A sharp gradient-corrected Lieb--Thirring inequality in one dimension with applications to density functional theory

The gist The authors prove sharp gradient-corrected Lieb–Thirring inequalities in one dimension and apply these results to derive non-asymptotic error bounds for the Thomas–Fermi–von Weizsäcker (TFW) approximation in density functional theory.

Gradient-Corrected Lieb–Thirring Inequalities

The core mathematical contribution involves proving a sharp gradient-corrected Lieb–Thirring inequality on the real line and the one-dimensional torus, which includes a parity dependent shell-correction term on the torus. The theorem states that for a one-particle density matrix γ in SN(R), T[γ] + 3G(ρ) ⩾ π/2 3 (1 + 2N squared Z R ρ 3). Equality is attained if and only if there exists µ > 0 such that up to translation, ρ(x) N = ρ R µ(x):= 1/π µ cosh(µx). On the torus, the inequality includes a shell term S(ρ), leading to T[γ] + 3G(ρ) ⩾ π/2 3 (1 + 2N squared Z T ρ cubed - 1N oddπ/2S(ρ)). The optimizers of these inequalities form a two-dimensional manifold called the Poincaré density manifold.

Error Bounds for the TFW Approximation

The results are then applied to study the Thomas–Fermi–von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. The paper derives non-asymptotic error bounds for this approximation, controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. Specifically, these bounds show that the TFW approximation is exact on the Poincaré manifold. Furthermore, in the periodic setting, it is shown that "the distance between the KohnSham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré density manifold. As a corollary, for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2".

Applications to Energy Approximations

The paper provides rigorous justifications for the TFW approximation in terms of kinetic energy functionals. Theorem 2.8 establishes a bound on the error between T R L and T R TFW in terms of a Poincaré deficit δ R P(ρ). An analogous result is given for the periodic setting, showing that the TFW approximation (with shell correction) is exact on these manifolds. This exactness implies that the energy error is zero if and only if the density comes from an optimizer of the gradient-corrected Lieb–Thirring inequality.

Inverse Kohn–Sham Potential Approximation

The study extends to the inverse Kohn–Sham potential. Theorem 2.17 estimates the difference between the exact inverse Kohn–Sham potential and its TFW counterpart in terms of the Poincaré deficit. This leads to the conclusion that for any ρ ∈ D1(T) ∩ H2(T), 1/N squared vKS(N ρ) = vTF(ρ) + Oρ(N −2). Corollary 2.19 explicitly computes the inverse potentials on the optimizing density manifold, showing that up to additive constants, vKS(N ρ)(x) = vTFW(N ρ)(x).

Local Kinetic-Energy Density Approximation

Theorem 2.14 compares the exact kinetic-energy density with its TFW approximation in the Yρ topology. The error is bounded by a term proportional to N q δ P(ρ) + δ P(ρ). This result shows that the error can be measured in the density-independent H−1 topology at the cost of multiplying it by the factor (CSob + M).

Conclusion on Optimizers and Potentials

The analysis concludes that for integer N, equality in the inequalities is achieved if and only if Q saturates the weighted inequality with a specific twist β N. This characterization allows for the explicit computation of the Kohn–Sham potentials at these manifolds, which are related to exactly and quasi-exactly solvable models like Pöschl–Teller and Lamé potentials. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new.

This research provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. It establishes that the TFW approximation is exact on the manifolds of optimizers for any number of particles. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new. The paper proves that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2. The work provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new. The paper proves that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2. The work provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new. The paper proves that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2. The work provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new. The paper proves that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2. The work provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new. The paper proves that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas–Fermi potential in the high-density limit N → ∞ with asymptotic rate of order N −2. The work provides a rigorous quantitative justification for an explicit gradient-corrected approximation in one-dimensional density-functional theory. The findings yield sharp error bounds and provide computable potentials associated with specific quantum mechanical models. The work demonstrates that the TFW approximation (with shell correction in the torus) is exact on the manifold of densities arising from a specific family of such exactly and quasi-exactly solvable potentials seems to be completely new.

1. Gradient-corrected Lieb-Thirring Inequality Solver: The AI can rigorously determine if a given density profile satisfies a sharp gradient-corrected Lieb-Thirring inequality by calculating its Poincaré deficit and comparing it to explicit bounds derived in Theorem 2.10 and Proposition 4.3 for both real line and torus settings.

2. Thomas-Fermi Approximation Error Estimator: The system can compute non-asymptotic error bounds for the Thomas–Fermi–von Weizsäcker (TFW) approximation by calculating the distance of a normalized density profile from the Poincaré density manifold, directly applying results from Theorem 2.8 and Theorem 2.10.

3. Inverse Kohn-Sham Potential Predictor: The AI can predict the inverse Kohn-Sham potential associated with a given density profile by computing an explicit estimate of the error between the exact inverse Kohn-Sham potential and its TFW counterpart, specifically obtaining bounds of order N to the power of minus two in the high-density limit N to infinity.

4. Poincaré Deficit Zero Detector: The system can identify if a given density profile is an optimizer of the gradient-corrected Lieb–Thirring inequality by checking if its cumulative mass coordinates satisfy the conditions derived in Lemma 5.4, thereby determining whether delta R P or delta T P equals zero and thus confirming membership in the Poincaré density manifold.

5. Optimal Density Profile Generator: The AI can generate density profiles that are known to be optimizers of the gradient-corrected LT inequality by using the explicit forms provided in Theorem 2.10 (equation 8) for integer particle numbers, specifically generating densities based on Jacobi elliptic functions and Jacobi amplitudes derived from the optimization conditions.

We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincaré density manifold. We then apply these results to study the Thomas--Fermi--von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincaré manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit N to infinity with asymptotic rate of order N-2. This rate is shown to be optimal.