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Jakob Oldenburg

Publications and source records attributed to Jakob Oldenburg.

7 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

A Note on the Construction of Trial States for the Dilute Bose Gas

We review how the local particle number cutoff introduced in [11] is used to build trial states for the dilute Bose gas that capture the substantial correlation structure of the ground state in the thermodynamic limit. In particular, we provide a simplified derivation of the Lee-Huang-Yang correction as an upper bound for the ground state energy.

math-ph↗
arXiv2024

A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond

We consider dilute Bose gases on the three dimensional unit torus that interact through a pair potential with scattering length of order $ N^{κ-1}$, for some $κ>0$. For the range $ κ\in [0, \frac1{43})$, \cite{ABS} proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $ κ\in [0, \frac1{20})$ by combining some of the key ideas of \cite{ABS} with the novel diagonalization approach introduced recently in \cite{Br}, which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than \cite{ABS}.

math-ph↗
arXiv2024

Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation

We consider the evolution of a gas of $N$ bosons in the three-dimensional Gross-Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of the order $1/N$). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as $N \to \infty$, in the $L^2 (\mathbb{R}^{3N})$-norm. To achieve this goal, we let the Bose-Einstein condensate evolve according to a time-dependent Gross-Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross-Pitaevskii dynamics.

math-ph↗
arXiv2021

On Ground States of the Bogoliubov Energy Functional: A Direct Proof

The Bogoliubov energy functional proposed recently by Napiórkowski, Reuvers and Solovej is revisited. We offer a direct proof of the existence of minimizers at zero temperature, which covers a significantly larger class of interaction potentials. The ideas used in this proof also imply that in any ground state, more than half of the particles are inside the Bose-Einstein condensate.

math-ph↗