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Quantitative Derivation of the Gross-Pitaevskii Equation
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, givingExpand
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Mean–Field Evolution of Fermionic Systems
The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems.Expand
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Effective Evolution Equations from Quantum Dynamics
In these notes we review the material presented at the summer school on "Mathematical Physics, Analysis and Stochastics" held at the University of Heidelberg in July 2014. We consider theExpand
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From the Hartree Dynamics to the Vlasov Equation
We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towardsExpand
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Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals.
An integral scheme for the efficient evaluation of two-center integrals over contracted solid harmonic Gaussian functions is presented. Integral expressions are derived for local operators thatExpand
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Mean-field Evolution of Fermionic Mixed States
In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions onExpand
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Mean-field dynamics of fermions with relativistic dispersion
We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. [“Mean-field evolution of fermionic systems,” Commun. Math. Phys. (to be published)] toExpand
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Interaction Corrections to Spin-Wave Theory in the Large-S Limit of the Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet can be naturally reformulated in terms of interacting bosons (called spin waves or magnons) as an expansion in the inverse spin size. We calculate the first orderExpand
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Interaction Corrections to Spin-Wave Theory in the Quantum Heisenberg Ferromagnet
Understanding the low-temperature properties of the Quantum Heisenberg Ferromagnet is central to a mathematical understanding of magnetism. The Quantum Heisenberg Ferromagnet can naturally beExpand
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The Dirac–Frenkel Principle for Reduced Density Matrices, and the Bogoliubov–de Gennes Equations
The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics,Expand
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