Psi_Q$ denotes the paper's original oscillatory sum, the quantity entering the atomic energy expansion (the mean-square and subsequence estimates apply to the paper's model sum $\Psi_0$), and $\gtrsim$ means 'bounded below by a positive constant times'. The density or distribution of this set is unknown.

(iii) Dirac–Schwinger error exponent. Prove the four-term energy expansion $E_S(Z) = E_1^{\rm TF} Z^{7/3} + \frac{q}{4} Z^2 - C_{\rm DS} Z^{5/3} + (\text{the oscillatory term of order } Z^{3/2}) + O(Z^{5/3-a})$ with an exponent $a > 1/6$ in the residual $O(Z^{5/3-a})$ measured after the oscillatory term. This is exactly the threshold at which $5/3 - a < 3/2$, so that the oscillatory term would dominate the residual and become individually observable in $E_S(Z)$.

(iv) Inverse spectral and dynamical questions from the closed-orbit trace formula. The paper's trace formula expresses the oscillatory part of the spectral counting function as a sum over classical closed orbits of the Thomas–Fermi flow. Determine to what extent dynamical invariants or orbit multiplicities can be recovered from the spectral oscillations, and whether such reconstructions are stable under perturbations of the Thomas–Fermi potential (equivalently, of the generating spherical measure $\mu$), with stability measured by continuity of the recovered quantities. The question is posed for a class of perturbations of the generating spherical measure $\mu$ and concerns both a quantified positive recovery of named invariants or orbit multiplicities and a negative result ruling such a recovery out.

(v) Weakening the Fourier-decay hypothesis. The analysis assumes the strong decay condition $\sum_n \sqrt{|n|}\,|\hat{\mu}(n)| < \infty$ on the spherical measure $\mu$ generating the potential. Extend the results to weaker decay, namely to all weights whose Fourier coefficients satisfy $|\hat{\mu}(n)| \le C\,|n|^{-3/2+\varepsilon}$ for some $\varepsilon > 0$ and constant $C$, moving toward the regime of the lattice-point counting problem.

The questions are posed at the level of precision that the paper's own $Z^{3/2}$ upper and lower bounds make sharp.

References

1. Antonio Córdoba, Charles L. Fefferman, Luis A. Seco, "Weyl sums and atomic energy oscillations", Revista Matemática Iberoamericana 11(1):165–226, 1995, DOI 10.4171/rmi/170, https://doi.org/10.4171/rmi/170 (primary source)
2. Rupert L. Frank, Konstantin Merz, Heinz Siedentop, "The Scott conjecture for large Coulomb systems: a review", Letters in Mathematical Physics 113, article 11, 2023, DOI 10.1007/s11005-023-01631-9 (preprint arXiv:2204.10081, 2022), https://arxiv.org/abs/2204.10081
3. A. Córdoba, C. Fefferman, L. Seco, "A trigonometric sum relevant to the nonrelativistic theory of atoms", Proceedings of the National Academy of Sciences 91(13):5776–5778, 1994, DOI 10.1073/pnas.91.13.5776, https://doi.org/10.1073/pnas.91.13.5776
4. Antonio Córdoba, Charles L. Fefferman, Luis A. Seco, "A number-theoretic estimate for the Thomas–Fermi density", Communications in Partial Differential Equations 21(7–8):1087–1102, 1996, DOI 10.1080/03605309608821218, https://doi.org/10.1080/03605309608821218
5. C. Fefferman, L. A. Seco, "On the Dirac and Schwinger corrections to the ground-state energy of an atom", Advances in Mathematics 107(1):1–185, 1994, DOI 10.1006/aima.1994.1060, https://doi.org/10.1006/aima.1994.1060
6. Charles L. Fefferman, Luis A. Seco, "Aperiodicity of the Hamiltonian flow in the Thomas–Fermi potential", Revista Matemática Iberoamericana 9(3):409–551, 1993, DOI 10.4171/rmi/142, https://doi.org/10.4171/rmi/142 (the companion paper proving that periodic trajectories of the classical Thomas–Fermi flow are, in a suitable sense, exceptional)
7. Victor Ivrii, "Asymptotics of the ground state energy of heavy molecules and related topics", arXiv:1210.1132 (2012, latest version 2017), https://arxiv.org/abs/1210.1132
8. Victor Ivrii, "Asymptotics of the ground state energy of heavy molecules and related topics. II", arXiv:1210.1329 (2012, latest version 2017), https://arxiv.org/abs/1210.1329 (this part treats the case of an external magnetic field)
9. Antonio Córdoba, "Weyl sums and atomic energy oscillations", chapter in "Suprematism in Harmonic Analysis" (Progress in Mathematics, vol. 356, Birkhäuser, Cham), pp. 201–246, 2024, DOI 10.1007/978-3-031-73737-4_9, https://doi.org/10.1007/978-3-031-73737-4_9 (an expository essay)