TY - JOUR AB - We give the first mathematically rigorous justification of the local density approximation in density functional theory. We provide a quantitative estimate on the difference between the grand-canonical Levy–Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the uniform electron gas energy of this density. The error involves gradient terms and justifies the use of the local density approximation in the situation where the density is very flat on sufficiently large regions in space. AU - Lewin, Mathieu AU - Lieb, Elliott H. AU - Seiringer, Robert ID - 14891 IS - 1 JF - Pure and Applied Analysis SN - 2578-5893 TI - The local density approximation in density functional theory VL - 2 ER - TY - JOUR AB - We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential. AU - Boccato, Chiara AU - Brennecke, Christian AU - Cenatiempo, Serena AU - Schlein, Benjamin ID - 6906 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime VL - 376 ER - TY - JOUR AB - The interaction among fundamental particles in nature leads to many interesting effects in quantum statistical mechanics; examples include superconductivity for charged systems and superfluidity in cold gases. It is a huge challenge for mathematical physics to understand the collective behavior of systems containing a large number of particles, emerging from known microscopic interactions. In this workshop we brought together researchers working on different aspects of many-body quantum mechanics to discuss recent developments, exchange ideas and propose new challenges and research directions. AU - Hainzl, Christian AU - Schlein, Benjamin AU - Seiringer, Robert AU - Warzel, Simone ID - 15072 IS - 3 JF - Oberwolfach Reports SN - 1660-8933 TI - Many-body quantum systems VL - 16 ER - TY - JOUR AB - We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer. AU - Deuchert, Andreas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 80 IS - 2 JF - Communications in Mathematical Physics TI - Bose–Einstein condensation in a dilute, trapped gas at positive temperature VL - 368 ER - TY - JOUR AB - We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations. AU - Leopold, Nikolai K AU - Petrat, Sören P ID - 6788 IS - 10 JF - Annales Henri Poincare SN - 1424-0637 TI - Mean-field dynamics for the Nelson model with fermions VL - 20 ER -