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Asbjørn Bækgaard Lauritsen

Publications and source records attributed to Asbjørn Bækgaard Lauritsen.

17 recordsLinked to original sources
arXiv2026

The inverse problem of time-dependent density functional theory on the torus

We prove that, for any given sufficiently regular one-particle density, there exists a unique external potential for which the solution to the many-body Schrödinger equation has this prescribed density. This implies in particular that one can exactly reproduce the time-dependent density of an interacting system using non-interacting electrons, which is at the heart of time-dependent density functional theory. Our result covers the Coulomb interaction and densities arising from extended nuclei.

math-ph↗
arXiv2026

Time-Dependent Density Functional Theory with Coulomb Interactions

We provide the first fully rigorous justification of time-dependent density functional theory in the continuum: Given a time-dependent density, we prove that there exists an external potential, unique up to a time-dependent constant, such that the corresponding Schrödinger equation reproduces the given density. This potential can be obtained with an iteration scheme. Our argument covers Coulomb interactions and does not need any Taylor expansion in time. It instead requires analyticity in space for the density and potential, which is compatible with extended nuclei.

cond-mat.mtrl-sci↗
arXiv2026

Mesoscopic Observables of the Dilute Fermi Gas at Low Energy

We consider a dilute quantum gas of interacting spin-1/2 fermions in the thermodynamic limit. For a trial state that resolves the ground state energy to the precision of the Huang--Yang formula, we compute the expectation values of one-body observables on a mesoscopic scale.

math-ph↗
arXiv2026

Geometric theory of constrained Schrödinger dynamics with application to time-dependent density-functional theory on a finite lattice

Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.

cond-mat.mtrl-sci↗
arXiv2026

Geometric Time-Dependent Density Functional Theory

We provide a new formulation of Time-Dependent Density Functional Theory (TDDFT) based on the geometric structure of the set of states constrained to have a fixed density. Orbital-free TDDFT is formulated using a hydrodynamics equation involving a new density-to-current functional map. In the corresponding Kohn--Sham equation, the density is reproduced using a non-local operator. Finally, we present numerical simulations for one-dimensional soft-Coulomb systems.

cond-mat.mtrl-sci↗
arXiv2024

Multi-band superconductors have enhanced critical temperatures

We introduce a multi-band BCS free energy functional and prove that for a multi-band superconductor the effect of inter-band coupling can only increase the critical temperature, irrespective of its attractive or repulsive nature and its strength. Further, for weak coupling and weaker inter-band coupling, we prove that the dependence of the increase in critical temperature on the inter-band coupling is (1) linear, if there are two or more equally strongly superconducting bands, or (2) quadratic, if there is only one dominating band.

math-ph↗
arXiv2024

Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion

We prove an upper bound on the energy density of the dilute spin-$\frac{1}{2}$ Fermi gas capturing the leading correction to the kinetic energy $8πa ρ_\uparrowρ_\downarrow$ with an error of size smaller than $aρ^{2}(a^3ρ)^{1/3-\varepsilon}$ for any $\varepsilon > 0$, where $a$ denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237--260).

math-ph↗
arXiv2024

Pressure of a dilute spin-polarized Fermi gas: Upper bound

We prove an upper bound on the pressure of a dilute fully spin-polarized Fermi gas capturing the leading correction to the pressure of a free gas resulting from repulsive interactions. This correction is of order $a^3ρ^{8/3}$, with $a$ the $p$-wave scattering length of the interaction and $ρ$ the particle density, depends on the temperature and matches the corresponding lower bound of [arXiv:2307.01113].

math-ph↗
arXiv2024

Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the $p$-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

math-ph↗
arXiv2023

The BCS Energy Gap at High Density

We study the BCS energy gap $Ξ$ in the high-density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential $V$ on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3 (2022)) on the critical temperature $T_c$ at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.

math-ph↗
arXiv2023

Universal behavior of the BCS energy gap

We consider the BCS energy gap $Ξ(T)$ (essentially given by $Ξ(T) \approx Δ(T, \sqrtμ)$, the BCS order parameter) at all temperatures $0 \le T \le T_c$ up to the critical one, $T_c$, and show that, in the limit of weak coupling, the ratio $Ξ(T)/T_c$ is given by a universal function of the relative temperature $T/T_c$. On the one hand, this recovers a recent result by Langmann and Triola (Phys. Rev. B 108.10 (2023)) on three-dimensional $s$-wave superconductors for temperatures bounded uniformly away from $T_c$. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions $d \in \{1,2,3\}$, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to $T_c$.

math-ph↗
arXiv2023

Stochastic cellular automaton model of culture formation

We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately geometrically round cultures get a competitive advantage. We first analyse the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e. freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity and the central European plain has less clear cultural borders.

physics.soc-ph↗
arXiv2023

Pressure of a dilute spin-polarized Fermi gas: Lower bound

We consider a dilute spin-polarized Fermi gas at positive temperature in dimensions $d\in\{1,2,3\}$. We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order $a^dρ^{2+2/d}$, where $a$ is the $p$-wave scattering length of the repulsive interaction and $ρ$ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237--260).

math-ph↗
arXiv2023

Universality in low-dimensional BCS theory

It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature $Ξ$ and the critical temperature $T_c$ is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density, and high density. The goal of this short note is to extend the universal behavior to lower dimensions $d=1,2$ and give an exemplary proof in the weak coupling limit.

math-ph↗
arXiv2021

Floating Wigner Crystal and Periodic Jellium Configurations

Extending on ideas of Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127, (2019)) we present a modified "floating crystal" trial state for Jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the Jellium energy coincide in dimensions $d\geq 2$, thus extending the result of Cotar and Petrache (arXiv: 1707.07664) and Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127, (2019)) that the three definitions coincide in dimension $d \geq 3$. We show that the Jellium energy is also equivalent to a "renormalized energy" studied in a series of papers by Serfaty and others and thus, by work of Bétermin and Sandier (Constr Approx, 47:39-74, (2018)), we relate the Jellium energy to the order $n$ term in the logarithmic energy of $n$ points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the Jellium energy of periodic configurations.

math-ph↗
arXiv2020

The BCS Energy Gap at Low Density

We show that the energy gap for the BCS gap equation is $ Ξ= μ\left( 8 e^{-2} + o(1)\right) \exp\left( \fracπ{2\sqrtμ a}\right) $ in the low density limit $μ\to 0$. Together with the similar result for the critical temperature [arXiv:0803.3324] this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential $V$. The results hold for a class of potentials with negative scattering length $a$ and no bound states.

math-ph↗