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Documents  Siedentop, Heinz | enregistrements trouvés : 6

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Research talks;Mathematical Physics

We present a 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 situations where the density is very flat on sufficiently large regions in space. (Joint work with Mathieu Lewin and Elliott Lieb) We present a 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 ...

82B03 ; 81V70 ; 49K21

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- xvi; 285 p.
ISBN 978-973-85432-7-0

Theta series in advanced mathematics , 0005

Localisation : Colloque 1er étage (SINA)

algèbre d'opérateurs # C*-algèbre # physique mathématique

46-06 ; 81-06 ; 00B25

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Research talks;Partial Differential Equations;Mathematical Physics

We study a specific Poincaré-Sobolev inequality in bounded domains, that has recently been used to prove a semi-classical bound on the kinetic energy of fermionic many-body states. The corresponding inequality in the entire space is precisely scale invariant and this gives rise to an interesting phenomenon. Optimizers exist for some (most ?) domains and do not exist for some other domains, at least for the isosceles triangle in two dimensions. In this talk, I will discuss bounds on the constant in the inequality and the proofs of existence and non-existence.
This is joint work with Rafael Benguria and Cristobal Vallejos (PUC, Chile)
We study a specific Poincaré-Sobolev inequality in bounded domains, that has recently been used to prove a semi-classical bound on the kinetic energy of fermionic many-body states. The corresponding inequality in the entire space is precisely scale invariant and this gives rise to an interesting phenomenon. Optimizers exist for some (most ?) domains and do not exist for some other domains, at least for the isosceles triangle in two dimensions. ...

35Q40 ; 49J40 ; 47J20

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Research talks;Analysis and its Applications;Partial Differential Equations;Mathematical Physics

We review some results on the joint mean-field and semiclassical limit of the fermionic N-body Schrödinger dynamics leading to the Vlasov equation, which is a model in kinetic theory for charged or gravitating particles. The results we present include the case of singular interactions and provide explicit estimates on the convergence rate, using the Hartree-Fock theory for interacting fermions as a bridge between many-body and Vlasov dynamics.

35Q83 ; 81V19 ; 82C40

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Research talks;Analysis and its Applications;Partial Differential Equations;Mathematical Physics

We are interested in the statistical mechanics of systems of N points with Coulomb interactions in general dimension for a broad temperature range. We discuss local laws characterizing the rigidity of the system at the microscopic level, as well as free energy expansion and Central Limit Theorems for fluctuations.

82C22 ; 81V19 ; 60F05 ; 82D05

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Research talks;Analysis and its Applications;Partial Differential Equations;Mathematical Physics

In this talk I will present the approach that using time evolution of Husimi measure of the N particle wave function to get the convergence of Schrödinger to Vlasov equation in the mean field and semiclassical regime. By a reformulation of the many particle Schrödinger equation, one can get a Vlasov ‘like’ kinetic equation for Husimi measure. Then the convergence will be obtained by doing appropriate error estimates in comparing these two dynamics. In this first stage result, the estimates have been obtained for regular solutions. This is a joint work with Jinyeop Lee and Matthew Liew. In this talk I will present the approach that using time evolution of Husimi measure of the N particle wave function to get the convergence of Schrödinger to Vlasov equation in the mean field and semiclassical regime. By a reformulation of the many particle Schrödinger equation, one can get a Vlasov ‘like’ kinetic equation for Husimi measure. Then the convergence will be obtained by doing appropriate error estimates in comparing these two ...

35Q83 ; 81V25 ; 81Q05

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