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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Summer Semester 2023

• 19.04.2023: Long Meng (Cermics, École des ponts Paritech).
  
  Title: A Rigorous Justification Of Mittleman’s Approach To The Dirac–Fock Model
  
  Abstract: In this talk, we study the relationship between the Dirac–Fock model and the electron-positron Hartree–Fock model. We justify the Dirac–Fock model as a variational approximation of QED when the vacuum polarization is neglected and when the fine structure constant α is small and the velocity of light c is large. As a byproduct, we also prove, when α is small or c is large, the no-unfilled shells theory in the Dirac–Fock theory for atoms and molecules. The proof is based on some new properties of the Dirac–Fock model.
  
• 26.04.2023: Prof. Chulkwang Kwak (Ewha university).
  
  Title: FPU to KDV
  
  Abstract: In this talk, we are going to consider the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. We particularly see that Harmonic analysis approaches allow us to observe dispersive properties of solutions to a reformulated FPU system, and with this observation, solutions to the FPU system can be approximated by counter-propagating waves governed by the Korteweg de-Vries (KdV) equation as the lattice spacing approaches zero. Additionally, we see different phenomena detected in the periodic FPU system.
  
• 04.05.2023: Mathieu Lewin (Ceremade, Université Paris-Dauphine PSL).
  
  Note: the talk will be given on Thursday 04.05 in the Mathematical Colloquium which takes place at 4:30 in A027.
  
  Title: Chemists and physicists have found how to approximate Schrödinger's equation; here is how mathematicians can contribute
  
  Abstract: Schrödinger's equation is a beautiful piece of mathematics. It fits on just one line and is supposed to accurately describe the behavior of most atoms and molecules of our world. But it is essentially impossible to simulate accurately, due to its very high dimensionality. In this talk I will explain how physicists and chemists have overcome this problem in an impressive way, within a framework called "Density Functional Theory". I will discuss the role that mathematical results have historically played in this revolution and then present more recent results.
  
• 10.05.2023: Peter Madsen (LMU Munich).
  
  Title: Representability and universal bounds in classical density functional theory.
  
  Abstract: We consider a system of indistinguishable classical particles in Euclidean space interacting through a short-range pair potential. Fixing the one-particle density profile of the system, we minimize the free energy over the set of states with this exact density. This can be useful e.g. to model interfaces between two different equilibrium phases of a system. In this talk, I will discuss the question of representability (which functions can arise as densities of many-body states?), which is a non-trivial problem when the interaction potential has a hard core. I will also explain how to obtain bounds on the free energy in terms of the density of the system. A main issue is contructing trial states with a fixed prescribed density. This is joint work with Mathieu Lewin and Michal Jex.
  
• 24.05.2023: Laurent Lafleche (ICJ, Université Claude Bernard Lyon 1).
  
  Title: Mean-field and semiclassical analysis: Quantum Wasserstein and Sobolev distances.
  
  Abstract: In the context of combined mean-field and semiclassical limits, such as the limit from the $N$-body Schrödinger equation to the Hartree--Fock and Vlasov equations, it is useful to obtain inequalities uniform in the Planck constant and the number of particles. It is therefore important to obtain analogous tools and inequalities in the context of quantum mechanics, such as operator versions of Wasserstein, Lebesgue and Sobolev distances, and the corresponding classical inequalities.
  The stability estimates for the Vlasov equation then yields different quantitative versions of the mean-field and semiclassical limits that do not have the same advantages, leading to different initial data, types of potentials, rates of convergence and time of validity of the estimates.
  
• 31.05.2023: Andreas Deuchert (University of Zurich).
  
  Title: A novel upper bound for the grand canonical free energy of the homogeneous Bose gas in the Gross–Pitaevskii limit
  
  Abstract: We consider a homogeneous Bose gas in the Gross–Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose–Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) the free energy of the interacting condensate is given in terms of a φ4 theory describing its particle number fluctuations, (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. This is joint work with Chiara Boccato, David Stocker.
  
• 01.06.2023: Sascha Lill (Università di Milano).
  
  Title: Friedrichs Diagrams—Bosonic and Fermionic
  
  Abstract: In Many-Body physics and QFT one often encounters tedious computations of commutators involving creation and annihilation operators. A diagrammatic language introduced by Friedrichs in 1965 allows for cutting down these computations tremendously, while representing the occurring operators in a particularly convenient visual form. We revisit a formula for bosonic commutators in terms of Friedrichs diagrams and prove its fermionic analogue. The talk is based on joint work with Morris Brooks from IST Vienna.
  
• 14.06.2023: Giao Ky Duong (LMU Munich).
  
  Title: TBA
  
  Abstract: TBA
  
• 28.06.2023: Yuri Suhov (Penn State University).
  
  Title: TBA
  
  Abstract: TBA
  
• 12.07.2023: Hoai-Minh Nguyen (Sorbonne Université).
  
  Title: TBA
  
  Abstract: TBA
  
• 19.07.2023: Nathan Metraud (University of the Basque Country (UPV/EHU) - Basque Center for Applied Mathematics (BCAM)).
  
