First Order Expansion in the Semiclassical Limit of the Levy–Lieb Functional

Arch. Rational Mech. Anal. (2025) 249:69 Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-025-02143-7 First Order Expansion in the Semiclassical Limit of the Levy–Lieb Functional Maria Colombo , Simone Di Marino & Federico Stra Communicated by G. Friesecke Abstract We prove the conjectured first order expansion of the Levy–Lieb functional in the semiclassical limit, arising from Density Functional Theory (DFT). In particular, we prove a general asymptotic first order lower bound in terms of the zero point oscillation functional and the corresponding asymptotic upper bound in the case of two electrons in one dimension. This is accomplished by interpreting the problem as the singular perturbation of an Optimal Transport problem via a Dirichlet penalization. Mathematics Subject Classification: 81Q05 82C70 49Q20 Contents 1. 2. 3. 4. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -limit of the unconstrained problem . . . . . . . . . . . . . . . . . . Rectangular truncation of Gaussians . . . . . . . . . . . . . . . . . . . Construction of the recovery sequence . . . . . . . . . . . . . . . . . 4.1. Structure of optimal plans, maps and potentials in one dimension . 4.2. Choice of the parameters . . . . . . . . . . . . . . . . . . . . . . 4.3. Construction of γ̄ε . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Deconvolution of plans . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Potential cost of the remaining mass and kinetic energy of its marginals 6.1. Marginals of kernels of product type along a linear map . . . . . . 6.2. Wasserstein estimate of Proposition 6.2 . . . . . . . . . . . . . . . 6.3. Kinetic energy of the remaining mass . . . . . . . . . . . . . . . . 7. Proof of Theorems 1.1 and 1.3 . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8 15 17 17 19 21 29 35 38 42 46 49 55 69 Page 2 of 57 Arch. Rational Mech. Anal. (2025) 249:69 1. Introduction A revolutionary approach to finding the ground state energy of a many-body electron system was developed in the 19s by Hohenberg and Kohn [24]: The Density Functional Theory (DFT). Their idea, subsequently formalized mathematically by Lieb and Levy [30], can be seen as breaking up the minimization over all wave functions into a first minimization over those wavefunctions having a given oneparticle density and then minimizing the resulting function over the one-particle density. In particular we define that   ε  ε 2 2 |∇ψ| + Vee |ψ| dx1 . . . dx N ψ ∈ H 1 (Rd N ), ψ → ρ, F (ρ) = inf Rd N 2 (1.1) where ψ → ρ means that the one electron density of ψ is ρ, that is, for all  |ψ|2 (x1 , . . . , xn ) d x1 . . . d xi . . . d x N , and, i = 1, . . . , N , we require ρ(xi ) = moreover, Vext (x1 , . . . , xn ) = i< j 1 . |xi − x j | Then, for any external potential Vext the corresponding energy of the ground state will be equal to    2 Vext dρ ρ ∈ M+ (Rd ), ρ(Rd ) = N . E(Vext ) = inf F  (ρ) + Rd In particular, F  (ρ) is a universal functional in the sense that depends only on the number of  electrons, whereas the dependence on the external field appears only in the term Vext dρ in the outer minimization. It becomes then fundamental for applications to approximate F  (ρ) and compute this value efficiently. In recent years a new approach gained importance, relying on the analysis of the Strictly Correlated Electrons (SCE) case. The interest in this approach resides in two main reasons: it is mathematically rigorous since it is a limiting procedure starting from the exact functional, and it is highly non-local in nature, thus it can be thought as complementary in some sense to the more classical Local Density Approximation (LDA) approach [27] (see also [28] for other universal approximating functionals). The idea is to have a parameter β > 0 which tunes the strength of the interaction between the electrons and the functional 2 Fβ (ρ) = inf 2  Rd N 2 |∇ψ|2 + βVee |ψ|2 dx1 . . . dx N 2 ψ ∈ H 1 (Rd N ), ψ  → ρ. The SCE limit considers the case β → ∞, the asymptotic expansion of Fβ (ρ) in β, to then deduce information about β < ∞. This limiting procedure has more 2 equivalent formulations. Using the homogeneity of Vee , Fβ is computed from 2 F  by scaling, namely Fβ (ρ) = β 2 F  (ρβ ) where ρβ (x) = β −d ρ(x/β). Another equivalent approach considers a varying kinetic energy coefficient: since 2 2 2 Arch. Rational Mech. Anal. (2025) 249:69 Page 3 of 57 69 Fβ (ρ) = β F  /β (ρ), the asymptotic expansion of Fβ as β → +∞ can be studied by means of the asymptotic expansion of F ε (ρ) as ε → 0, which is mathematically convenient since the zeroth-order term of the expansion does not need to be renormalized. This is formally the same as changing the value of  in the original definition of the Levy–Lieb functional, but the rigorous justification of this non-physical procedure (since  is a physical constant) relies on the previous explanation. Seidl was the first to conjecture the SCE limit case for ε → 0 in [39] and further investigated in [23]. This new functional found its use, for example in [6,32,33,43] it states that   Vee dγ : γ ∈  N (ρ) , lim F ε (ρ) = FO T (ρ) := inf 2 2 2 ε→0 R3N where  N (ρ) is the set of probabilities in Rd N which induce ρ, namely such that its push-forward through any projection ei : (x1 , . . . , x N ) → xi is ρ. We denote with 0 (ρ) the set of γ ∈  N (ρ) which are minimizers for FO T (ρ). Moreover, in the physics literature, ansatz of minimizers in the 1D case and radial case were conjectured in [39,40]: in the 1D case the conjecture was proven to hold in [9], while in the radial case various counterexamples were found in [3,11,38]. The functional FO T has been studied in the last years also with regard to the limit of infinitely many particles: the zeroth order expansion for N → ∞ was investigated in [15], proving the mean field limit, while the first order was proven independently in [26] and [16,17], with ideas coming from the seminal papers [36,37], and in connection with the Lieb-Oxford inequality [29,31]. We refer the reader also to the recent review [20]. Already in [39], the next order in the asymptotic expansion of F ε for ε → 0 was conjectured in some cases, while the most general formulation appears in [22]: √ (1.2) F ε (ρ) = FO T (ρ) + εFZ P O (ρ) + O(ε3/4 ). In both [22,39] there is an explicit conjecture for the zero point oscillation functional FZ P O (ρ) in the case d = 1, involving the eigenvalues of the Hessian of an effective potential described in the following. As in [8], we introduce the potential u which solves the dual problem for FO T ; its existence and regularity have been studied in [7,10,18,21]. This is known to be unique if the support of ρ is con (...truncated)