arXiv subjects

Thiago Carvalho Corso

Publications and source records attributed to Thiago Carvalho Corso.

16 recordsLinked to original sources

arXiv2026

A maximal Hohenberg-Kohn theorem for non-interacting systems via potential theory

We show that for Schrödinger operators in a connected domain, the Hohenberg-Kohn theorem holds within the class of Laplace form-bounded external potentials if and only if the single-particle density is strictly positive quasi-everywhere. Furthermore, we show that this condition is satisfied for non-interacting Schrödinger operators whenever a ground-state exists. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and thereby the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded distributions. The main ingredient to establish these results is a characterization of regular states, whose proof relies on tools from classical potential theory. Moreover, this characterization reveals that, in the continuum setting, the fundamental mechanism underlying the Hohenberg-Kohn theorem is the (quasi)-unique continuation of the density rather than of the many-body wavefunction.

math-ph↗

arXiv2026

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

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.

math-ph↗

arXiv2026

A non-degeneracy theorem for interacting fermions in one dimension

In this paper, we show that the ground-state of many-body Schrödinger operators for electrons in one dimension is non-degenerate. More precisely, we consider Schrödinger operators of the form $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ acting on $\wedge^N \mathrm{L}^2([0,1])$, where the external and interaction potentials $v$ and $w$ belong to a large class of distributions. In this setting, we show that the ground-state of the system with Fermi statistics and local boundary conditions is non-degenerate and does not vanish on a set of positive measure. In the case of periodic and anti-periodic (or more general non-local) boundary conditions, we show that the same result holds whenever the number of particles is odd and even, respectively. This non-degeneracy result seems to be new even for regular potentials $v$ and $w$. As an immediate application of this result, we prove eigenvalue inequalities and the strong unique continuation property for eigenfunctions of the single-particle one-dimensional operators $h(v) = -Δ+v$. In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of $H_N(v,w)$.

math.SP↗

arXiv2025

Simulation of the magnetic Ginzburg-Landau equation via vortex tracking

This paper deals with the numerical simulation of the 2D magnetic time-dependent Ginzburg-Landau (TDGL) equations in the regime of small but finite (inverse) Ginzburg-Landau parameter $ε$ and constant (order $1$ in $ε$) applied magnetic field. In this regime, a well-known feature of the TDGL equation is the appearance of quantized vortices with core size of order $ε$. Moreover, in the singular limit $ε\searrow 0$, these vortices evolve according to an explicit ODE system. In this work, we first introduce a new numerical method for the numerical integration of this limiting ODE system, which requires to solve a linear second order PDE at each time step. We also provide a rigorous theoretical justification for this method that applies to a general class of 2D domains. We then develop and analyze a numerical strategy based on the finite-dimensional ODE system to efficiently simulate the infinite-dimensional TDGL equations in the presence of a constant external magnetic field and for small, but finite, $ε$. This method allows us to avoid resolving the $ε$-scale when solving the TDGL equations, where small values of $ε$ typically require very fine meshes and time steps. We provide numerical examples on a few test cases and justify the accuracy of the method with numerical investigations. We end the paper showing that, in the mixed flow case, the limiting ODE system is able to capture the crystallization process in which, for large times, the vortices arrange into a stable pattern.

math.NA↗

arXiv2025

A quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we focus on the regularity of this map as a function of the density and the coupling constant in front of the interaction term. More precisely, we first prove a quantitative version of the Hohenberg-Kohn theorem, thereby showing that this map is Lipschitz continuous with respect to the natural Sobolev norms in the space of densities and potentials. In particular, this implies that the inverse (Kohn-Sham) problem is not only well-posed but also Lipschitz stable. Using this result, we then show that the density-to-potential map is in fact real analytic with respect to both the density and the interaction strength. As a consequence, we obtain a holomorphic extension of the universal constrained-search functional to a suitable subset of complex-valued densities. This partially extends the DFT framework to non-self-adjoint Schrödinger operators. As further applications of these results, we also establish the existence of an exchange-only part of the exchange-correlation potential, and justify the Görling-Levy perturbation expansion for the correlation energy.

math-ph↗

arXiv2025

A rigorous formulation of Density Functional Theory for spinless fermions in one dimension

In this paper, we present a completely rigorous formulation of Kohn-Sham density functional theory for spinless fermions living in one dimensional space. More precisely, we consider Schrödinger operators of the form $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ acting on $\wedge^N \mathrm{L}^2([0,1])$, where the external and interaction potentials $v$ and $w$ belong to a suitable class of distributions. In this setting, we obtain a complete characterization of the set of pure-state $v$-representable densities on the interval. Then, we prove a Hohenberg-Kohn theorem that applies to the class of distributional potentials studied here. Lastly, we establish the differentiability of the exchange-correlation functional and therefore the existence of a unique exchange-correlation potential. We then combine these results to provide a rigorous formulation of the Kohn-Sham scheme. In particular, these results show that the Kohn-Sham scheme is rigorously exact in this setting.

