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Foredrag ved Søren Mikkelsen, Peter Madsen, Benjamin Støttrup og Kasper Studsgaard Sørensen


03.09.2018 kl. 13.30 - 16.15


13:30-14:00 Søren Mikkelsen (Aarhus): "Semiclassical commutator bounds."

Abstract: "In a series of papers Benedikter et. al. ([1]-[4]) prove that a initial state of a fermionic system which is close to a Slater determinant,remains close to a Slater determinant when evolved under the assumption of semiclassical bounds on two commutators. In their papers the assumption is not rigorously proven but they have arguments supporting the assumption. We will in this talk discuss under which condition we can prove their assumption and discuss the strategy and methodes used to prove it.


[1] Niels Benedikter, Vojkan Jaksic, Marcello Porta, Chiara Sario, Benjamin Schlein, Mean-field evolution of fermionic mixed states. Comm. Pure Appl. Math. 69 (2016), 12: 2250-2303 [2] Niels Benedikter, Marcello Porta and Benjamin Schlein: Mean-fi eld evolution of fermionic systems. Comm. Math. Phys 331 (2014), 1087-1131.

[3] Niels Benedikter, Marcello Porta and Benjamin Schlein: Mean- field dynamics of fermions with relativistic dispersion. J. Math. Phys. 154 (2014), 021901.

[4] Niels Benedikter, Marcello Porta and Benjamin Schlein: Hartree-Fock dynamics for weakly interacting fermions. Mathematical results in quantum mechanics, World Sci. Publ., Hackensack, NJ 2015 177-189."

14:15-14:45 Peter Madsen (Aarhus): "Semi-classics of large fermionic systems in homogeneous magnetic fields."

Abstract: "We consider a system of $ N $ fermions in the presence of a (strong) homogeneous magnetic field and with a semi-classical parameter $ \hbar $. The system is confined by an external potential, and the intensity of the interaction scales like $ 1/N $. We describe the bottom of the spectrum of the corresponding Hamiltonian asymptotically to first order, as $ \hbar $ tends to zero, and the magnetic field strength and the number of particles become large. The limit is described by a magnetic Thomas-Fermi type functional, and we also obtain convergence of the position densities of approximate ground states of the system to the minimizers of the Thomas-Fermi functional, in a weak sense. The lower bound on the bottom of the spectrum is obtained using coherent states and a de Finetty type theorem. "

15:00-15:30 Benjamin Støttrup (Aalborg): Magnetic pseudodifferential operators represented as generalized Hofstatder-like matrices."

Abstract: "First, we reconsider the magnetic pseudodiff erential calculus and show that any "decent" bounded magnetic pseudodifferential operator can be represented as a generalized Hofstadter matrix; as a by-product, we prove a Calderon-Vaillancourt type result for a certain class of nondecaying symbols. Second, we make use of the matrix representation and prove sharp spectral results when the magnetic fi eld strength varies. Namely, we show that the spectrum is in general only $1/2$- Holder continuous in the Hausdorff  distance. This is joint work with Horia D. Cornean, Henrik Garde and Kasper S. Sørensen "

15:45-16:15 Kasper Studsgaard Sørensen (Aalborg): "Some spectral properties of magnetic pseudodifferential operators."

Abstract: "We first show that the minimum and maximum values of the spectrum of a large class of bounded magnetic pseudodi fferential operators with non-decaying symbols are Lipschitz continuous when the magnetic perturbation comes from a constant magnetic field. We then use this to show that gap edges of such operators also are Lipschitz continuous, as long as the gap does not close.

This is joint work with Horia Cornean, Henrik Garde & Benjamin Støttrup "



Institut for Matematiske Fag


Skjernvej 4A, AUD 5.034