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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Winter Semester 2023

• 25.10.2023: Jingxuan Zhang (Yau Mathematical Sciences Center, Tsinghua University).
  
  Title: Propagation properties of long-range interacting bosons
  
  Abstract: We prove localization estimates for time-dependent many-body Schrödinger equations on bosonic Fock space over lattice domains, ensuring non-existence of super-ballistic propagation in bosonic systems with long-range interactions. The main conditions are that the hopping terms in the Hamiltonian have sufficiently fast polynomial decay, the pairing terms consist solely of density-density couplings, and the initial state satisfies a uniform density bound. Our estimates are thermodynamically stable and give rise to an explicit upper bound for the maximal velocity of propagation. Based on joint work with Marius Lemm and Carla Rubiliani.
  
• 08.11.2023: Jan Derezinski (Faculty of Physics, University of Warsaw).
  
  Title: Singular boundary conditions: theory and examples
  
  Abstract: 1-dimensional Schr\"odinger operators whose potentials are singular at one of endpoints are very common in applications. I will describe how to describe their boundary conditions. I will give various examples such as Bessel Hamiltonians, radial Coulomb Hamiltonian, perturbed Bessel Hamiltonians. It is often natural to organize them in holomorphic families, which have sometimes surprising properties.
  
• 16.11.2023: Jan Derezinski (Faculty of Physics, University of Warsaw).
  
  Title: Decay of phonons in Bose gas.
  Unusual day:: Colloqium talk, Thursday 4:30 in A027
  Abstract: Interacting Bose gas at zero temperature is often described by the Bogoliubov approximation. It involves quasiparticles, called phonons, with a rather curious dispersion relation responsible for superfluidity. The Fermi Golden Rule predicts that the lifetime of phonons is proportional to the 5th inverse power of momentum. This was first computed by Beliaev and goes under the name of the Beliaev damping. I will describe in a mathematically systematic way the chain of steps that leads to the formula for Beliaev damping. Some of these steps can be made rigorous, some of them are only heuristic. Anyway, in my opinion, this is one of the most beautiful computations of theoretical physics.
  
• 17.11.2023: Esteban Cárdenas (University of Texas at Austin).
  
  Title: Quantum Boltzmann dynamics around the Fermi ball
  Unusual day:: Room B252
  Abstract: The quantum Boltzmann equation was proposed almost a hundred years ago as a quantum-mechanical correction to the classical Boltzmann equation. However, its derivation from first principles has yet to find a mathematical proof. In this talk, I present some results on the emergence of quantum Boltzmann dynamics for states close to the Fermi ball, in the high density regime. We study the momentum distribution of a gas of N fermions in the particle-hole picture, and calculate to second order perturbation theory the effective dynamics of the system. This talk is based on joint work with Thomas Chen.
  
• 22.11.2023: Stavros Komineas (Department of Mathematics and Applied Mathematics, University of Crete).
  
  Title: Skyrmion dynamics in antiferromagnets
  
  Abstract: An extention of the sigma model gives the continuum limit for an antiferromagnet. We discuss skyrmion solutions when the Dzyaloshinskii-Moriya interaction (a chiral term) is present and study numerically and analytically propagating skyrmions (topological solitary waves). We extend the study of dynamics to oscillating skyrmions, and we find the possibility of skyrmion collapse (despite its topological nature) and formation of a singularity in finite time. When an external field is applied, the sigma model is modified in an essential way especially regarding dynamics. We derive dipole-type conservation laws for the linear momentum of solitons and skyrmions which establish a link between topology and dynamics.
  
• 20.12.2023: Mathieu Lewin (CNRS & Université Paris-Dauphine, PSL).
  
  Title: TBA
  
  Abstract: TBA
• 31.01.2024: Morris Brooks (Institute of Mathematics, University of Zurich).
  
  Title: Diagonalization of dilute quantum gases
  
  Abstract: In this talk I present an elementary derivation of the celebrated Lee-Huang-Yang formula for Bose gases in the Gross-Pitaevskii Regime, which unifies various approaches that have been developed in recent years. We will focus on the simplest possible case in order to highlight the conceptual novelties
• 21.02.2024: Cristina Caraci (Institute of Mathematics, University of Zurich).
  
  Title: Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime.
  
  Abstract: We consider a system of N bosons in the unit box in the Gross-Pitaevskii regime. We establish a precise bound for the ground state energy E(N) of the system. While the leading, of order N, contribution to the energy has been known since the pioneering works of Lieb-Seiringer-Yngvason in the early 2000, and the second order corrections (of order one) have been more recently first determined by Boccato-Brennecke-Cenatiempo-Schlein, our estimate also resolves the next term in the asymptotic expansion of E(N), which is of the order (log N)/N, confirming predictions by Wu in 1959. The talks is based on a joint work with Alessandro Olgiati, Diane Saint Aubin and Benjamin Schlein