  Title: Quadratic Fermionic Hamiltonians
  
  Abstract: Quadratic Hamiltonians are important in many-body quantum fields theory. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case. Following Berezin, they are quadratic in the fermionic field and in this way well-defined as self-adjoint operators acting on the fermionic Fock space. In 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, under some conditions, and it is demonstrated to be reminiscent of the celebrated Shale-Stinespring condition on Bogoliubov transformations. Moreover, we show that we can implement Bogoliubov transformations through a novel elliptic operator-valued non-linear differential equations. This allows for their (N-) diagonalization under much weaker assumptions than before. Joint work with Jean-Bernard BRU

Winter Semester 2022

• 02.11.2022: Cristina Caraci (UZH Zürich).
  
  Title: The excitation spectrum of two-dimensional Bose gases in the Gross-Pitaevskii regime
  
  Abstract: I will discuss spectral properties of two dimensional Bose gases confined in a unit box with periodic boundary conditions. We assume that N particles interact through a repulsive two-body potential, with scattering length that is exponentially small in N, i.e. the Gross-Pitaevskii regime. We proved that bosons in this regime exhibit complete Bose-Einstein condensation and we established the validity of the prediction of Bogoliubov theory. In particular, we determined the ground state energy expansion of the Hamilton operator up to second order correction, and the low-energy excitation spectrum. This is a joint work with Serena Cenatiempo and Benjamin Schlein.
  
  
• 09.11.2022: Théotime Girardot (Aarhus University).
  
  Title: A Lieb-Thirring Inequality For Extended Anyons
  
  Abstract: We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. We consider N 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter.
  
  
• 16.11.2022: Simone Rademacher (LMU Munich).
  
  Title: The effective mass problem for the Landau-Pekar equations
  
  Abstract: We provide a definition of the effective mass for the classical polaron described by the Landau–Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar. This is joint work with Dario Feliciangeli and Robert Seiringer
  
  
• 23.11.2022: Tobias Ried (MPI MiS Leipzig).
  
  Title: A variational approach to the regularity of optimal transportation
  
  Abstract: In this talk I want to present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced by Goldman—Otto. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a one-step improvement lemma, and feeds into a Campanato iteration on the C^{1,\alpha}-level for the displacement.
  The variational approach is flexible enough to cover general cost functions by importing the concept of almost-minimality: if the cost is quantitatively close to the Euclidean cost function |x-y|^2, a minimiser for the optimal transport problem with general cost is an almost-minimiser for the one with quadratic cost. This allows us to reprove the C^{1,\alpha}-regularity result of De Philippis—Figalli, while bypassing Caffarelli’s celebrated theory. (This is joint work with F. Otto and M. Prod’homme)
  
  
• 30.11.2022: Lea Boßmann (LMU Munich).
  
  Title: Edgeworth expansion for the weakly interacting Bose gas
  
  Abstract: We consider the ground state and the low-energy excited states of a system of weakly interacting bosons. We derive an Edgeworth expansion for the fluctuations of bounded one-body operators around the condensate, which yields corrections to a central limit theorem. Based on joint work with Sören Petrat.
  
  
• 14.12.2022: Sabiha Tokus (Uni. Tübingen).
  
  Title: Self-adjoint extensions of gapped operators
  
  Abstract: Whereas many typical operators we encounter in non-relativistic quantum mechanics are bounded from below and thus admit a self-adjoint extension according to Friedrichs’ construction, this is not true for a generic Dirac-type operator which does not have a lower bound. This raises the question of finding a self-adjoint extension for classes of symmetric operators that are not semi-bounded. With inspiration from work by Esteban, Dolbeault, Loss and Séré, we find a class of operators, satisfying a gap condition, for which we can construct a distinguished self-adjoint extension and a corresponding min—max principle, similarly as in the Friedrichs extension. This is joint work with Lukas Schimmer and Jan Philip Solovej.
• 18.01.2023: Sören Petrat (Jacobs University).
  
  Title: The Binding Energy in the Weakly Interacting Bose Gas Beyond Bogoliubov Theory
  
  Abstract: I will give a summary of recent results on asymptotic expansions for non-relativistic bosons in the mean-field limit. These expansions are around Bogoliubov theory and provide approximations for the Bose gas to any order in the small parameter N^(-1/2), where N is the number of particles. The expansions were proven for both the dynamics, and the low-lying energies and eigenstates for particles with pair interaction. We have proven an additional result for the dynamics of bosons that are coupled linearly to a Nelson-type quantum field. The talk will in particular emphasize the explicit nature of the expansion, i.e., how the expansion can be used to compute physical quantities such as the binding energy, in a way that is independent of the particle number N. This is joint work with Lea Bossmann, Marco Falconi, Nikolai Leopold, David Mitrouskas, Peter Pickl, Robert Seiringer, and Avy Soffer.
• 08.02.2023: August Bjerg (University of Copenhaguen).
  
  Title: On the scattering lengths of finite and infinite Thomas-Fermi atomsfie
  
  Abstract: After an introduction to the scattering length we discuss the Schrödinger operators associated to finite and infinite Thomas-Fermi atoms. The focus will be on the 1-dimensional problem. In the finite case the scattering length will be simply that of minus the Thomas-Fermi potential, whereas for the infinite atoms we define it through a choice of self-adjoint extension of the associated Schrödinger operator. The key take-away is that if we choose a sequence of finite atoms with increasing atomic number but such that their scattering length is fixed then the associated operators converge in the strong resolvent sense towards that of the infinite atom with the corresponding scattering length. If time permits we might have a look at the asymptotic behavior of the scattering lengths of the Thomas-Fermi potentials or briefly discuss the (physically more relevant) radially symmetric 3-dimensional analogue of the problems treated. Based on work in progress with Jan Philip Solovej.