math-ph↗

arXiv2025

Lieb-Thirring inequalities for the shifted Coulomb Hamiltonian

In this paper we prove sharp Lieb-Thirring (LT) inequalities for the family of shifted Coulomb Hamiltonians. More precisely, we prove the classical LT inequalities with the semi-classical constant for this family of operators in any dimension $d\geq 3$ and any $γ\geq 1$. We also prove that the semi-classical constant is never optimal for the Cwikel-Lieb-Rozenblum (CLR) inequalities for this family of operators in any dimension. In this case, we characterize the optimal constant as the minimum of a finite set and provide an asymptotic expansion as the dimension grows. Using the same method to prove the CLR inequalities for Coulomb, we obtain more information about the conjectured optimal constant in the CLR inequality for arbitrary potentials.

math-ph↗

arXiv2025

v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus

In this paper, we show that the ground-state density of any non-interacting Schrödinger operator on the one-dimensional torus with potentials in a certain class of distributions is strictly positive. This result together with recent results from [Sutter el al (2024), J. Phys. A: Math. Theor. 57 475202] provides a complete characterization of the set of non-interacting v-representable densities on the torus. Moreover, we prove that, for said class of non-interacting Schrödinger operators with distributional potentials, the Hohenberg-Kohn theorem holds, i.e., the external potential is uniquely determined by the ground-state density. In particular, the density-to-potential Kohn-Sham map is single-valued, and the non-interacting Lieb functional is differentiable at every point in this space of $v$-representable densities. These results contribute to establishing a solid mathematical foundation for the Kohn-Sham scheme in this simplified setting.

math-ph↗

arXiv2025

A generalized three lines lemma in Hardy-like spaces

In this paper we address the following question: given a holomorphic function with prescribed $L^p(\mathbb{R})$ and $L^q(\mathbb{R})$ norm (with $1\leq p,q \leq \infty$) along two parallel lines in the complex plane, then what is the maximum value that this function can achieve at a given point between these lines. Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring inequalities.

math.CV↗

arXiv2024

On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We achieve this through a variational characterization of the $L^1$ norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].

math-ph↗

arXiv2024

Trace estimates for harmonic functions along circular arcs with applications to domain decomposition on overlapping disks

In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and numerical range of the Dirchlet-to-Dirichlet (DtD) operator sending data on the boundary of the disk to the restriction of its harmonic extension along circular arcs inside the disk. The estimates we derive here have applications in the convergence analysis of the Schwarz domain decomposition method for overlapping disks in two dimensions. In particular, they allow us to establish a rigorous convergence proof for the discrete parallel Schwarz method applied to the Conductor-like Screening Model (COSMO) from theoretical chemistry in the two-disk case, and to derive error estimates with respect to the discretization parameter, the number of Schwarz iterations, and the geometry of the domain. Our analysis addresses challenges beyond classical domain decomposition theory, especially the weak enforcement of boundary conditions.

math.AP↗

arXiv2024

Numerical simulation of the Gross-Pitaevskii equation via vortex tracking

This paper deals with the numerical simulation of the Gross-Pitaevskii (GP) equation, for which a well-known feature is the appearance of quantized vortices with core size of the order of a small parameter $\varepsilon$. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit $\varepsilon\to0$, through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite-dimensional GP equation for small, but finite, $\varepsilon$. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of $\varepsilon$ typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.

math.NA↗

arXiv2023

The density-density response function in time-dependent density functional theory: mathematical foundations and pole shifting

We establish existence and uniqueness of the solution to the Dyson equation for the density-density response function in time-dependent density functional theory (TDDFT) in the random phase approximation (RPA). We show that the poles of the RPA density-density response function are forward-shifted with respect to those of the non-interacting response function, thereby explaining mathematically the well known empirical fact that the non-interacting poles (given by the spectral gaps of the time-independent Kohn-Sham equations) underestimate the true transition frequencies. Moreover we show that the RPA poles are solutions to an eigenvalue problem, justifying the approach commonly used in the physics community to compute these poles.

math-ph↗

arXiv2023

A mathematical analysis of the adiabatic Dyson equation from time-dependent density functional theory

In this article, we analyze the Dyson equation for the density-density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterize the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial density-density response function and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation (RPA) and the adiabatic local density approximation (ALDA). In particular, these results show that neither of these approximations can shift the ionization threshold of the Kohn-Sham system.

math-ph↗

arXiv2023

Two-term asymptotics of the exchange energy of the electron gas on symmetric polytopes in the high-density limit

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

math-ph↗

arXiv2023

Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations

We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the LDA, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. GGA corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and PBE we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.

physics.chem-ph↗